computational materials science · 1. introduction modeling and simulation tools are crucial for...

Computational Materials Science 127 (2017) 141–160

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Reliable Molecular Dynamics: Uncertainty quantification using intervalanalysis in molecular dynamics simulation

http://dx.doi.org/10.1016/j.commatsci.2016.10.0210927-0256/� 2016 Elsevier B.V. All rights reserved.

⇑ Corresponding author.E-mail addresses: [email protected], [email protected]

(Y. Wang).

Anh V. Tran, Yan Wang ⇑Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, GA 30332, USA

a r t i c l e i n f o

Article history:Received 10 May 2016Received in revised form 16 September2016Accepted 14 October 2016

Keywords:Molecular dynamicsInteratomic potentialUncertainty quantificationInterval analysis

a b s t r a c t

In molecular dynamics (MD) simulation, atomic interaction is characterized by the interatomic potentialas the input of simulation models. The interatomic potentials are derived experimentally or from firstprinciples calculations. Therefore they are inherently imprecise because of the measurement error ormodel-form error. In this work, a Reliable Molecular Dynamics (R-MD) mechanism is developed toextend the predictive capability of MD given the input uncertainty. In R-MD, the locations and velocitiesof particles are not assumed to be precisely known as in traditional MD. Instead, they are represented asintervals in order to capture the input uncertainty associated with the atomistic model. The advantage ofthe new mechanism is the significant reduction of computational cost from traditional sensitivity anal-ysis when assessing the effects of input uncertainty. A formalism of generalized interval is incorporatedin R-MD, as an intrusive uncertainty quantification method, to model the propagation of uncertainty dur-ing the simulation. Error generating functions associated with embedded atomic method (EAM) inter-atomic potentials are developed to capture the bounds of input variations to demonstrate intervalinteratomic potentials. Four different uncertainty propagation schemes are proposed to capture theuncertainty of the output. An example of uniaxial tensile loading of single-crystal aluminum is used todemonstrate the R-MD mechanism.

� 2016 Elsevier B.V. All rights reserved.

1. Introduction

Modeling and simulation tools are crucial for engineers todesign and develop new materials efficiently. Uncertainty isalways involved in model selection, calibration, and validation pro-cesses. Reliable simulation predictions require us to quantify inputuncertainty of models. There are two elements of uncertainty inmodeling and simulation: aleatory uncertainty and epistemicuncertainty. Aleatory uncertainty is the inherent randomness inthe phenomenon being observed, and the impossibility of exhaust-ing all descriptions deterministically. Epistemic uncertainty can begenerally related to the lack of perfect knowledge about theinvolved physical processes [1].

Molecular dynamics (MD) is one of the most widely used ato-mistic simulation tools. In MD simulation, the aleatory uncertaintycorresponds to any fluctuation of the simulated system, e.g. thenatural thermal fluctuation that can be described by Boltzmanndistribution at an equilibrium microscopic state. The epistemicuncertainty includes, but is not limited to, the imprecise

interatomic potentials, the finite size effect, the boundary condi-tion imposed on the simulation cell, and the cutoff radius of theinteratomic potentials. The aleatory uncertainty associated withthe thermal fluctuation is generally inseparable from MD simula-tion, and sometimes is induced by the ensemble integrator. Forexample, in Langevin thermostat, this thermal fluctuation isaccounted by the friction-noise in the stochastic differential equa-tions [2]. The epistemic uncertainty in MD simulations is mostlycaused by the imperfection of the interatomic potential. Theseinteratomic potentials are typically derived from first principlescalculations or approximated based on experimental data. Theseresults are contaminated by both systematic and random errors.The systematic errors of first principles calculations come from dif-ferent approximations and assumptions in the models, such asBorn-Oppenheimer approximation, Hartree-Fock approximation,and the assumed finite linear combination of the variational solu-tion based on the set of basis functions [3]. On the other hand, thesystematic errors of experimental results involve measurementbias and calibration errors. Based on the results, an interatomicpotential model is formulated with a set of parameters to minimizea measurable error, which usually in turn is converted to a least-square error problem. Because of the non-negative residual incurve fitting and approximation error techniques used in deriving

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Fig. 1. Schematic illustration of R-MD in 2D.

142 A.V. Tran, Y. Wang / Computational Materials Science 127 (2017) 141–160

the interatomic potentials, MD simulations include both model-form uncertainty and parameter uncertainty. Furthermore, theepistemic uncertainty from interatomic potential in MD simula-tions is amplified because the number of interacting pairs, whichscales at least as N2, where N is the number of atoms. Therefore,quantifying uncertainty in MD simulations is a critical problem,in both assessing the accuracy and reliability of the simulationprediction.

Uncertainty quantification (UQ) problems are divided into twomain paradigms, intrusive and non-intrusive methods on proba-bilistic and non-probabilistic frameworks. In non-intrusive UQtechniques, the simulation is viewed as a black box, and the simu-lator is modeled as a one-to-one non-linear function that mapsfrom the input domains to the output or quantities of interests.Popular techniques, including stochastic collocation, Monte Carlo,and global sensitivity analysis, rely on statistical techniques tobuild comprehensive output distributions based on the assumedinput distributions. Generalized polynomial chaos expansion is awidely used technique, and can be utilized either intrusively ornon-intrusively. As an intrusive technique, it has been applied tosolve stochastic differential equations and partial differential equa-tions with random inputs. As a non-intrusive technique, it is typi-cally used together with Smolyak sparse grid and nested sets instochastic collocation methods.

Other intrusive UQ techniques, such as local sensitivity analysisand interval-based approaches, aim to provide the output proba-bility density function or its bounded support for expensive simu-lation by incorporating and propagating the uncertainty internallyusing minimal number of runs. In interval-based approaches, theuncertainty is coupled into the input and represented by intervals.The simulator is thus extended to handle the interval inputs andpropagate the uncertainty throughout the simulation. The outputuncertainty, which is also represented as intervals, is computedat every time step at a relatively cheap computational cost.

Various UQ methods have been applied to multi-scale simula-tion for materials. Comprehensive literature reviews are availablein [4,5]. Frederiksen and Jacobsen [6] applied Bayesian update totrain the interatomic potentials parameters with experimentaldata sets by minimizing the square error between experimentaldata and simulation results. Jacobson et al. [7] constructedresponse surfaces with Lagrange interpolation to study the sensi-tivity of macroscopic properties with respect to interatomic poten-tial parameters. Cailliez and Pernot [8] calibrated Lennard-Jonespotential for Argon based on Bayesian calibration/predictionframework. Rizzi et al. [9,10] assumed uniform distribution forthe four-site, TIP4P, water model parameters and constructed thegeneralized polynomial chaos representation by non-intrusivespectral projection and Bayesian inference approaches, then lateron, calibrated these force-field parameters based on Bayesianinference. Angelikopoulos et al. [11] applied the Bayesian calibra-tion to calibrate the water-carbon interactions based on water con-tact angles in water wetting of graphene, the aggregation offullerenes in aqueous solution, and the water transport across car-bon nanotubes. Rizzi et al. [9] applied polynomial chaos expansionto study the effect of input uncertainty in MD. Cailliezf et al. [12]applied the efficient global optimization algorithms in parameterspace to calibrate the potential parameters for TIP4P model, basedon probabilistic kriging metamodels. Wen et al. [13] studied theeffect of different spline interpolations on the potential predictionsby calculating the quasi-harmonic thermal expansion and finite-temperature elastic constant of a one-dimensional chain in tabu-lated interatomic potentials. Hunt et al. [14] developed a softwarepackage for non-intrusive propagation of uncertainties in inputparameters, using surrogate models and adaptive sampling meth-ods, such as Monte Carlo, Latin Hypercube, and Smolyak sparse

grids, based on generalized polynomial chaos expansion. Li et al.[15] discussed the cut- and random sample-high dimensionalmodel representation to quantify the uncertainty induced bypotential surfaces.

As an intrusive approach on non-probabilistic framework, werecently proposed an interval-based reliable MD (R-MD) mecha-nism [16,17] that incorporates Kaucher interval arithmetic [18]into classical MD to quantify output uncertainty. Classical intervalarithmetic provides a complete solution by capturing all possibili-ties for simple algebraic operations, such as addition, subtraction,multiplication, and division. Kaucher interval arithmetic general-izes and extends [19] classical interval arithmetic with bettertopology and algebraic properties. Compared to classic intervalarithmetic, Kaucher interval arithmetic is preferred for three rea-sons. Firstly, the over-estimation problem is significantly reduced.Secondly, the self-dependency problem, which also results in anover-estimation of a function, where dependent variables arerepeated more than once, is mitigated. Thirdly, the negation andreciprocal operations with respect to addition and multiplicationexist. In contrast to the Kaucher interval space, the classical inter-val space only forms a semi-group algebraic structure because ofthe lack of invertibility. In R-MD, the input uncertainty associatedwith interatomic potentials is captured in interval forms, either asintervals or as interval functions. Consequently, the atomistic posi-tions, velocities, and forces are also interval-valued. Fig. 1 plots aschematic sketch of simple 2D R-MD simulation cell, where theatomistic positions are interval-valued. The exact atomistic posi-tions and velocities are unknown, but bounded by intervals. In thispaper, the details of how Kaucher interval arithmetic is applied insimulation including interval potential, interval force computation,and interval statistical ensemble are described. In Section 2, wereview the algebraic operations of Kaucher interval arithmetic. InSection 3, the formulation of interval potential and interval forceare discussed, and four R-MD uncertainty propagation schemesare implemented in the framework of Large-scale Atomic/Molecu-lar Massively Parallel Simulator, also known as LAMMPS [20]. Anapplication to tensile uniaxial deformation of aluminum singlecrystal is demonstrated in Section 4, including UQ results, compar-isons between different schemes, finite-size effects, and compar-ison with sensitivity analysis results as a part of verificationprocess. Following are the discussion in Section 5 and conclusionin Section 6.

2. Kaucher interval arithmetic

The classical interval space, denoted as IR, is a collection of clas-sical interval, where the upper bound is strictly greater than or

A.V. Tran, Y. Wang / Computational Materials Science 127 (2017) 141–160 143

equal to the lower bound, that is, x P x. The Kaucher interval spaceis a collection of Kaucher intervals, denoted as KR, is an extensionof classical interval space IR, where the aforementioned constraintof upper and lower bounds is removed. The dual operator simplyswaps the lower and the upper bounds of an arbitrary intervalsimultaneously, as dual x; x½ � :¼ x; x½ �. In order to distinguish arith-metic operation on KR to IR, the addition, subtraction, multiplica-tion, and division operator are denoted with a circumscribed

operator, and defined as follows. x; x½ � � y; yh i

¼ xþ y; xþ yh i

.

x; x½ �� 2 x� y; x� yh i

. x; x½ � � y; yh i

is defined in Table 1.

x; x½ �ø y; yh i

¼ x; x½ � � 1=y;1=yh i

. In Table 1, the generalized interval

space KR is decomposed into four subspaces:P :¼ fx 2 KR j x P 0; x P 0g contains positive intervals,Z :¼ fx 2 KR j x 6 0 6 xg contains proper interval that includezero, �P :¼ fx 2 KR j � x 2 Pg contains negative intervals,and dual Z :¼ fx 2 KR j dual x 2 Zgcontains improper intervalsthat include zero.

Let xþ ¼maxfx;0g, x� ¼maxf�x;0g, x _ y ¼maxfx; yg, themultiplication table can be further simplified by Lakeyev’s formula[21] as

x; x½ � � y; yh i

¼ ðxþyþÞ _ ðx�y�Þ � ðxþy�Þ _ ðx�yþÞ; ðxþyþÞ _ ðx�y�Þh�ðxþy�Þ _ ðx�yþÞ

ið1Þ

The Kaucher interval space KR is equipped with a norm defined as

kxk :¼maxfjxj; jxjg ð2Þwith the following properties:

kxk ¼ 0 if and only if x ¼ 0 ðseparates pointsÞ;kxþ yk 6 kxk þ kyk ðsubadditivity or triangle inequalityÞ;kaxk ¼ jajkxk with a 2 R ðabsolute homogeneityÞ:A induced distance metric onKR is then can be defined as an exten-sion of norm between x and y as

dðx; yÞ :¼max jx� yj; jx� yjn o

; ð3Þ

which is related with the norm by dðx;0Þ ¼ kxk anddðx; yÞ ¼ kx� yk. As shown in [18], KR is a complete metric spaceunder the defined metrics dð�; �Þ.

Definition 1. An interval x is proper if the upper bound is greaterthan or equal to the lower bound, that is, x 6 x. An interval andcalled improper if that lower bound is greater than or equal to theupper bound, that is, x P x. If two bounds equal each other, x ¼ x,then x is a degenerated, pointwise or singleton interval, and has areal value.

Definition 2. An interval is called sound if it does not include inad-missible solutions; an interval solution is called complete if itincludes all possible solutions. In general, a sound interval is a sub-set of the true solution set, whereas a complete interval is asuperset.

Table 1Definition of Kaucher multiplication.

y 2 P y 2 Zx 2 P xy; xy

h ixy; xyh i

x 2 Z xy; xy½ � min xy; xyn o

;max xy; xynh

x 2 �P xy; xyh i

xy; xyh i

x 2 dualZ xy; xyh i

0

3. Reliable Molecular Dynamics (R-MD)

Using Kaucher intervals, R-MD incorporates the input uncer-tainties in the interatomic potential with the interval forms. Thesensitivity of the interested quantities with respect to inputs isassessed on-the-fly at every time step. Compared to the non-intrusive solutions, the intrusive UQ techniques significantlyreduce the computational time to estimate the uncertainty of theoutput.

3.1. Representations of atoms’ position, velocity and forces intervals

There are mainly two ways to represent an interval x ¼ x; x½ �,either as upper bound x and lower bound x, or midpointmidðxÞ ¼ 1

2 xþ xð Þ and radius radðxÞ ¼ 12 x� xð Þ. The advantage of

the lower-upper bounds representation is that the computationaltime is optimized, because most of the interval calculations areperformed under this representation. However, it would requireheavy modifications to incorporate the intervals into MD simula-tion. On the other hand, the midpoint-radius representation canbe built as a simple extension based on the classical MD packages.The midpoint values of atoms’ interval positions, velocities, andforces intervals can be assigned as the values in classical MD atevery time step, and the radii can be computed based on the loweror upper bounds accordingly. With the radii of the interval posi-tions, velocities, and forces, the uncertainty of the quantities ofinterest can be quantified accordingly. The drawback of this tech-nique is that it is not computationally optimal because of therepetitive converting processes to lower-upper boundsrepresentation.

A caveat for midpoint-radius representation is the over-constraint problem of choosing radius values for atoms’ intervalpositions, velocities, and forces. This problem is described inFig. 2a. Because the values from classical MD are not necessarilythe exact midpoint, two radii are differentiated. One is innerradius, the other is outer radius. The inner radius is the smaller dis-tance from midpoint to one of the interval bounds, whereas theouter radius is the larger distance. If the inner radius is chosen,the interval is sound and the solution underestimates the trueuncertainties. If the outer radius is chosen, the interval is completeand the solution over-estimates the true uncertainties. In the innerradius case, the soundness is chosen over the completeness. In theouter radius case, the completeness is chosen over the soundness.Based on the introduction of radius variable, Fig. 2b presents thepossibility of decoupling radius variables from the interval vari-ables for the atomistic positions, velocities, and forces. This is anadvantage of the midpoint-radius representation because insteadof building interval objects to handle, the decoupling allows thedirect calculation of the radius variable, which in turns can be usedto quantify the uncertainty of the output.

3.2. Interval interatomic potential

The interatomic potentials are approximated based on physicalmodels and therefore they are inherently imprecise due to approx-imation errors. Typically, the model and its parameters are chosen

y 2 �P y 2 dual Z

xy; xyh i

xy; xyh i

oixy; xyh i

0

xy; xyh i

xy; xyh i

xy; xy½ � max xy; xyn o

;min xy; xyn oh i

(a) The over-constraint problem by the choice of innerand outer radius in midpoint-radius representations of an interval.

(b) The decoupling technique to accelerate computation inmidpoint-radius representation.

Fig. 2. Illustration of (a) the relationship between inner-outer radius with sound-complete solution and (b) decoupling of radius values in R-MD.


to fit for a particular purpose using curve fitting techniques, ormulti-linear regression analysis to minimize the residue, or maxi-mum likelihood estimator, associated with the interested outputquantities. In MD, these quantities are usually lattice constant,phonon-dispersion curve, stacking fault energy, melting point tem-perature, thermoelastic properties, as well as other mechanical andthermal properties of a material.

In R-MD, interval-valued interatomic potentials are applied tomodel the input uncertainty. For example, for Lennard-Jonespotential, the well depth e and the location at which the inter-atomic potential between two atoms is zero r can be generalizedas e; e½ � and r;r½ �. For tabulated potentials such as embeddedatomic model (EAM), some analytical error generating functionscan be devised to represent the error bounds of potentials andthe interpolation error of the potentials. With interval-valuedinteratomic potentials, the forces are also intervals. The update ofpositions and velocities is based on Kaucher interval arithmetic,which is described in Section 2.

3.3. Interval force calculations

The force computation plays a pivotal role in R-MD. Theoreti-cally, it is a simplification of a NP-problem. Consider a 2D R-MDsimulation, where an arbitrary atom has N neighboring atoms, assketched in Fig. 3a. Mathematically, the interval force can be com-puted rigorously by considering the interaction between vertices ofeach individual rectangles, as illustrated in Fig. 3b. For 2D, in oneneighboring list, this approach leads to 4N possibilities to computea total resulting force for one atom at every time step, and thusvery exhaustive. The problem is worse in R-MD 3D, because the

number of possibilities increases to 8N . Without approximationsand simplifications, this problem leads to an exhaustive search tofind a good estimation of interval forces. To simplify, the inter-atomic forces are computed based on the centroids of the prismswith weak-strong variation assumptions. This statement will beexplained in further details for EAM potential in the next section.

3.3.1. An example of interval-valued potential: Interval EAM potentialIn this section, our previous work in [16] to introduce

uncertainty in EAM potentials is summarized. The EAM is asemi-empirical, many-atom potential and well suited for metallicsystems. Instead of solving the many-electron Schrödinger equa-tion, the total energy of a solid at any atomic arrangements canbe calculated based on the local-density approximation. Thisapproximation views the energy of the metal as the energyobtained by embedding an atom into the local electron densityprovided by the remaining atoms of the system. In addition, thereis an electrostatic interaction, which is the two-body term. Eachatom can be viewed as an impurity in the host of other atoms,and when an impurity is introduced, the total potential is a sumof host and impurity potentials. Daw and Baskes [22,23], followingStott-Zaremba corollary, proposed an approximation to the totalpotential as

Etot ¼Xi

FðqiÞ þ12

Xi;ji–j

/ðrijÞ ð4Þ

/ðrijÞ is the short-range pairwise potential describing the electro-static contribution. qi is the density of the host at the position ribut without atom i. The host density can be further approximated

(a) N atom in a neighboring list

(b) Interaction between onevertex with other vertices inanother prism in R-MD

Fig. 3. 2D R-MD force simplification problem.


by a sum of the atomic density f of the constituents asqi ¼

Pj–if jðrijÞ, where f j, the local density function, is a simple func-

tion of the position of other atoms positions. F is the embeddingenergy function, which is the energy to place an atom in the electronenvironment at the local electron density qi.

In the R-MD mechanism, the interatomic potential uncertaintyis modeled by two forms of error generating functions, e1ðrijÞ ande2ðqÞ, which are associated with qðrijÞ or /ðrijÞ, and FðqÞ, respec-tively, depending on whether the function’s domain is the electrondensity q or the interatomic distance r. The interval electron den-

sity function is q;qh i

ðrijÞ ¼ qðrijÞ � eðqÞ1 ðrijÞ, the interval pairwise

potential is /;/h i

ðrijÞ ¼ /ðrijÞ � eð/Þ1 ðrijÞ, and the interval embed-

ding energy function is F; F� �ðqÞ ¼ FðqÞ � eðFÞ2 ðqÞ. We refer to e1ðrÞ

as type I error generating function which associates with r domain,and e2ðqÞ as type II error generating function which associates withq domain. As the interatomic distance between two atom i and jrij !1, their interaction becomes weaker and approach to 0asymptotically. On the other hand, the electron density functionqðrijÞ and the pairwise potential /ðrijÞ of rij must remain boundedas rij ! 0. In addition, the inclusion properties of interval mustbe kept for both the original functions and their first derivatives.Consequently, the type I error generating function e1ðrÞ is requiredto satisfy the following six conditions. The error function and itsfirst derivative must be bounded in the vicinity of zero, i.e., (1)limr!0e1ðrÞ <1 and (2) limr!0

@e1ðrÞ@r <1. The function and its first

derivative must decay asymptotically, i.e., (3) limr!1e1ðrÞ ¼ 0 and(4) limr!1

@e1ðrÞ@r ¼ 0. The value of the function and its derivative

must be included in the interval, i.e., (5)

8r 2 ½0;1Þ : f ðrÞ 2 f ðrÞ � e1ðrÞ½ � and (6) @f ðrÞ@r 2 @qðrÞ�@e1ðrÞ

@r

h i, where

f ðrÞ denotes the function either qðrijÞ or /ðrijÞ. Based on the sixrequired conditions, two analytical and admissible forms arefound. They are a rational function where the denominator is onedegree higher than the numerator as

e1ðrÞ ¼ a1r þ a0b2r2 þ b1r þ b0

ð5Þ

and an exponential function as

e1ðrÞ ¼ ae�br where b > 0: ð6Þ

Similarly, the analytical choice of type II error generating func-tion associated with FðqÞ is limited by five conditions. (1) The errorfunction must have a negative finite slope at q ¼ 0 for FðqÞ � e2ðqÞ,i.e.�1 < @½FðqÞ�e2ðqÞ�

@q < 0. (2) The function must have a positive slope

at large electron density for FðqÞ � e2ðqÞ. (3) The function mustdecay asymptotically when two atoms are far away, i.e.limq!1e2ðqÞ ¼ 0. (4) The function must be non-negative function,i.e., e2ðqÞP 0, 8q 2 ½0;1Þ. (5) The local extrema (in this case, max-ima) at q ¼ q0 because FðqÞ � e2ðqÞ attains its minimum q0 as well,

thus @½FðqÞ�e2ðqÞ�@q jq¼q0

¼ 0 means that @e2ðqÞ@q jq¼q0

¼ 0. The condition (5)

corresponds to the so-called effective pair scheme, which requiresthe embedding function to attain its minimum at the density ofequilibrium crystal [24]. One analytical choice of the error generat-ing function of type II that satisfies all of these conditions is

e2ðqÞ ¼ aqq0

� �bq0

e�bðq�q0Þ; b >1q0

; bq0 R f0;1g: ð7Þ

For EAM potential, the classical force for atom ith based on itsneighborhood (jth atoms) is calculated as

F!i ¼ �

Xj–i

@FiðqÞ@q

��q¼qi

� @qjðrÞ@r

��r¼rijþ @FjðqÞ

@q

��q¼qj

� @qiðrÞ@r

��r¼rij

"

þ@/ijðrÞ@r

��r¼rij

#r!

i � r!

j

� �rij

ð8Þ

where F!is the vector force, FðqÞ is the embedding energy, qðrÞ is the

local electron density function, /ðrijÞ is the pairwise potential. Forinterval EAM potential, the upper and lower bounds of the intervalforce are extended based on the classical force calculation and cal-culated separately as

F!

i ¼�Xj–i

@FiðqÞ@q

��q¼qi

� @ eðFÞ2 ðqÞ@q

��q¼qi

0@

1A � @qjðrÞ

@r

��r¼rij� @ eðqÞ1 ðrÞ

@r

��r¼rij

0@

1A

24

þ @FjðqÞ@q

��q¼qj

� @ eðFÞ2 ðqÞ@q

��q¼qj

0@

1A � @qiðrÞ

@r

��r¼rij

@ � eðqÞ1 ðrÞ@r

��r¼rij

0@

1A

þ @/ðrÞ@r

��r¼rij� @ eð/Þ1 ðrÞ

@r

��r¼rij

0@

1A35 � ri!� rj

!� �rij

ð9Þ


F!

i ¼ �Xj–i

@FiðqÞ@q

��q¼qi

þ @ eðFÞ2 ðqÞ@q

��q¼qi

0@

1A

24

� @qjðrÞ@r

��r¼rijþ @ eðqÞ1 ðrÞ

@r

��r¼rij

0@

1A @FjðqÞ

@q

��q¼qj

þ @ eðFÞ2 ðqÞ@q

��q¼qj

0@

1A

� @qiðrÞ@r

��r¼rijþ @ eðqÞ1 ðrÞ

@r

��r¼rij

0@

1Aþ @/ðrÞ

@r

��r¼rijþ @ eð/Þ1 ðrÞ

@r

��r¼rij

0@

1A35

�ri!� rj

!� �rij

ð10Þ

If the error generating functions are zeros almost everywhere, theinterval force converges to the classical force, which is a singletoninterval. Fig. 4a illustrates the computation of atomistic intervalradius at each time step. The atomic interactions between atom iand its neighbors are assumed to vary within some ranges andbecome slightly stronger or weaker based on the prescribed intervalpotential functions. Therefore, the total force acting on atom i at onetime step can be captured by an upper bound and a lower bound, oran interval force vector. The upper bound and lower bound of theforce interval can have different magnitudes, as well as differentdirections, as illustrated Fig. 4b. Compared to our previous work[16,17], here the signum function is not included to allow morevariations in the interval interatomic forces.

3.4. Uncertainty propagation schemes

Four schemes are developed to quantify and propagate theuncertainty. They are termed midpoint-radius, lower–upperbounds, total uncertainty principle, and interval statistical ensem-ble schemes.

3.4.1. Midpoint-radius schemeAt each time step, the lower and upper bounds of the pairwise

interval force can be separately computed according to theinterval-valued potentials. Based on the midpoint-radius represen-tationof an arbitrary interval inKaucher interval spaceKR, the innerand outer radii of force intervals can be computed respectively as

radinner f ; fh i

¼min f � f ; f � f

n o; if f ; f

h iis proper

max f � f ; f � fn o

; if f ; fh i

is improper

8<:

radouter f ; fh i

¼max f � f ; f � f

n o; if f ; f

h iis proper

min f � f ; f � fn o

; if f ; fh i

is improper

8<:

ð11Þ

(a) pairwise force between atom iand its neighbors: upper and lowerbounds calculation

Fig. 4. The computational process of the interva

where f denotes the nominal force from traditional MD calculation,

f and f denote the lower and upper bounds of the interval forces,respectively. This formula is consistent with the fact that a properinterval has positive interval radius, and an improper interval hasnegative interval radius. The radius of pairwise interval force is thencomputed based on the choice of inner or outer radius. The radius ofeach pairwise interval force associated with atom i can be summedtogether within its neighborhood to produce the radius of the totalinterval force for atom i at one time step. The corresponding radiifor atomistic interval velocities and interval positions can then beupdated accordingly. At each time step, the midpoint value of pair-wise interval force is reassigned as the usual nominal pairwise forcevalue f in classical MD simulation, as shown in Fig. 2a. With theradius and midpoint of the atomistic interval forces, velocities,and positions, one can reconstruct the according intervals of atomi at that time step. The advantage of this scheme is that the numberof operations can be reduced based on the fact that the midpoint ofinterval positions and interval velocities are equal to the classicalMD values and the calculations of radii and midpoints can bedecoupled, as described in Section 3.1. The decoupling processallows for shortening computational time by computing the radiiof the atomistic intervals directly. The computational procedure issummarized in Algorithm 1.

One technical issue of the midpoint-radius uncertainty propa-gation scheme is that the inner radii tend to underestimate theuncertainty of the forces, positions, and velocities, whereas theouter radii tend to over-estimate the uncertainty. Therefore, tworuns of simulation, one with inner radii option, the other withouter radii option, are recommended. Although the actual intervalbounds are unknown, the results from these two simulation runscan give a good estimate of error bounds. The choice of inner radiiis associated with higher soundness, whereas the choice of outerradii is associated with higher completeness, because the solutionset is broader in the later case.

A caveat for this implementation scheme is that forisothermal ensemble, such as isothermal-isobaric (NPT)statistical ensemble, the uncertainty associated with temperatureis pessimistically large even though the epistemic uncertaintyassociated with temperature should be small, because itmostly fluctuates around a constant. The temperature is computedby

Kinetic Energy ¼XNk¼1

mv2k

2¼ 1

2dNkBT ð12Þ

(b) total force acting on atom i at onetime step

l force acting one atom at every time step.


where N is the number of atoms, kB is the Boltzmann constant, T isthe temperature, and d is the dimensionality of the simulation.Because of the over-estimation issue, the uncertainty associatedwith temperature is set as 0 in this scheme to preserve the isother-mal properties of the statistical isothermal-isobaric ensemble, andhence only the uncertainties in atoms’ positions and forces con-tribute to the pressure uncertainty. The output of this scheme isthe radii values of the interval output.

Algorithm 1. Implementation of midpoint-radius scheme.

1:
for every time step do 2: compute the total upper bound force f 3: compute the total lower bound force f 4: compute the total nominal force f
5:
compute the pairwise force interval radius radðf Þ basedon the choice of inner or outer radius
6:
radðvÞ radðvÞ þ radðf Þ=m 7: radðxÞ radðxÞ þ radðvÞ � Dt 8: quantifying uncertainty by interval radius of atomistic
positions, velocities, and forces
9: end for
3.4.2. Lower-upper bounds scheme

The second approach to implement R-MD mechanism is to uti-lize the lower and upper bounds representation of Kaucher inter-vals to model the uncertainty of each atoms in the MDsimulation. Therefore this approach is referred to as lower-upperbounds scheme throughout this work. In this approach, the upperand lower bound values of the atomistic positions, velocities, andforces, as well as their nominal values are retained at every timestep. The nominal value of the atomistic interval velocities areupdated based on the classical MD, whereas the lower and upperbounds of the atomistic interval velocities are updated from thenominal velocities and the lower and upper bounds of the totalinterval force. In the same manner, the nominal value of the ato-mistic interval positions are updated based on the nominal valueof the atomistic interval velocities, whereas the bounds of the ato-mistic interval positions are updated according to both the nomi-nal value of the atomistic interval positions and the bounds ofthe atomistic interval velocities. This computational scheme isdescribed in Algorithm 2 for velocity-Verlet integrators. The outputof this scheme is interval outputs, with lower and upper bounds.

Algorithm 2. Implementation of lower-upper bounds scheme.

1:
for every time step do 2: compute the total upper bound force f 3: compute the total lower bound force f 4: v v þ f=m 5: v v þ f=m 6: v v þ f=m 7: x xþ v � Dt 8: x xþ v � Dt 9: x xþ v � Dt 10: compute interval outputs 11: end for
3.4.3. Total uncertainty principle scheme

In this computational implementation, besides using the errorgenerating function to quantify the uncertainty in the atomistic

total force, the radii of atoms’ interval velocities are assumed tobe a fixed percentage of the magnitude of velocities. The totaluncertainty principle scheme is conceptually equivalent to thetemperature scaling process in isothermal-isobaric ensemble bycoupling the simulation system to a thermostat. The scheme isbased on the so-called total uncertainty principle, which states thatthe total uncertainty level of the system during two consecutiveobservations remains the same. As a result, the uncertainty needsto be scaled back from time to time during simulation. The physicalmeaning of this process is that the temperature of the simulationcell is measured at every time step, so that the total uncertaintyof the simulation is roughly at the same scale throughout the sys-tem. The scaling process is mathematically expressed as

radðvÞ ¼ a% � v; for Kaucher interval subscheme ð13Þ

radðvÞ ¼ a% � jvj; for classical interval subscheme ð14Þwhere v denotes the classical MD values of the atoms’ velocities,which is also the nominal values of the atomistic interval velocities.Eqs. (13) and (14) are associated with Kaucher intervals and classi-cal intervals, respectively. The total uncertainty principle scheme isimplemented according to Algorithm 3. The total uncertaintyscheme with the strictly non-negative radius of interval velocities,described by Eq. (14), as the total uncertainty scheme with classicalintervals. The other uncertainty scheme, described by Eq. (13), isreferred to as the total uncertainty scheme with Kaucherintervals. The total principle uncertainty scheme can be thought asan extension to the midpoint-radius scheme described inSection 3.4.1.

Algorithm 3. Implementation of total uncertainty principlescheme.

1:
for every time step do 2: compute the total upper bound force f 3: compute the total lower bound force f 4: compute the total nominal forcef
5:
compute the pairwise force interval radius radðf Þbased on the choice of inner or outer radius
6:
radðvÞ a% � v or radðvÞ a% � jvj 7: radðvÞ radðvÞ þ radðf Þ=m 8: radðxÞ radðvÞ � Dt 9: quantifying uncertainty by interval radius of atomistic
positions, velocities, and forces
10: end for
3.4.4. Interval statistical ensemble scheme: interval isothermal-isobaric (NPT) ensemble

As a result of incorporating the uncertainty into each atomisticposition, velocity and force, the system control variables also havetheir own uncertainty and should be consequently modeled asintervals. For example, in NPT ensemble, the pressure and temper-ature of the system are the control variables and could be repre-sented as Kaucher intervals. Furthermore, if the simulation cell iscoupled to a chain of thermostats and barostats, then the positions,velocities, and forces of thermostats and barostats can also bemodeled as Kaucher intervals and updated via Kaucher intervalarithmetic. The advantage of this approach is that the scheme pre-serves the dynamics of the statistical ensemble, and therefore theuncertainty of the system can be quantified more rigorously. Werefer to this scheme as the interval statistical ensemble scheme.The computational procedure of the interval statistical ensemblescheme is summarized in Algorithm 4.


In the NPT statistical ensemble, the simulation cell is coupled tothe Nosë-Hoover chain (NHC) of thermostats and barostat. The cor-responding interval governing equations of motions [25] areextended to Kaucher intervals space KR as

_ri; _rih i

¼pi;pi

h imi

�pg ;pg

h iWg

� ri; ri� �

;

pi;pi

h i¼ F i;F i� �� pg

Wg� pi;pi

h i� 1Nf

Tr pg ;pg

h iWg

�pi�pn;pn

h iQ

� pi;pi

h i

_h; _hh i

¼pg ;pg

h iWg

� h; hh i

;

_pg ; _pg

h i¼ V Pint;Pint

� �� IPext � h;h

h i� R;R� �� hT

;hTh i

� 1Nf

XNi¼1

pi;pi

h i2mi

0B@

1CAI �

pn1;pn1

h iQ1

� pg ;pg

h i;

_nk; _nkh i

¼pnk

;pnk

h iQk

for k ¼ 1; . . . ;M;

_pn1 ; _pn1

h i¼

XNi¼1

pi;pi

h i2mi

� 1Wg

Tr pg ;pg

h iT� pg ;pg

h i� �

� ðNf þ d2ÞkText � pn1;pn1

h i�

pn2;pn2

h iQ2

;

_pnk ; _pnk

h i¼

p2nk�1

; p2nk�1

h iQk�1

� kText

0@

1A� pnk

; pnk

h i

�pnkþ1 ;pnkþ1

h iQkþ1

for k ¼ 2; . . . ;M � 1;

_pnM ; _pnM

h i¼

pnM�1 ; pnM�1

h i2QM�1

� kText

0B@

1CA; ð15Þ

where ri; ri� �

and pi;pi

h iare the interval positions and momentum

of atom i, respectively, h;hh i

is the interval cell matrix, pg ;pg

h iis

the interval modularly invariant form of the cell momenta, nk; nkh i

and pnk; pnk

h iare respectively the thermostat interval variable and

its conjugated interval momentum of the kth thermostat of theNosë-Hoover chain of length M. The constant mi, Wg , and Qk arethe mass of atom i, barostat, and kth thermostat, respectively. Themass of the barostat and thermostats are used to tune the frequencywhere those variables fluctuate. The tensor I is the identity matrix.The constant Nf ¼ 3N is the system degrees of freedom. Text and Pext

denote the external temperature and external hydrostatic pressure,respectively. The matrix R is defined by

R ¼ h�10 ðt � IPextÞh�T0 ð16Þ

The extended interval time integration schemes are also imple-mented, following the time-reversible measure-preserving Verletintegrators derived by Tuckerman et al. [26].

Algorithm 4. Implementation of interval statistical ensemblescheme.

1:
for every time step do 2: compute the total upper bound force f 3: compute the total lower bound force f 4: compute the total nominal forcef
5:
update interval velocities v ;v½ � and nominal velocitiesv based on the interval statistical ensemble
6:
update interval positions x; x½ � and nominal positions x
based on the interval statistical ensemble
7: compute interval outputs 8: end for
4. An example of R-MD: uniaxial tensile loading of an aluminumsingle crystal oriented in h100i direction

The R-MD mechanism and four uncertainty propagatingschemes are implemented on LAMMPS [20]. For each atom, thelower and upper bounds of positions, velocities, and forces, areadded and retained in the computer temporary memory for everytime step. Based on the interval positions, velocities, and forces, wefollowed the virial formula to compute the interval or radius of themicroscopic symmetric pressure tensor. The implementation of theinterval statistical ensemble is based on the modified C-XSClibraries [27] to include Kaucher interval arithmetic.

In the rest of this section, the simulation setting of a case studyof uniaxial tensile loading of an aluminum single crystal is intro-duced in Section 4.1. Section 4.2 introduces the interval EAMpotential for aluminum based on the error generating functionsdescribed in Section 3.3.1. Section 4.3 present the numericalresults of the study. Section 4.4 compares the results of differentuncertainty propagation schemes against each other. In Section 4.5,different schemes and their effectiveness are verified according tothe soundness and completeness levels. In Section 4.6, the resultsare further verified by studying the finite-size effects of four differ-ent schemes.

4.1. Simulation settings

A R-MD simulation of stress-strain relation for fcc aluminumsingle crystal loaded in h100i direction [28–30] is adopted. Thesimulation cell contains 10 lattice constants in x, y, and z direc-tions, and 4000 atoms. The simulation time step is 1 fs, and peri-odic boundary condition is imposed for all directions of thesimulation cell. The modified interval EAM aluminum interatomicpotential used here is developed based on Mishin et al. [24], whichwas originally derived from both experiments and first principlescalculations. The simulation cell is equilibrated for 20 ps, and thelattice is allowed to expand at each simulation cell boundary to atemperature of 300 K and a pressure of 0 bar. After the equilibra-tion, all the uncertainties associated with the system control vari-ables are reset to 0, and all the atomistic intervals are degeneratedinto singleton intervals. Next, the simulation cell is deformed in x

direction at a strain rate of _e ¼ 10�10 s�1 under the NPT ensemble.The symmetric microscopic pressure tensor is then computed as

Pij ¼PN

k¼1mkvkivkj

VþPN

k¼1rki f kjV

ð17Þ

wherem is the mass of the atoms, r, v, and f are the atoms’ positions,velocities, and interatomic forces, respectively. The pressure tensorcomponent Pxx is taken as the stress r and engineering strain valuese are output into a separate file, which later is post-processed in

0 0.05 0.1 0.15 0.20

2

4

6

8

10

Strain

Stre

ss (

GP

a)

Variations of Aluminum potentials in MD

Winey et al. (2009)Mishin et al. (1999)Voter and Chen (1987)Zhou et al. (2004)Liu et al. (2004)Mendelev et al. (2008)

Fig. 5. Uniaxial tensile deformation of aluminum simulation cell using variousinteratomic potential.

Table 2Error generating function parameters used in the simulation sensitivity analysis.

Function Errortype

Functional form Parameters

Pairwise potential /ðrijÞ Type I e1ðrÞ ¼ ae�br a ¼ 2:0000,b ¼ 0:7675

Electron density qðrÞ Type I e1ðrÞ ¼ ae�br a ¼ 0:0139,b ¼ 0:4993

Embedding function FðqÞ Type IIe2ðqÞ ¼ a q

q0

� �bq0e�bðq�q0Þ a ¼ 0:2700,

b ¼ 1:5000


MATLAB. As a self-consistency check, we also ran another simula-tion with 40 40 40 lattice constants with 1 fs time step, andanother 10 10 10 with 0.1 fs to compare with the simulationresults. Our interested quantity is the stress, which is also one theoutputs of the simulation, and is directly proportional to Pxx. Thus,the goal of this case study is to quantify the uncertainty associatedwith the element Pxx in the microscopic pressure tensor.

Fig. 5 presents the results of stress r ¼ Pxx versus strain e usingaluminum EAM interatomic potentials from Winey et al. [31], Mis-hin et al. [24], Voter and Chen [32], Zhou et al. [33], Liu et al. [34],Mendelev et al. [35]. Fig. 5 shows that the choice in potentials inMD simulation is indeed a major source of uncertainty, anddemonstrates the need for quantifying uncertainty in MDpotentials.

4.2. Interval EAM potential for aluminum based on Mishin’s potential

One example of type I error generating function associated withqðrÞ is shown in Fig. 6a, where e1ðrÞ ¼ ae�br , along with the sensi-tivity analysis results in Fig. 6b. Table 2 specifies the numericalparameters and the associated functional form to generate theuncertainty in the modified Mishin’s aluminum potential [24]. Anerror generating function for qðrÞ in the exponential form is shownin Fig. 7a, and the sensitivity analysis result is shown in Fig. 7b.

An example of type II error generating function is shown inFig. 8a, the sensitivity of stress-strain relation because of the erroris shown in Fig. 8b. The result shows that the curve is much lesssensitive with respect to type II errors than to type I errors.

0 1 2 3 4 5 6 7−20

0

20

40

60

80Φ(r) as a function of r

r axis

Φ(r

) ax

is

original1st derivative

(a) Enclosed φ(r) and its first deriva-tive by the error generating functione(φ)1 (rij)

Fig. 6. (a) Error generating function in /ð

4.3. Numerical results

In this section, we concentrate on Mishin et al. [24] EAM poten-tial, where the cutoff radius is 6.28721 Å, and study three sets ofparameters for error generating functions, as tabulated in Table 3.We also set q0 ¼ 1:0 for all the schemes and a ¼ 0:001 for the totaluncertainty scheme.

We compare and analyze the error qualitatively with tables andcontrast the patterns of different schemes in Section 4.4. Fig. 9ashows the lower and upper bounds of pressure Pyy using the inter-val statistical ensemble during the equilibration phase, with theparameters described in Table 3. Fig. 9b presents the magnifiedview of Fig. 9a in the window of 16–20 ps. The correspondingterms for Pxx and Pzz are similar to Fig. 9. As an illustration,Fig. 10a shows the interval positions of atoms as prisms at the timet ¼ 12 ps within the simulation cell where the midpoints of atoms’interval positions are computed from the Verlet integrals, wherethe parameters from Table 2 have been used. The periodic bound-ary is plotted as the box. In the implementation, the interactionbetween particles is modeled by the extended Newton’s thirdlaw as

F ij; F ij� �þ F ji; F ji

� � ¼ 0 ð18Þ

or equivalently

radðF ijÞ ¼ �radðF jiÞ ð19Þ

in Kaucher interval form, where half of the interval radii should bepositive, and the other half are negative. To verify this, the radii his-togram of the atom positions in x-direction at 12 ps is also plottedas in Fig. 10b. It shows that the mean l ¼ 2:45064 � 10�6 � 0 asexpected. The distribution appears to be normal. The radii his-tograms of the atom position in y- and z-directions are similar toFig. 10b.

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Strain

Stre

ss (

GP

a)

Sensitivity Analysis of φ(r)

Original EAM potModified EAM pot +e

(φ)1 (r)

Modified EAM pot −e(φ)1 (r)

(b) The sensitivity analysis results ofφ(r) with respect to the error gener-ating function e

(φ)1 (rij)

rÞ and (b) sensitivity analysis results.

0 1 2 3 4 5 6 7−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3ρ(r) as a function of r

r axis

ρ(r)

axi

s


(a) Enclosed ρ(r) and its first deriva-tive by the error generating functione(ρ)1 (rij)

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Strain

Stre

ss (

GP

a)

Sensitivity Analysis of ρ(r)


(ρ)1 (r)

Modified EAM pot −e(ρ)1 (r)

(b) The sensitivity analysis results ofρ(r) with respect to the error gener-ating function e

(ρ)1 (rij)

Fig. 7. (a) Error generating function in qðrÞ and (b) sensitivity analysis results.

0 0.5 1 1.5 2−15

−10

−5

0

5F(ρ) as a function of ρ

ρ axis

F(ρ

) ax

is


(a) Enclosed F (ρ) and its first deriva-tive by the error generating functione(F )1 (ρ)

0 0.05 0.1 0.15 0.2 0.250

2

4

6

8

10

12

Strain

Stre

ss (

GP

a)

Sensitivity Analysis of F (ρ)


(F )2 (ρ)

Modified EAM pot −e(F )2 (ρ)

(b) The sensitivity analysis results ofF (ρ) with respect to the error gener-ating function e

(F )1 (ρ)

Fig. 8. (a) Error generating function in FðqÞ and (b) sensitivity analysis results.

Table 3Error generating function parameters used in numerical study.

eð/Þ1 ðrÞ eðqÞ2 ðrÞ eðFÞ2 ðqÞa b a b a b

1.2500 � 10�2 9.2103 � 10�1 1.3930 � 10�4 1.2668 � 100 1.0800 � 10�3 1.5000 � 100

0 0.5 1 1.5 2x 104

−6000

−4000

−2000

0

2000

4000

6000

8000

Time (fs)

Pre

ssu

re (

bar

s)

Pressure fluctuation during the equilibration process

Nominal pressureUpperbound PressureLowerbound Pressure

(a) Microscopic pressure Pyy fluctua-tion with nominal values, lower andupper bounds during equilibrationphase.

1.6 1.65 1.7 1.75 1.8 1.85 1.9 1.95 2x 104

−800

−600

−400

−200

0

200

400

600

Time (fs)

Pre

ssu

re (

bar

s)

Pressure fluctuation during the equilibration process

Nominal pressureUpperbound PressureLowerbound Pressure

(b) Magnified view for microscopicpressure Pyy in Figure 9a between16ps and 20ps during equilibrationphase

Fig. 9. Relationship between the lower bound Pyy , the upper bound Pyy and the nominal Pyy in the pressure tensor component of y-direction.


(a) Orthographic view of simulation cell in MATLAB where atoms are pre-sented as prisms according to their interval centers and radii at time 12psduring the deformation process in the interval statistical ensemble scheme.

(b) Histogram of the radii of atomistic interval positions in x-direction inFigure 10a shows a mean very close to 0 in midpoint-radius representationscheme, illustrating Newton’s third law in interval form.

Fig. 10. (a) Visualization of R-MD mechanism in MATLAB for interval statistical ensemble scheme and (b) the histogram of the radii of atomistic interval in x direction with anormal distribution fit whose mean very close to 0.


4.4. Comparisons of numerical results for different schemes

To measure the deviation between the results from these fourimplementation schemes and those of the classical MD simulationsand compare the errors between different schemes, the maximumdeviation from the nominal values to the interval bounds,

AðeÞ¼ rad½rðeÞ;rðeÞ�j j; formidpoint-radiusmax rðeÞ�r0ðeÞj j; rðeÞ�r0ðeÞj jf g; for lower-upperbounds

ð20Þ

is used to describe the absolute error, where rðeÞ is the lower boundand r is the upper bound of the stress, respectively, r0ðeÞ is theresult of the classical MD simulations for the same simulation cellsize. The maximum deviation AðeÞ is equivalent to the l1-norm ofr;r½ � � r0 in R2 for any e of the simulation.

Fig. 11a presents the general comparison graph among fourimplementation schemes, where the total uncertainty scheme isfurther splitted into two subschemes, one with classical intervals,and the other with Kaucher intervals. As indicated in Fig. 11a,the total uncertainty scheme with classical intervals clearly coversall the possibilities of the solutions, whereas other schemes followvery closely with the classical MD simulation result. Fig. 11b pro-vides a magnified view of Fig. 11a, where the ranges of strainand stress are 0:0901 6 e 6 0:0943 and 5:4486 6 r 6 5:6817. We

note that for all the schemes, the classical MD simulation result liesbetween the lower and upper bounds. More interestingly, for thetotal uncertainty scheme with Kaucher intervals, the lower andupper bounds sometimes swap the positions, but still capture theclassical MD simulation solution in their ranges. This observationis contrast with the total uncertainty scheme with classical inter-vals, where an over-estimated solution is expected, as presentedin Fig. 11a. After the plastic deformation, the result in interval sta-tistical ensemble schemes does not follow the classical MD simula-tion result as closely as other non-propagating schemes, such asthe midpoint-radius scheme and the lower-upper bounds scheme,but the absolute errors are comparable in the elastic portion duringthe deformation process. Fig. 12a shows another set of stress-strainresults with larger error generating function parameters, where

eð/Þ1 ðrÞ : a ¼ 1:2500 � 10�1, b ¼ 6:9078 � 10�1; eðqÞ2 ðrÞ : a ¼ 1:3930�10�3, b ¼ 8:8305 � 10�1; eðFÞ2 ðqÞ : a ¼ 1:0800 � 10�2; b ¼ 1:5000�100. Fig. 12b presents the magnified view of Fig. 12a, where theranges of strain and stress are 0:0901 6 e 6 0:0943 and4:900 6 r 6 6:0000, similar to the window of Fig. 11b for compar-ison purposes. The patterns of these stress-strain curves are analo-gous to those of Fig. 11.

Fig. 13a compares the absolute value of outer radius AðeÞ for therange 0 6 e 6 0:1. Fig. 13d presents the same plot, but the totaluncertainty with Kaucher intervals scheme is removed to further

0 0.05 0.1 0.15 0.20

2

4

6

8

10

12

Strain

Stre

ss (

GP

a)

Comparison of R−MD using various scheme

Stat.Int. upperStat.Int. lowerUp.Low. upperUp.Low. lowerMid. Rad. upperMid. Rad. lowerTot.Unc.Classical upperTot.Unc.Classical lowerTot.Unc.Kaucher upperTot.Unc.Kaucher lower10×10×10 Original

(a) General comparison between the results of all schemes showing bands withdifferent widths covering the classical MD simulation result using parametersin Table 2.

0.0905 0.091 0.0915 0.092 0.0925 0.093 0.0935 0.0945.45

5.5

5.55

5.6

5.65

Strain

Stre

ss (

GP

a)

Comparison of R−MD using various scheme

Stat.Int. upperStat.Int. lowerUp.Low. upperUp.Low. lowerMid. Rad. upperMid. Rad. lowerTot.Unc.Classical upperTot.Unc.Classical lowerTot.Unc.Kaucher upperTot.Unc.Kaucher lower10×10×10 Original

(b) Magnified view of Figure 11a where 0.0901≤ ε ≤0.0943 and 5.4486≤σ ≤5.6817 showing the total uncertainty scheme with Kaucher intervals swapbounds during the simulation.

Fig. 11. Comparisons of the stress outputs computed by four different schemes. The figures show that in all implementation schemes, the classical MD simulation result iscontained in the range of the implementation results.


contrast the error between the other three schemes. All of themshow that the uncertainty of the simulation slightly decreases dur-ing the elastic deformation. The uncertainty in the interval statisti-cal ensemble decreases with a slower rate, compared to themidpoint-radius and lower-upper bounds scheme. There is a goodagreement between the midpoint-radius and the lower-upperbounds scheme results, because in the midpoint-radius, the mid-point of interval stress is assigned as the classical MD value, andin the lower-upper bounds scheme, the upper and lower boundsare also calculated based on the classical MD value as well. It isthen expected that the uncertainty estimated by these two meth-ods are comparable. One common feature between the midpoint-radius scheme and the interval statistical ensemble scheme is thatthe uncertainty does not oscillate as heavily as in the total uncer-tainty ensemble scheme with (Fig. 13a) and the lower-upper

bounds scheme (Fig. 13d). This observation is explained by thehypothesis that compared to the non-propagating implementationschemes, the schemes that propagates the uncertainty tend to pro-duce smoother transition between time step. Last but not least, inthree schemes excluding the total uncertainty scheme, the uncer-tainty of the stress is predicted to grow larger after the yield point,especially in the interval statistical ensemble scheme (Fig. 13c).However, the interval statistical ensemble scheme predicts muchfluctuation of the stress before the yield point (Fig. 13d), whichperhaps makes more physical intuition.

Table 4 compares the computational time and slow-down factorbetween the different schemes to the classical MD simulation. Thecomputational time is obtained by running the simulation with dif-ferent schemes on 4 processors with MPI support. The slow-downfactor is calculated the ratio of the computational time using R-MD

Fig. 12. Comparisons of the stress outputs with larger variation, computed by four different schemes. The used parameters are eð/Þ1 ðrÞ : a ¼ 1:2500 � 10�1, b ¼ 6:9078 � 10�1;eðqÞ2 ðrÞ : a ¼ 1:3930 � 10�3, b ¼ 8:8305 � 10�1; eðFÞ2 ðqÞ : a ¼ 1:0800 � 10�2; b ¼ 1:5000 � 100.


to the classical MD simulations, showing a factor of between 4 and5 times slower compared to the classical MD simulations. Thisresult shows the advantage of the proposed intrusive UQ method.Non-intrusive UQ techniques, such as generalized polynomialchaos expansion, and stochastic collocation, could have muchmoresignificant slow-down ratios. It demonstrates that the intrusive UQtechniques are much more computationally affordable to high-fidelity simulations, where the computational time can vary fromdays to months.

4.5. Verification and validation

We repeat the sensitivity analysis as in Figs. 6–8 in Section 3.3.1with the error generating function parameters as tabulated inTable 3. Fig. 14a presents an overview of the four implementationresults along with the sensitivity analysis results. The totaluncertainty scheme with classical intervals provide a complete

but also over-estimated solution. However, it does capture thevariations around the yield point and after the plastic deformation.The results of sensitivity analysis is compared with the results ofdifferent schemes. We follow the soundness and completenessconcepts described in Section 2. Fig. 15 explains the relationshipbetween the estimated, the sensitive and the true range of uncer-tainty in UQ problem. The sensitive range, which is the solution ofsensitivity analysis, is a subset of the true range. Theoretically, thetrue range can be obtained by Monte Carlo sampling methods.Practically, such methods are very computationally expensiveand thus are not applicable. To evaluate the effectiveness of differ-ent schemes, the results obtained by R-MD are contrasted with thesensitivity analysis results using classical MD simulation withmodified potentials by the error generating functions describedin Section 3.3.1. A solution set is sound if it does not include aninadmissible or impossible solution; a solution set is called com-plete if it covers all possible solutions [36]. In Fig. 15, if the esti-

0 0.02 0.04 0.06 0.08 0.10

0.02

0.04

0.06

0.08

0.1

0.12

Strain

Stre

ss (

GP

a)

Maximum deviation of stress from nominal to interval end bounds

Interval statistical ensemble schemeLower−upper bounds schemeMidpoint−radius schemeTotal Uncertainty scheme with Kaucher intervals

0 0.05 0.1 0.15 0.20.0205

0.021

0.0215

0.022

0.0225

0.023

0.0235

0.024

Strain

Stre

ss (

GP

a)


Lower−upper bounds schemeMidpoint−radius scheme

0 0.05 0.1 0.15 0.2−1.5

−1

−0.5

0

0.5

Strain

Stre

ss (

GP

a)


Interval statistical ensemble schemeLower−upper bounds schemeMidpoint−radius scheme

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160.016

0.018

0.02

0.022

0.024

0.026

0.028

Strain

Stre

ss (

GP

a)


Interval statistical ensemble schemeLower−upper bounds schemeMidpoint−radius scheme

Fig. 13. Plots of AðeÞ ¼ maxfjr� r0j, jr� r0jg in different schemes.

Table 4Comparison of computational time for various schemes in R-MD.

Classical MD Midpointradius Lower-upper Tot. uncertainty Int. ensemble

Computational time (s) 467.941 1989.812 1911.058 1932.983 2170.92Slow-down factor 1 4.252 4.086 4.1308 4.639


mated range XE is a subset of the true range XT , that is XE #XT , thenXE is sound. If XT #XE, the solution set XE is complete. As intervalscan also be thought of as the sets of possible solutions, the termsolution set is hereby used interchangeably with the interval orthe range.

In measure theory, the Lesbegue measure l is a standard way ofassigning a measure to a subsets in Rn. For 1D, 2D and 3D, thephysical meaning of Lebesgue measure is the length, the areaand the volume of the set. The soundness is assigned as the ratioof Lesbegue measure of the overlapped range XE \ XT and the Les-begue measure of the estimated range. Similarly, the completenessis assigned as the ratio of Lesbegue measure of the overlappedrange XE \ XT and the Lesbegue measure of the true range.

Soundness Index ¼ lðXE \ XTÞlðXEÞ ;

Completeness Index ¼ lðXE \ XTÞlðXTÞ ð21Þ

The soundness and completeness indices are mathematicallybounded between 0 and 1, as obviously XE \ XT is the subset of XE

and XT . The soundness close to 1 means the estimated solutionset contains nearly all possible solutions, but also is likely to under-estimate true solution set. Analogously, the completeness indexclose to 1 indicates a complete but over-estimated solution sets.In this case, the true solution set XT is approximated by the sensitiverange XS, which is defined as the smallest proper interval that con-tains 7 classical MD simulation results. The equivalent mathemati-cal expression is XT � XS. The sensitive range XS includes 1 run withoriginal interatomic potential and 6 other runs with modified inter-atomic potentials by the error generating functions, whose param-eters are tabulated in Table 3. Consequently, the soundness andcompleteness indices are approximated as

Soundness Index � lðXE \ XSÞlðXEÞ ;

Completeness Index � lðXE \ XSÞlðXSÞ ð22Þ

Fig. 14. The schematic plot and actual plot of output uncertainty. In this simulation, the sensitive range is the min and max of classical MD simulation results with theoriginal interatomic potential and the alternated potentials by adding or subtract error generating functions to one of its three functions.


Fig. 16a and b plots the soundness and completeness of the fourschemes, respectively, with the total uncertainty scheme is furtherdivided into classical and Kaucher intervals. Fig. 16a and b also indi-cates that the R-MD results overlap with the sensitivity analysis.One of the differences with Section 3.3.1 sensitivity analysis is thatthe original result no longer always lies in the middle of the

�eðFÞ2 ðqÞ, �eð/ÞðrÞ, �eðqÞðrÞ as in Section 3.3.1. Indeed, among 7 runs,the result of the classical simulation run with the original inter-atomic potential ends up with 10.40% getting the maximum valueand 14.40% getting the minimum value in 7 values accounting forthe sensitivity analysis results. The total uncertainty scheme withclassical intervals achieves the completeness of 1 thoroughly dur-

ing the simulation, implying that it always covers all of the truesolution set. The soundness of the interval statistical ensemblescheme, along with midpoint-radius and lower-upper boundsschemes, are more consistent compared to the total uncertaintyscheme. Their results represent mostly between 50% and 100% ofthe sensitive range. The completeness is, however, less consistentand fluctuates much, meaning they do not cover all the possibilitiesin the sensitive ranges, and thuswill not cover all the possibilities inthe true ranges as well. The total uncertainty scheme with Kaucherintervals captures the uncertainty most effectively after the defor-mation process, with high completeness (around 80–90% inFig. 16b) and reasonably high soundness (30–40% in Fig. 16a).

Fig. 15. Schematic plot of estimated, sensitive and true range of uncertainty. Thesensitive range XS obtained by parametric study or sensitivity analysis is always asubset of the true range XT .


Mathematically, the sensitive range is a subset of the true range,XS #XT , which also means lðXSÞ 6 lðXTÞ. Eqs. (21) and (22) can bemanipulated algebraically further to establish the relationship

lðXE \ XSÞlðXEÞ 6 lðXE \ XTÞ

lðXEÞ ; ð23Þ

and

lðXE \ XSÞlðXSÞ ¼ lðXE \ XTÞ � lðXE \ ðXT n XSÞÞ

lðXTÞ � lðXT n XSÞ ð24Þ

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

S

Ran

ge C

over

age

(bet

wee

n 0

and

1)

Soundness between different

Stat.Int.Up.LowMid.RadTot.Unc.ClassicalTot.Unc.Kaucher

(a) Soundness indices of di

0 0.02 0.04 0.06 0.080

0.2

0.4

0.6

0.8

1

S

Ran

ge C

over

age

(bet

wee

n 0

and

1)

Completeness between differen

Stat.Int.Up.Low.Mid.Rad.Tot.Unc.ClassicalTot.Unc.Kaucher

(b) Completeness indices of

Fig. 16. Comparison of soundness and completeness

because of the additivity properties of Lesbegue measure on disjointsets XT ¼ XS [ ðXT n XSÞ and XS \ ðXT n XSÞ ¼£) lðXTÞ ¼ lðXSÞþlðXT n XSÞ. Consequently, the actual soundness is expected to behigher than Fig. 16a if the true range XT is known. However, thereis not enough numerical evidence to draw a conclusive commentabout the completeness in Fig. 16b. The completeness could eitherincrease or decrease depending on how lðXE\ðXT nXSÞÞ

lðXT nXSÞ compares withlðXE\XT ÞlðXT Þ . Yet, one can be certain that if the output is not complete

compared to the sensitivity analysis result, the output is also notcomplete compared to the true solution set.

4.6. Finite-size effect

We performed finite-size effect analysis for four implementa-tion schemes with 10 10 10 (4000 atoms), 12 12 12(6912 atoms), 14 14 14 (10,976 atoms) and 16 16 16(16,384 atoms) lattice constants of the simulation cell with param-

0.1 0.12 0.14 0.16 0.18 0.2train

schemes and sensitivity analysis

fferent uncertainty schemes.

0.1 0.12 0.14 0.16 0.18 0.2train

t schemes and sensitivity analysis

different uncertainty schemes.

indices between different uncertainty schemes.

0.07 0.071 0.072 0.073 0.074

4.3

4.4

4.5

4.6

4.7

Strain

Stre

ss (

GP

a)

Finite−size effect of R−MD using statistical interval ensemble

10×10×10 upper

10×10×10 lower

10×10×10 original

12×12×12 upper

12×12×12 lower

12×12×12 original

14×14×14 upper

14×14×14 lower

14×14×14 original

16×16×16 upper

16×16×16 lower

16×16×16 original

Fig. 17. Finite-size effect of R-MD and MD, where solid lines denote the 10 10 10, dotted lines denote 12 12 12, dashdot lines denote 14 14 14, dashed linesdenote 16 16 16.

10 11 12 13 14 15 160.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

Strain

A∗

=1 0 .1

0.1

0A

(ε)

dε

(GPa)

Finite-size effect analysis based on A∗

Midpoint−radius schemeLower−upper bounds schemeTotal Uncertainty scheme with Kaucher intervalsStatistical Interval Ensemble

Fig. 18. Error analysis of A for different schemes with respect to different sizes ofthe simulation cell.


eters as in Table 3. The results of R-MD are then compared with theresults of the classical MD simulations with the corresponding sizeof 10 10 10, 12 12 12, 14 14 14 and 16 16 16.Fig. 17 presents magnified view of the finite-size effect of the inter-val statistical ensemble between 0:07 6 e 6 0:075 and4:25 6 r 6 4:70. The solid lines denote the 10 10 10, the dot-ted lines denote 12 12 12, dashdot lines denote 14 14 14,dashed lines denote 16 16 16 simulation results. Also, on thesame graph, the diamond markers denote the upper bound, thesquare makers denote the lower bound and the circles denotethe classical MD simulation results.

To measure the convergent of different uncertainty quantifica-tion schemes, the strain-averaged absolute stress error between0 6 e 6 0:1, denote as A,

A ¼ 10:1

Z 0:1

0AðeÞde ð25Þ

where AðeÞ is the maximum deviation from the nominal to theinterval end bounds in Eq. (20) because of the input uncertaintyin the interatomic potential, is used to verify the finite-size effectof different schemes. The reason that e ¼ 0:1 is picked as the upperbound of the integral is that after the simulation cell runs into plas-tic deformation, it is hard to quantify and predict the error exactly,and between the range ½0; 0:1� the stress r versus strain e curves aremore consistent so that they can be compared qualitative against

each other. Mathematically, A is proportional to the L1-norm ofAðeÞ because AðeÞ measures how far the interval bounds is com-pared to the nominal values. Concisely, A is a compound L1 � l1-norms to qualitative describe the finite-size effect of the simulationresults. We expect other compound norms of the same kind, that is,Lp � lq-norms, to behave similarly, and thus, the quantity A can bejustified as an arbitrary measure for the finite-size effects. The valueof A is obtained by integrating the AðeÞ numerically by trapezoidalrule. Fig. 18 presents the integral strain-averaged absolute stresserror A with respect to different sizes of the simulation cell,showing a convergent error with respect to the size of simulationcells for all the schemes. The total uncertainty scheme with Kaucherintervals has one more parameter a to model the temperature mea-surement at every time step, thus it is not expected to exactly fol-low other schemes. However, the total uncertainty scheme withKaucher interval still shows a convergent pattern with respect tothe increase size in the simulation cell. We then conclude all fourimplementation schemes yield reliable results.

5. Discussion

There are different advantages and disadvantages betweenintrusive and non-intrusive techniques that one should considerbefore applying any of these methods. Disadvantages in intrusiveUQ techniques are as follows. Simulator is required to be modifi-able. Analysts need to have comprehensive understanding of howto apply intrusive UQ techniques to that simulator. Extra time isalso needed to develop UQ solution. However, these drawbackscan be mitigated by introducing an open framework or applicationprogram interfaces in simulation packages by developers to assistthe analysts. The main advantages of intrusive UQ techniques aretheir ability to estimate the variation ranges of outputs (via inter-val analysis) or the probability density function itself (via intrusivepolynomial chaos expansion) without sampling inputs and repeat-ing the simulation. The simulator developers and UQ analysts canwork together to have UQ embedded in simulator so that thedetailed knowledge of UQ is not required as the user of simulator.On the other hand, disadvantages of non-intrusive UQ techniquesare as follows. First, simulation needs to be repeated throughoutthe input parameter space, even though this problem can be alle-viated by using sparse grids. Second, there are many cases where


the input parameters of simulation are functions. Capturing uncer-tainty associated with these functions is still a challenge to non-intrusive UQ techniques. The advantages of non-intrusive UQinclude the possibility of building a computationally efficient sur-rogate model or a meta-model of simulations so that the outputcan be quickly evaluated. It also allows to capture the output prob-ability density function in details. Non-intrusive UQ techniques canbe developed as independent packages to couple with anysimulations.

The interval UQ technique is an intrusive and non-probabilisticthat promptly provides the range estimation for the bounds of out-put probability density function with only one or two runs. For thealgorithm with midpoint-radius representation, the output resultis an interval radius, of which midpoint is identical with classicalMD simulation. With the choice of inner or outer radius, the resultswill be different, and therefore two runs are recommended. Onerun is with the inner radius for all atomistic positions, velocities,and forces, whereas another run is with outer radius. Between bothof them, the choice of inner radius provides higher soundness, butlower completeness, compared to the choice of outer radius. Forthe algorithm with lower-upper bounds representation, the outputresult is an interval which contains the quantified uncertaintyinformation, and thus only one run is needed in this case. Betweentwo interval representations, the lower-upper bounds representa-tion is superior to the midpoint-radius representation, because theformer only requires one run, based on the fact that two boundsare computed simultaneously. Additionally, all of the interval Kau-cher arithmetic computations are carried out based on the lower-upper bounds representation as in Lakeyev’s formula and Kauchermultiplication table, which are described in Section 2. Therefore,the lower-upper bounds representation saves computational timecompared to midpoint-radius representation.

Beside the low number of runs needed to obtain a UQ solution,the interval technique also allows to capture and incorporate theuncertainty associated with the input potential functions into sim-ulation, by modeling the error generating function. In this study,two forms of analytical error generating function are proposed.Other forms of error generating functions can also be similarlyimplemented. Even though not considered in this study, theinterval technique can also be applied to capture aleatory uncer-tainty associated with thermal fluctuation. For example, the alea-tory uncertainty of Langevin thermostat can be quantified byinterval-valued atomistic force, velocity, and position to solve thestochastic differential equations. By quantifying the uncertaintyinternally, the intrusive UQ method offers an efficient way toassess sensitivity without sampling. The interval approach can beviewed as an efficient alternative technique to sampling-basedsensitivity analysis where variation ranges are estimated. How-ever, if the complete information of probability density functionsis required, one should resort to the probabilistic approaches.

As the simulation schemes change, new intrusive UQ solutionsare required to be developed to adapt with the changes in simula-tion. During this study, we have implemented and provided a basicplatform that can be extended to general intrusive UQ solutions inLAMMPS package. Still, there is a bargaining trade-off between thetime invested to develop an intrusive UQ solution and the timespent on repetitive simulations for non-intrusive UQ techniques.Another limitation of the interval technique at the current stageis its inability to quantify output probabilistically, such as jointdensity, correlations, and moments. A more advanced concept thatfuses both interval and statistical method is probability boundsanalysis [37] that envelops the cumulative density function byusing interval methods. Based on the uncertain cumulative densityfunction, one can derive the uncertain probability density function,and thus other statistical quantities. In addition, quantifications ofcorrelation between intervals [38] and between random sets [39]

have been studied to assess interdependency similarly to the prob-abilistic approach. These issues remain as open questions for futureresearch.

The classical interval has been shown to be a conservativechoice, in the sense that it almost always produces a completebut over-estimated range. Therefore, the Kaucher intervals areused as a substitute an extension because of better algebraic prop-erties. The Kaucher interval arithmetic yields a much smaller epis-temic uncertainty compared to the classical interval arithmetic bythe introduction of improper intervals, and hence tends to reducemuch the impact of over-estimation and self-dependency problemin classical intervals, and perhaps keep the epistemic uncertaintywithin some acceptable bounds. For MD simulation with EAMpotential functions, the Kaucher interval technique is considerablyefficient in term of computational time, because this methodologydoes not require the simulation to repeat numerous times. In ourstudy, the computational time is between 4.1 and 4.6 times morethan the classical MD simulations, which is an important improve-ment in term of computational time.

One of the criteria to evaluate the effectiveness of a UQ solutionis to compare the estimated uncertainties with the true ones. If theestimated uncertainty is considerably greater compared to the trueuncertainty, the solution does not hold much valuable, meaningfulor conclusive information. For output with probability densityfunction, such as polynomial chaos expansion, Monte Carlo, andLatin hypercube sampling methods, one can, for example, measurethe distance between two distributions by any Lp norm, or Kull-back–Leibler divergence. For output with range or probability den-sity function bounded support estimation, represented as intervals,one can compare the estimate interval with the true intervalderived from any sampling methods. The effectiveness of intervalUQ method is then quantified by soundness and completenessindices, which are bounded between 0 and 1. Physically, theseindices measure the ratios of the width of the ‘‘shared” intervalto the width of estimated and true intervals. Since the true intervalis not available, we approximate the true interval by finite sam-pling runs with alternated potentials. The approximated soundnessand completeness indices are then compared to the true one byalgebraic relations. For the uniaxial tensile of aluminum singlecrystal example with NPT ensemble in this study, the UQ solutionare shown to be more sound and complete during the elastic defor-mation regime than the plastic deformation regime.

So far, we have considered the uncertainty of the force by com-puting the total lower and upper bounds interval force separately.Another way to compute the force uncertainty is to consider itscontribution from each individual pairs. This scheme has also beenattempted in the study, and the same issue for this implementationscheme is that the temperature uncertainty is very large, evencompared to the total uncertainty scheme with classical intervals.The same treatment for temperature is also applied, that is, theuncertainty of pressure is only calculated by the interval forceand interval position term, but not the kinetic term. Still, the epis-temic uncertainty are too large to declare a meaningful approachfor the UQ problem. In fact, the pressure UQ problem with con-straint on the temperature uncertainty in this case study is a spe-cial case of another more general problem, which is a UQ problemwith uncertainty constraint on related quantities.

Four different implementation schemes have been developed toquantify the uncertainty in MD simulations. The total uncertaintyscheme with classical intervals produces the most conservative,yet over-estimated results, as expected. The total uncertaintyscheme with Kaucher intervals is less conservative, and followsmore closely with the interval statistical ensemble and otherresults. The a parameter is set to 0.001 through trials and errors.Reducing this a parameters will also reduce the width of theinterval stress. Perhaps the most interesting findings in these sim-


ulation runs is that the uncertainty does not always grow larger.Indeed, it becomes smaller toward the yield point. However, onlyin the interval statistical ensemble, the stress uncertainty heavilyfluctuates after the yield point. In our opinion, this behavior trulyreflects the simulation uncertainty because in the interval statisti-cal ensemble, the system dynamics is preserved from the begin-ning to the end of the simulation. Therefore, the intervalstatistical ensemble is recommended for the propagating schemes.Regarding the completeness and soundness of the interval statisti-cal ensemble, the interval stress is fairly sound, based on the factthat its estimated range provides at least 50% of the true solutionset (Fig. 16a). Yet, it is not complete, but also covers around 50%of the true solutions on average (Fig. 16b). For non-propagatingschemes, the total uncertainty schemes with classical intervals isrecommended, because it demonstrates the worst-case scenario.For most of the deformation process, this level of conservativeyields a very low soundness on the solution. However, right afterthe yield point deformation process, it represents almost 60% ofthe solution set (Fig. 16a). If one is concerned with the behaviorof the stress right after the peak, then the total uncertainty schemeis suggested. The completeness of the total uncertainty schemewith classical intervals is uniformly 1, implying that it always cov-ers more than all possibilities.

6. Conclusions

In this paper, we introduce a novel concept of MD that usesinterval analysis to quantify the uncertainty in MD simulation.The uncertainty in tabulated EAM potential is captured by analyt-ical forms of error generating functions. Based on the uncertaintyof the inputs, four different implementation schemes are proposedand developed to quantify the uncertainty of the simulation.Among these four schemes, the total uncertainty and the lower-upper bounds are non-propagating, and the midpoint-radiusscheme and the interval statistical ensemble are propagatingschemes. For non-propagating schemes, at every time step, theuncertainty of the simulation system is estimated, but not carriedforward to quantify the uncertainty at the next time step. Thephysical interpretation is that these non-propagating schemescompute the uncertainty of the output due to the uncertainty ofthe interatomic potentials at every time step. For propagatingschemes, the uncertainty of the simulation system is quantifiedand propagated toward the end of the simulation. The midpoint-radius scheme and total uncertainty scheme utilizes themidpoint-radius representation of intervals, whereas the lower-upper bound scheme and the interval statistical ensemble schemeuse the lower-upper bounds representation of intervals. In thenon-propagating schemes, such as lower-upper bounds and totaluncertainty schemes, the oscillations of uncertainty are expected.At different time steps in these non-propagating schemes, theuncertainty of the system is quantified by the interval force, whichoriginally comes from the interatomic potentials uncertainty.Because the uncertainty does not propagate from the previous timestep to the current time step, it fluctuates more heavily comparedto that of propagating schemes. In propagating schemes, such asinterval statistical ensemble and midpoint-radius schemes, theuncertainty propagates from the beginning of the simulation tothe end. Based on the findings in this work, the interval statisticalensemble is recommended for investigating the uncertainty quali-tatively and quantitatively, because it preserves the simulationdynamics and only generalizes the real number to intervals. Onthe other hand, the total uncertainty scheme with classical inter-vals is also recommended due to their completeness. Even thoughthe estimated solution range are over-estimated most of the time,this scheme can be very useful during critical events (the yield

point in this case). However, one should process with cautionrespect to the a parameter. In this study, a was tuned to 0.001mostly by trials and errors. The advantage of R-MD is that theuncertainty can be quantified based on one or two runs of the sim-ulations, compared to hundreds or thousands runs of other non-intrusive UQ techniques. The experience, along with the resultsfound in this paper, shows that the interval statistical ensembleis the most promising direction to quantify and propagate theuncertainty. Further work will include the implementation of moreinterval interatomic potential such as Lennard-Jones, Stillinger-Weber, and others.

Acknowledgement

The project is supported in part by U.S. National Science Foun-dation under Grant No. CMMI-1306996. The first author thanksWerner Hofschuster at University of Wuppertal for many helpfulconversations about C-XSC interval libraries. The authors alsothank two anonymous reviewers for their valuable suggestionsduring the review process to improve the paper.

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http://dx.doi.org/10.1103/PhysRevB.59.3393

http://dx.doi.org/10.1103/PhysRevB.59.3393

http://link.aps.org/doi/10.1103/PhysRevB.59.3393
















































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