optimization-based property preserving methods,...

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Sandia National Laboratories is a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA-0003525.

Optimization-BasedPropertyPreservingMethods,or

Pavel Bochev, Sandia National Laboratories

GoingBoldlyBeyondCompatibleDiscretizations

Supported by DOE ASCR

http://www.pnnl.gov/computing/cm4/

Future Directions

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M.D’Elia,J.Cheung,P.Kuberry,D.Littlewood,M.Perego,K.Peterson,D.Ridzal (SNL),M.Shashkov (LANL),M.Gunzburger (FSU),A.Shapeev (SkolTech),S.Moe (U.WA),M.Luskin,D.Olson(U.MN)

Key collaborators

Sponsors

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HeterogeneousNumericalMethods

2

M1 M2

M3 M4Mod

el

N3

N4

N1N2

N5

Num

eric

al p

arts

HNM

N3

N4

N5

N2N1

DOE applications requirediverse “mathematicalparts”: PDEs, integralequations, classical DFT,potential-based atomistic…

Diverse math. modelsrequire diverse “numericalparts”: mesh based (FE,FV, FD), meshless (SPH,MLS), implicit, explicit,Eulerian, Lagrangian…

HNM = Collection ofdiverse numerical partsfrom multiple disciplinesfunctioning together as aunified simulation tool

Property-Preserving Methods

HNMCoupling

HNMSolving

Parts must be stable, accurate andpreserve key physical properties.

The parts must function together as a unified simulation tool (HNM).

HNMs must be stable, accurate andpreserve key physical properties

We must be able to solve our HNMs efficiently

This trichotomy defines a hierarchy of challenges

An HNM Trichotomy

Simulation of these complex systems requires new types of methods:

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Ahierarchyofchallenges…

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1. Achieving Stability & Accuracy (Structural aspects)• Game changer: FE exterior calculus, DEC, mimetic FD,…• Typically achieved by topological means:

- Fields = cochains, topologically dual grids, etc..

2. Preserving Physical Properties (Qualitative aspects)• Max. principle, bounds, Geometric Conservation Law, correlations…• Challenges: trying to preserve these properties directly is either

- Restrictive: Cartesian mesh, angle conditions, etc, and/or,- Entangles accuracy with the property preservation, e.g., limiters.

• Stable and accurate does not imply property preserving

+

--

+

GRAD DIV

Violation of physical bounds in a PN diode simulation

3. Assembling the Parts into HNMs and Solving Them• Traditional monolithic, operator-splitting and partitioning approaches are

rapidly approaching a point of diminishing returns for complex problems; see DOE report “Multifaceted Mathematical Approach for Complex Systems”

Game changer?

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Optimization-basedmodeling(OBM)• Couch coupling, splitting and preservation of properties into an optimization problem:

☛ Reverses the roles of the coupling conditions, properties and the models.☛ Divide and conquer approach:- separates accuracy from physical properties (local bounds, conservation, etc..)- separates components: they are independent and communicate via virtual controls 𝜽

minimize!u∈Rn

Jij (ui,h;uj,h )i≠ j

∑ + Ji (ui,hT −ui,h )

i

∑ subject to Li,h (ui,hT ) = f (

!θ ) and P(uh ) ≥ 0

coupling accuracy propertiestargets

• Comprises 4 basic building blocks

P : Rn →Rm

Ji : Rn →R

Jij : Rn×n →R

Li,h (ui,h ) = fi,h a numerical scheme generating an optimally accurate targeta nonlinear operator describing the desired physical propertiesa measure of target-to-state misfita measure of state-to-state misfit (transmission, coupling, etc.)

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Basicpatterns

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1. Coupling/splitting

2. Preservation of properties

§ Optimization-based domain decomposition, M. Gunzburger, 1997 § Decomposition of everything, J.L. Lyons, 2001§ Additive splitting, Bochev, Ridzal, 2008-10, 2016

§ “Enforcing discrete maximum principle”, M. Shashkov et al, 2007 - using constrained optimization§ “Enforcement of constraints & max. principle in VMS”, T. Hughes, 2009 - applies Shashkov’s idea to VMS§ “Flux-corrected remap (FCR)”, M. Shashkov et al, 2010 - using local optimization strategy; motivated by FCT§ “Optimization-based remap and transport”, Bochev, Shashkov, Ridzal,…2011-2014 – motivated by FCR

§Local-to-Nonlocal coupling”, Bochev, D’Elia, Littlewood, 2014-17§ Atomistic-to-continuum coupling, Olson, Bochev, Luskin, 2013-15§ Interface control DD, Discacciati, Gervasio, Quarteroni, 2013-15

minimize!u∈Rn



i

∑ subject to Li,h (ui,h ) = f (!θ ) and P(uh ) ≥ 0


minimize!u∈Rn



i

∑ subject to Li,h (ui,hT ) = f (

!θ ) and P(uh ) ≥ 0


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Part1:optimization-basedcouplingandsplitting

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M. D’Elia, P. Kuberry, D. Littlewood, M. Perego, D. Ridzal (SNL), M. Gunzburger (FSU), A. Shapeev (SkolTech), M. Luskin, D. Olson (U. MN)


HNMCoupling

HNMSolving

• Semi-Lagrangian transport of passive tracers• Volume correction (Geometric Conservation Law)• Monotone and conservative Finite Element transport • Property-preserving recovery and compression• Property-preserving data transfer (remap)

• Abstract decomposition theory• Navier-Stokes solvers• Advection-diffusion solvers

• Local-to-Nonlocal couplings• Atomistic-to-Continuum• Interface problems

Current OBM portfolio

The use of optimization ideas to couple heterogeneous numerical methods and to preserve the relevant physical properties could be a game changer.

Collaborators:

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Local-to-nonlocalcouplings:why?

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• Nonlocal continuum mechanics: allows interactions at distance without contact:- Can accurately resolve small scale features, e.g., crack tips or dislocations.

• Nonlocal dielectric models: more accurate description of solvation processes:- Essential in, e.g., electrokinetic nanofluidic channels, where local response

modification to Poisson-Boltzmann are qualitatively incorrect.

• Atomistic models: accurate simulation of defects (vacancy, impurity…)

• Classical DFT: evaluation of chemical potentials of charged species- More accurate than Poisson-Nernst-Planck models at mesoscale.

J. Bardhan. “Gradient models in molecular biophysics: Progress, challenges, opportunities.”. Journal of the mechanical behavior of materials, 22.5-6: 2013

R. Nilson and S. Griffiths. Influence of atomistic physics on electro-osmotic flow: An analysis based on DFT. The Journal of Chemical Physics, 125(16), 2006.

However, full nonlocal simulation can be very expensive!Coupling to local (PDE) models can improve efficiency.

true for suspensions adjusted using NaOH (!log{aH"} #8 ! 11).

Particle size along [001], determined by XRD, typically doesnot reflect coarsening due to oriented attachment. This is be-cause junction on {112} forms chains the long axes of whichare inclined to [001], parallel to 112*.

3.3. Titania Hydrothermally Coarsened in Solutions ofOrganic Compounds

Addition of organic compounds to suspensions prior to hy-drothermal treatment dramatically effects coarsening. TEMdata indicate suppression of the oriented attachment coarseningmechanism when glycine is added to 0.001 M HCl suspensions.In addition, XRD reveal that the presence of glycine in solutionresults in a dramatic suppression of the period of fast growth,or fastest growth along [001] (Fig. 8). Other organic acids, suchas acetic acid and adipic acid, also suppress the oriented at-tachment mechanism. The details of the interactions betweenorganic molecules and surfaces will be reported separately(Penn and Banfield, in prep.).

4. DISCUSSION

Two primary coarsening mechanisms were observed. Thefirst involves single particle growth via addition of Ti ions tosurfaces from solution at rates that depend on temperature,additives, and the crystal structure and energies of crystallo-graphically distinct surfaces. The second mechanism involvesgrowth by addition of solid particles to surfaces. This occurs ina precise, crystallographically controlled manner, resulting incoherent interfaces and leading to the development of singlehomogeneous crystals. Twinning and other intergrowths canoccur because attachment requires structural accord only in thetwo dimensions defined by the junction plane (see Penn andBanfield, 1998a, and Uyeda et al., 1973). Furthermore, misori-entations produced by twists and tilts at the plane of attachmentcan produce defects, ranging from pure tilt to pure screwdislocations (see Penn and Banfield, 1998b and Chun et al.,1995).

4.1. Morphology Evolution

Reduction in surface energy is the primary driving force forsimple particle growth and morphology evolution is driven bythe further reduction in energy due to minimization of the areaof high surface energy faces. Preliminary growth from equidi-mensional primary particles is characterized by rapid develop-ment of morphology, resulting in formation of distinctly fac-eted crystallites dominated by {101} surfaces. This is followedby a period in which growth along [001] far exceeds growthalong $101% and, thus, (001) faces are shrinking. The fact thatgrowth along $101% nearly stops during the period of rapid[001] growth suggests a solubility limited system. Work byWillis (1992) on titanium ion speciation in hydrothermal aque-ous solutions supports this possibility (titanium ion concentra-tions were typically calculated to be in the ppb range, even athigh temperature and low pH).

Detailed examination of the advancing crystal faces and anunderstanding of the relative surface energies of crystal facesleads to an expectation that particle morphologies will bedominated by {101} and that growth along [001] will be rapidin comparison to {101}. Donnay-Harker (1937) rules predictthat the surface energy of the (001) faces is approximately 1.4times that of the {101} faces. Considering growth to be per-pendicular to a particular set of faces, crystal morphology willbe defined by the slowest growing faces because the fastest

Fig. 5. Graph of average anatase, hydrothermally coarsened in DIwater and in 0.001 M HCl, particle dimensions vs. time.

Fig. 6. TEM micrograph of a single crystal of anatase that was hydrothermally coarsened in 0.001 M HCl.

1552 R. L. Penn and J. F. Banfield

Penn, et al. Geochim. et Cosmochim. Acta,63,1999

Anatase TiO2 nanoparticles in DI water + HCl to control pH


HNMCoupling

HNMSolving

http://www.lehman.edu/faculty/tkurtzman/research.html

De Novo drug discovery

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Local-to-nonlocalcouplings:how?

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Problem:

Optimization reformulation: (Pattern # 1)

Ωn

Ωl

Ωlmore efficient more accurate

Advantages• Eliminates ghost forces and is consistent for all polynomial orders.• Is provably stable & admits rigorous error analysis.• Allows code reuse: models communicate via virtual controls only.• None of the existing couplings, e.g., Arlequin, blending, etc. have all of these properties!

Virtual controls

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Analysis

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Rigorous LtN coupling theory based on an equivalent reduced-space problem:

TheoremThe reduced space functional defines a norm on the space of virtual controls:

- Well-posed sub-models imply well-posed optimization problem!- In the local subdomain solution is nice Þ interaction radius e can be very small.

- Well-posedness of the reduced space problem is a consequence of

(θn,θl ) ∗:= un (θn )−ul (θl ) 0,Ωb

TheoremCoupling error is bounded by the modeling error on the local subdomain:

unEX −uLtN

0,Ω≤ un

EX −ulEX

0,Ωl

~ O(ε 2 )

un (θn ),ul (θl )( )0,Ωb≤ δ un (θn )

0,Ωbul (θl ) 0,Ωb

strong Cauchy-Schwartz inequality

• M. D’Elia, P. Bochev, Mat. Res. Soc. Cambridge Univ. Pres. 2015.

Key highlights

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Extensiontolocalandnonlocalelasticity

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Consistency tests

Mesh

(BFGS)

LtN solution

Models

Navier-Cauchy elasticity

linear quadratic

! Peridynamic/model/of/solid/mechanics/with/linearized/Linear/Peridynamic/Solid/(LPS)/constitutive/model

! Solve/the/strong/from/using/the/meshless/method/of/Silling/and/Askari

Local/and/Nonlocal/Elasticity/Models

7

Nonlocal)model

Silling,/S.A.//Reformulation/of/elasticity/theory/for/discontinuities/and/long*range/forces.//Journal*of*the*Mechanics*and*Physics*of*Solids,/48:175*209,/2000.Silling,/S.A.,/and/Askari,/E.//A/meshfree/method/based/on/the/peridynamic/model/of/solid/mechanics.//Computers*and*Structures,/83:1526*1535,/2005.Silling,/S.A.//Linearized/theory/of/peridynamic/states,*Journal*of*Elasticity*99,/85*1111,/2010.

Local)model! Navier*Cauchy/equations/of/classical/elasticity! Apply/the/standard/finite/element/approach/to/solve/the/variational form




7

Nonlocal)model



Linear Peridynamic Solid




7

Nonlocal)model






7

Nonlocal)model






7

Nonlocal)model



S. A. Silling. Reformulation of elasticity theory for discontinuities and long-range forces. J. Mech. Phys. Solids, 48(1):175 – 209, 2000

M.L. Parks, D.J. Littlewood, J.A. Mitchell, and S.A. Silling, PeridigmUsers’ Guide, Tech. Report SAND2012-7800, SNL, 2012.

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Materialdamagesimulations

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D’Elia, Bochev, Littlewood, Perego, Optimization-based coupling of local and nonlocal models: Applications to peridynamics. Handbook of Nonlocal Continuum Mechanics for Materials and Structures, Springer 2017.

Tension test with pre-crack

Virtual controls

Virtual controls Virtual controls

crack

Bar with pre-crack loaded in tension

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Interfaceproblems

12

−∇⋅∇φi = f in Ωi

φi = 0 on Γi

&'(

+φ1 = φ2

n1 ⋅∇φ1 = −n2 ⋅∇φ2

&'(

on γProblem:

min J(φ1,φ2;θ1,θ2 ) =1

2n1 ⋅∇φ1 ds

γ1h

∫ + n2 ⋅∇φ2 dsγ2

h

∫$

%&&

'

())

2

+1

2φ1 −E(φ2 )

γ1h

2+ φ2 −E(φ1)

γ2h

2( )+δ θ1 γ1

h

2+ θ2 γ2

h

2( )

Optimization reformulation: (Pattern #1)

φ1 = φ2 on γ

n1 ⋅∇φ1 = n2 ⋅∇φ2 on γ

control penalty

s.t. ∇φi,∇ψi( )Ω1= f ,ψi( )Ω1

+ θi,ψi γih constraints=subproblems

• Physics-motivated constraints/objectives can be defined on non-coincident interfaces!• Can avoid complex and/or expensive mesh operations at every time step.

Advantages

Data transfer between non-coincident interfaces remains a challenge€

Ω1h Ω2

h

€

Γ1h

€

Γ2h

γ1h

γ2h

Virtual Neumann controls


HNMCoupling

HNMSolving

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Linearconsistency

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Theorem

Assume that the discrete interfaces have matching boundaries. Then the optimization-based mesh tying formulation recovers exactly globally linear solutions ϕ* of the model problem. γ1

h γ2h

γ1h (a) = γ2

h (a)

γ1h (b) = γ2

h (b)

2D

gaps

overlaps

gaps

overlaps

P. Kuberry, P. Bochev, K. Peterson, An optimization-based approach for elliptic problems with interfaces. SISC., 2017P.Bochev, P. Kuberry and K. Peterson, A virtual control coupling approach for problems with non-coincident discrete interfaces. Proceedings of LSSC 17

ϕ(x,y) = 3x+2y

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Additiveoperator-splitting

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Problem:


€

Q u,v( ) = Q1 u,v( ) + Q2 u,v( )

1. Component problems are easier to solve that the monolithic problem2. Can write Q as a sum of parts that are easier to solve

minimizeu1,u2 ,θ

J(u1,u2 ) =1

2u1 −u2 U

2

subject to Q1 u1, v1( )− θ, v1( )V = f , v1 ∀v1 ∈V

Q2 u2, v2( )+ θ, v2( )V = 0 ∀v2 ∈V

$%&

'&

Advantages• Component models communicate via virtual controls 𝜽 but remain independent!• Optimization problem completely equivalent to original one: no splitting error.• Leads to an alternative to traditional operator-split iterative solvers• Applicable to a wide range of multiphysics problems.

Seek u∈U s.t. Q u, v( ) = f , v ∀v ∈V, f ∈V *

U,V, H - Hilbert spacesV * - dual of V

V, H,V *{ } - Gelfand triple

Q ⋅, ⋅( ) :U ×V →R

P. Bochev and D. Ridzal, Optimization-based additive operator decomposition with applications, CAMWA, 2016.


HNMCoupling

HNMSolving

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Additiveoperator-splittingtheory

Assumptions on the “multiphysics” operator:

supv∈V

Q u, v( )v

V

≥ γ uU

and supu∈U

Q u, v( )u

U

> 0

Q u, v( ) ≤ γ uU

∀u∈U, ∀v ∈V

%

&''

(''

€

⇒ u U ≤1

γf

V *

Assumptions on the additive components:

€

Q u,v( ) = Q1 u,v( ) + Q2 u,v( )• with weakly coercive component forms:

supv∈V

Qi u, v( )v

V

≥ γ i uU

and supu∈U

Qi u, v( )u

U

> 0

Qi u, v( ) ≤ γ i uU

∀u∈U, ∀v ∈V

%

&''

(''

€

⇒ Qi u,v( ) = f ,v is well -posed

• Component problems are easier to solve that the monolithic problem

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Lagrange multiplier solution: saddle-point optimality system

€

u1 − u2, ˆ u 1 − ˆ u 2( )U

+ Q1ˆ u 1,λ1( ) + Q2

ˆ u 2,λ2( ) = 0 ∀ˆ u 1, ˆ u 2 ∈U

ˆ θ ,λ2 − λ1( )V

= 0 ∀ ˆ θ ∈V

Q1 u1,ˆ λ 1( ) + Q2 u2,

ˆ λ 2( ) + θ, ˆ λ 2 −ˆ λ 1( )

V= f , ˆ λ 1 ∀ ˆ λ 1,

ˆ λ 2 ∈V

'

(

) )

*

) )

The optimality system is well-posed problem with a unique solution (u1,u2,θ,λ1,λ2).

Moreover, if u is a solution of the original variational equation, then u=u1=u2

➪ Control penalty is not required for well-posedness of the optimality system!➪ As a result, original and reformulated problems are completely equivalent

Notable facts

Theorem

There’s no splitting error!

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Discretization

supvh∈V h

Q uh, vh( )vh

V

≥ γ uh

U and sup

uh∈Uh

Q uh, vh( )uh

U

> 0

U h ⊂U, V h ⊂VAssume is a pair of inf-sup-stable spaces for the form Q:

• The KKT optimality system is well-posed with a unique solution • Discrete reformulated and monolithic problems are equivalent:• The following quasi-optimal error estimate holds:

Theorem

This turns out to be sufficient for the well-posedness of the discrete reformulated problem

(u1h,u2

h,θ h,λ1h,λ2

h )

uh = u1h = u2

h

uih −ui U

i=1,2

∑ + λih −λi V

i=1,2

∑ ≤C infUh

vih −ui U

i=1,2

∑ + infV h

µih −λi U

i=1,2

∑$

%&&

'

())

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Solving the KKT system

U −U 0 Q1T 0

−U U 0 0 Q1T

0 0 0 −V VQ2 0 −V 0 00 Q2 V 0 0

"

#

$$$$$

%

&

'''''

u1

u2

θλ1

λ2

"

#

$$$$$

%

&

'''''

=

000f0

"

#

$$$$$

%

&

'''''

Let be the solution of the full KKT system. Then• and• The triple solves the reduced KKT system

Theorem

Q1 0 −V0 Q2 V0 0 (Q1

−1 +Q2−1)V

"

#

$$$

%

&

'''

u1

u2

θ

"

#

$$$

%

&

'''=

f0

−Q1−1f

"

#

$$$

%

&

'''

(u1, u2,θ,λ1 ,λ2 )

λ1 = λ2 = 0 u = u1 = u2

(u1, u2,θ )

Q1 +Q2( )u = f

Monolithic problem

KKT

Can we really solve this 5X larger problem faster and more efficiently than the monolithic one?

The upper triangular reduced KKT system provides a basis for an efficient iterative solver

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Additivesplittingiterativesolvers

19

Q1−1 +Q2

−1( )Vθ = −Q1−1f

Q1u1 = f +Vθ Q2u2 = −Vθ

1. Use GMRES to solve the last equation

- Application of decouples trivially into linear system solves with and .Q1−1 +Q2

−1( ) Q1 Q2

- And, by assumption are easier to invert than !Q1, Q2 Q1 +Q2( )

2. Recover the state by solving either or - Since both yield the solution of the monolithic problem!- Note that this also allows to further simplify the KKT system to the split problem

u = u1 = u2

• However, derivation of this system is not obvious at first!• The variational setting and its discretization are left to a guesswork and serendipity.• Optimization automates and formalizes the discovery of decompositions.

Q1 −V

Q2 V

"

#$$

%

&''

u1

θ

"

#$$

%

&''=

f

0

"

#$$

%

&''

In principle one could bypass optimization and split the problem via auxiliary variables

P. Bochev and D. Ridzal, Optimization-based additive operator decomposition with applications, CAMWA, 2016.P. Bochev and D. Ridzal, An optimization-based approach for the design of robust solution algorithms. SINUM 47/5, 2009.

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Application:Oseen’s equations

20

Problem:

Optimization reformulation:

−νΔu+ (b ⋅∇)u+∇p

∇⋅u

%

&'

(

)*=

f

0

%

&'(

)*

• We apply optimization-based splitting to develop efficient solvers for Oseen equations• Approach is agnostic to the type of splitting, only requires well-posed subproblems.⇒ Can be applied with any additive splitting, including the ones mentioned above.

• Scalable iterative solvers remain a challenge. Splitting is often used in these solvers:

- physics-based splitting: vector Laplacian + convection term (Hamilton et al NLA, 2010)- dimension-based splitting: 1D scalar advection-diffusion (Benzi et al, ANM 2011)- However, iterative algorithm design depends on the splitting employed.

Result from linearization of the Navier-Stokes equations (NSE) by fixed point

or a Newton-type method

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−σΔu+ (b ⋅∇)u+ 2∇p

2∇⋅u

%

&'

(

)*+

(σ −ν )Δu−∇p

−∇⋅u

%

&'

(

)*=

f

0

%

&'(

)*

Idea: split the PDE into an “Easy” Oseen + Stokes Choosing a sufficiently large splitting parameter σ ensures

that each subproblem is dominated by the Laplacian

Weak forms of the components

Q1(u, p;v, q) =σ (∇u,∇v)+ (b ⋅∇u, v)− 2(p,∇⋅v)+ 2(q,∇⋅u)

Q2 (u, p;v, q) = (ν −σ )(∇u,∇v)+ (p,∇⋅v)− (q,∇⋅u)

Each subproblem is weakly coercive on U×V: the split satisfies our assumptions

U =V = H01(Ω)× L0

2 (Ω); H = L2 (Ω) (u, p;v, q)U = (∇u,∇v)+ (p, q); u, pU

2= ∇u

0

2+ p

0

2

Monolithic problem

Q(u, p;v, q) =Q1(u, p;v, q)+Q2 (u, p;v, q)

Q(u, p;v, q) =ν (∇u,∇v)+ (b ⋅∇u, v)− (p,∇⋅v)+ (q,∇⋅u)

Q is also weakly coercive on U×V: the monolithic problem

satisfies our assumptions

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22

minimize J(u1, p1;u2, p2 ) =1

2∇u1 −∇u2 0

2+ p1 − p2 0

2( )

s.t. Q1 u1, p1;v1, q1( )− ξ, v1( )1 − r, q1( )0

= f , v1 ∀v1, q1 ∈U

Q2 u2, p2;v2, q2( )+ ξ, v2( )1 + r, q2( )0= 0 ∀v2, q2 ∈U

%&'

('

%

&

''

(

''


θ = ξ, r{ }∈ H01(Ω)× L0

2 (Ω)

Virtual distributed control

Splitting interpretation via auxiliary variables

−σΔu+ (b ⋅∇)u+ 2∇p−Δξ

2∇⋅u− r

%

&'

(

)*+

(σ −ν )Δu−∇p+Δξ

−∇⋅u+ r

%

&'

(

)*=

f

0

%

&'(

)*

As in the abstract case• Derivation of this system is not obvious at first!• The variational setting and its discretization are left to a guesswork and serendipity.• Optimization automates and formalizes the discovery of decompositions.

Auxiliary variables=

Virtual controls!

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Numericalexamples

23

0 1 2 3 4 5 6 7 80

0.5

1Velocity PDE

y

x

0 1 2 3 4 5 6 7 80

0.5

1Velocity Opt 1

x

y

0 1 2 3 4 5 6 7 80

0.5

1Velocity Opt 2

x

y

Backward facing step channel geometry and a typical mesh

0 1 2 3 4 5 6 7 80

0.5

1

Equivalence of reformulated and monolithic problems

0 1 2 3 4 5 6 7 80

0.5

1

x

Pressure PDE

y

0

0.05

0.1

0 1 2 3 4 5 6 7 80

0.5

1

x

Pressure Opt 1

y

0

0.05

0.1

0 1 2 3 4 5 6 7 80

0.5

1

x

Pressure Opt 2

y

0

0.05

0.1

uh −uih

0

2+ ph − pi

h

0

2

i=1,2

∑ =1.0×10−7Solutions of optimization-based decomposition match monolithic solution to within the GMRES tolerance set to 1.0E-08

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Numericalexamples

24

Optimization-based vs. “naïve” preconditioned monolithic solver

30

40

50

60

70

80

90

1 2 3 4 5

Itera

tions

Mesh scaling

Optimization Monolithic

Q1 +Q2( )Monolithic solver

preconditioned by : Q1

Q1 ~−σΔu+ (b ⋅∇)u+ 2∇p

2∇⋅u

%

&'

(

)*

“Easy” Oseen

ParametersGMRES tol=10-6, , .ν = 5×10−3 σ =1

Refinement level (#5 = 407K DoF)

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Numericalexamples

25

Optimization-based vs. preconditioned monolithic solver

Q1 +Q2( )Monolithic solver

preconditioned by : Q1

Q1 ~−σΔu+ (b ⋅∇)u+ 2∇p

2∇⋅u

%

&'

(

)*

“Easy” Oseen

ParametersGMRES tol=10-6, , Level 8 mesh.σ =1

0100200300400500600700800900

1000

0.1 0.01 0.001 0.0001

Itera

tions

Viscosity scaling

Optimization Monolithic These preliminary results do not explore further tuning of the optimization solver through the splitting parameter

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Part2:optimization-basedproperty-preservation

26M. D’Elia, M. Perego, K. Peterson, D. Ridzal (SNL), M. Shashkov (LANL), S.Moe (U. WA)


HNMCoupling

HNMSolving

• Abstract decomposition theory• Navier-Stokes solvers• Advection-diffusion solvers

• Local-to-Nonlocal couplings• Atomistic-to-Continuum• Interface problems

Current OBM portfolio

• Semi-Lagrangian transport of passive tracers• Volume correction (Geometric Conservation Law)• Monotone and conservative Finite Element transport • Property-preserving recovery and compression• Property-preserving data transfer (remap)

Collaborators:

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Passivetracertransport

27

Problem:

Optimization reformulation: recall that

∂tρ +∇⋅ρu = 0

∂tρq+∇⋅ρqu = 0

#$%

Critical for multiple DOE missions: sea-ice, atmosphere (climate-ACME), subsurface flow…

P : Rn →Rm

J : Rn →R

Lh (uh ) = fh a numerical scheme generating an optimally accurate targeta nonlinear operator describing the desired physical propertiesa measure of target-to-state misfit

minimizeuh∈Rn

J(uhT −uh ) subject to P(uh ) ≥ 0 OBM separates property preservation

from accuracy considerations

• OBM is a divide and conquer alternative to direct property-preserving methods, • A property-preserving OBM formulation for has 3 key ingredients:L(u) = f

ρ

q

u

- density- tracer mixing ratio- velocity

∂q

∂t+u ⋅∇q = 0 where:⇒


HNMCoupling

HNMSolving

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BuildingtheOBMformulation

28

Conservation of total mass

P(ρ, q) =

ρ dVΩ

∫ −M = 0

ρqdVΩ

∫ −Q = 0

ρimin ≤ ρi ≤ ρi

max

qimin ≤ qi ≤ qi

max

%

&

''''''

(

)

******

3. A constraint operator

1. A numerical scheme generating an optimally accurate target:

Physical local bounds for ρ Preservation of constant tracers, i.e., “compatibility”.

Preservation of linear correlations between:

2. A measure of target-to-state misfit:

q1(x) = aq2 (x)+ bConservation of total tracer

• Semi-Lagrangian spectral element transport scheme: SL-SE• Semi-Lagrangian finite volume transport scheme: SL-FV

J uhT −uh( ) = 1

2uh

T −uh ℓ2

2

Physical local bounds for q

• Semi-Lagrangian schemes are the method of choice in many DOE applications because they allow much larger time steps than the CFL-limited Explicit Eulerian methods.

• Considering both SL-SE and SL-FV allows us to demonstrate that enforcement of properties as constraints is impervious to the underlying discretization type.

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29

Why SL + Spectral Elements?

cubed-sphere mesh

Dennis J, Edwards J, Evans K, Guba O, Lauritzen P, Mirin A, St.-Cyr A, Taylor M, Worley P. 2012. CAM-SE: A scalable spectral element dynamical core for the Community Atmosphere Model. IJHPCA. 26:74-89.

Afewwordsaboutthetargets

The ACME atmospheric dynamical core code HOMME (High Order Modeling Environment) uses SL-SE for tracer transport:

• SE: Diagonal mass matrix + Spectral accuracy• SL: avoids severe CFL restrictions of high-order methods• SL+SE: Simple, efficient and accurate!

Why SL + Finite Volumes?

Cell-centered schemes are ubiquitous in DOE codes. However,• They use limiters (monotone reconstruction) to control local bounds.• Limiters: use local “worst case” scenarios to enforce the bounds.• Limiters entangle accuracy with preservation of bounds, which

obscures sources of discretization errors.

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Anodal SLSpectralElementtarget

30

∂tρ +∇⋅ρu = 0

∂tρq+∇⋅ρqu = 0

#$%

∂tρ +u ⋅∇ρ = −ρ∇⋅u

∂tq+u ⋅∇q = 0

$%&

Dρ = −ρ∇⋅u

Dq = 0

$%&

Key idea: convert PDEs into ODEs along Lagrangian particle paths

dx

dt= u(x(t), t)

Step 1: Find departure points

dx

dt= u(x(t), t) and x(tn+1) = pij

pij{ }

Arrival points = Gauss points

pij{ }

Departure points

p = x(tn+1)

p = x(tn )

Step 2: Solve the initial value problems in [tn,tn+1]:

Dρ = −ρ∇⋅u and ρ(tn ) = ρh ( !pij, tn )

Dq = 0 and q(tn ) = qh ( !pij, tn )

Step 3: Update values at arrival points ρh (p, tn+1) = ρ(tn+1) qh (p, tn+1) = q(tn+1)

Initial values = SE reconstruction at departure points

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Example:departurepointsforrotation

31

u(x(t), t) =0.5− y

0.5− x

"

#$$

%

&''

dx

dt= u(x(t), t) and x(tn+1) = pij

Solved by RK4

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Acell-centered SLFiniteVolumetarget

32

!C(!(t))

C(!(t))

!i (t)

qi (t)!!i (t)

!qi (t)

!i (t +!t)

qi (t +!t)

C(!(t +"t))d

dtρ dx

Ci

∫ = 0

d

dtρqdx

Ci

∫ = 0

Key idea: For Lagrangian volumes

Step 2: Remap Lagrangian quantities from arrival to departure grid, then:

• Reconstruct

• Reconstruct

Step 1: Find departure grid ( trace back arrival grid vertices) C(Ω(t))

ρi

qi

mi = ρi dxCi

∫

Qi = ρi qi dxCi

∫

!"#

!"#

Step 3: Update values on the Eulerian (arrival) grid (simply copy Lagrangian values!)C(Ω(t))

Dukowicz and Baumgardner (2000) JCP

mi (t +Δt) =mi (t)

Qi (t +Δt) =Qi (t)

mi (t +Δt) = mi ρi =mi

µi

Qi (t +Δt) = Qi qi =Qi

mi

Compute Lagrangian quantities on departure grid

µi = dxCi

∫

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33

Recall that OBM is a Divide and Conquer approach that separatesaccuracy from physical properties

• Optimal orders of convergence• Avoid severe CFL restrictions of explicit Eulerian schemes

Thus, the principal design goals for the target schemes are

Accuracy:

• Diagonal mass matrix • Less complex than, e.g., tent-pitching schemesSimplicity:

• They do not conserve mass and total tracer• They do not preserve local solution bounds

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Physical bounds

These properties will be enforced through optimization!

The target schemes are not burdened to satisfy physical properties

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34

We will now combine the target schemes and the physical properties together

Step 1: Run the target scheme to compute optimization target at tn+1

miT = ρi dx

Ci

∫

QiT = ρi qi dx

Ci

∫

!"#

SL-FVSL-SE

Step 2: Determine local solution bounds for the optimization variables

SL-FVSL-SE

qijmin =min

p∈Κq(pij, tn )

qijmax =max

p∈Κq(pij, tn )

Solution is constant along Lagrangianpaths ⇒ taking min/max in a select neighborhood of the departure points is sufficient to determine solution bounds:

How?

qhT (pij, tn+1) = q(tn+1)

ρhT (pij, tn+1) = ρ(tn+1)!

"#

ρijmin ≤ ρ(pij, tn+1) ≤ ρij

max

qijmin ≤ q(pij, tn+1) ≤ qij

max

!"# Qi

min ≤ !Qi ≤Qimax

mimin ≤ !mi ≤mi

max!"#

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35

Step 3: Solve two Quadratic Programs (QP) for the relevant variables

cell massminρij

ρij − ρijT( )

2

Ci

∑ subject to

ρ dxΩ

∫ = M; and ρijmin ≤ ρij ≤ ρij

max

minqij

qij − qijT( )

2

Ci

∑ subject to

qdxΩ

∫ =Q; and qijmin ≤ qij ≤ qij

max

minmi

mi − miT( )

2

Ci

∑ subject to

mi

Ci

∑ = M; and mimin ≤ mi ≤mi

max

minQi

Qi − QiT( )

2

Ci

∑ subject to

Qi

Ci

∑ =Q; and Qimin ≤ Qi ≤Qi

maxcell tracernodal tracer

nodal density SL-FVSL

-SE

Step 4: Update solution at at tn+1 and proceed to the next time step

ρh (p, tn+1) = ρ(tn+1)

qh (p, tn+1) = q(tn+1)

mi (t +Δt) = mi ρi =mi

µi

Qi (t +Δt) = Qi qi =Qi

mi

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Theoptimizationproblems

36

All four QPs have identical algebraic form:

qn+1 = argminq

qT Mq+ cT q+ c0 subject to wT q =wT q n

qmin ≤ q ≤ qmax

"#$

%$

Conservation

Local bounds

☞ Example of a “singly linearly constrained QP with simple bounds”☞ QP structure admits a fast O(N) optimization algorithm.

Theorem (Existence of optimal solutions)The feasible set of the optimization problem is non-empty. Moreover, the QP problem has a unique optimal solution.

M = φijφkl dxΩ

∫ = diag(Mij ); c = −2Mq̂; c0 = q̂T Mq̂; w→ Gauss-Lobato weights

M, w depend on the target scheme (SE or FV). For example, in the SE case:

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FastOptimizationAlgorithm

37

L(q,λ,µ1,µ2 ) = Mij qij − q̂ij( )2

node

∑ −λ wij qij − qij,n( )node

∑ − µ1,ij (qij − qijimin )

node

∑ − µ2,ij (qij − qijimax )

node

∑

qij = q̂ij +λ +µ1,ij −µ2,ij

qijimin ≤ qij ≤ qiji

max

µ1,ij ≥ 0, µ2,ij ≥ 0

µ1,ij (qij − qijimin ) = 0,

µ2,ij (qij − qijimax ) = 0

$

%

&&&

'

&&&

and wij qij − qij,n( )node

∑ = 0

The Lagrangian

The Karush-Kuhn-Tucker (KKT) conditions

Without the equality constraint the QP splits into N one-dimensional QPs with simple bounds:

Without the equality constraint the KKT conditions are fully separable and can be solved for any fixed value of λ.

qij,n+1 = argminqij

Mij qij − q̂ij( )2

subject to qijimin ≤ qij ≤ qiji

max

qij,n+1=med(qijimin, q̂ij, qiji

max )

P. Bochev, D. Ridzal, and M. Shashkov. Fast optimization-based conservative remap of scalar fields through aggregate mass transfer. Journal of Computational Physics, 246(0):37 – 57, 2013.

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FastOptimizationAlgorithm

38

qij = qij +λ; µ1,ij = 0; µ2,ij = 0 if qijimin ≤ qij +λ ≤ qiji

max

qij = qijimin; µ2,ij = 0; µ1,ij = qij − qiji −λ if qiji

min ≥ qij +λ

qij = qijimax; µ1,ij = 0; µ2,ij = qij − qij +λ if qij +λ ≥ qiji

max

$

%&&

'&&

Step 1: solve the first set of KKT conditions to find q as a function of λ

qij (λ) =med(qijimin, qij +λ, qiji

max );

Step 2: solve the single equality constraint for λ

Solve wij qij (λ)− qij,n( )node

∑ = 0

- Piecewise linear, monotonically increasing function of single scalar variable λ- Can solve to machine precision by a simple secant method - Globalization is unnecessary: λ0=0 is an excellent initial guess:

- solves the QP without the equality constraint, i.e., “almost” a solution- Locality barely violates the mass conservation constraint

qij (λ0 )

⇒ qij (λ0 )

Trivial, communication-free O(N) computation

qij (λ0 ) =med(qijimin, qij, qiji

max );

Mass-form OBR algorithmSecond, we adjust � in an outer iteration in order to satisfy

CX

i=1

�mi (�) = 0 .

When we find the �⇤ such thatPC

i=1 �mi (�⇤) = 0 holds, we will havesolved the full KKT conditions.

The functionPC

i=1 �mi (�) is a piecewise linear, monotonically increasingfunction of a single scalar variable �. Therefore, a secant method canbe e�ciently employed as the outer iteration.

0�

. . . given �p , �c , rp1 Compute �mi (�c )

median(emmini � mi , �m

Ti + �c , emmax

i � mi ) 8i .

Compute residual rc PC

i=1 �mi (�c ).

2 Set ↵ (�p � �c )/(rp � rc ). Set rp rc .

3 Set �p �c . Set �c �c � ↵rc .

In all our examples, the algorithm requires 5 secant iterations!

,D. Ridzal Feature-preserving solution transfer 22

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SL-SE:Examples

39R. J. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SINUM 33 (1996) 627–665.

80x80 bi-cubic elements; CFL=0.7

Rotational Deformational Divergent (*)

Long time accuracy

CFL 14.08

20 cycles

15.8489 25.1189 39.8107 63.095710

−4

10−3

10−2

10−1

100L1 Convergence Rates for Limiters With Deformational Flow

No LimitingQM Interp−Loose BoundsQM Interp−Tight boundsQM Reconstructionh3An SL-SE method with limiters would

typically truncate the order of convergence to 2 even for L1 errors.

We see essentially no degradation in the 3rd order L1 error rate (compared to “raw” solution convergence).

Convergence: Gaussian Hill

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SL-FV:Examples

40

Smooth Gaussian hills on a cubed sphere mesh

Again we see no degradation of accuracy due to use of optimization.

Convergence Test

t = 0 t = 2.5 t = 5

OBT⇤ Unlimitedmesh steps l

2

l1 l

2

l1

3

� 600 0.386 0.465 0.368 0.4251.5

� 1200 0.182 0.268 0.172 0.2250.75

� 2400 0.0626 0.113 0.0559 0.08430.375

� 4800 0.0167 0.0425 0.0144 0.0233

Rate 1.51 1.16 1.56 1.403 1.5 0.75 0.375

10−2

10−1

100

l2 OBT

l∞

OBT

l2 Unlim

l∞

Unlim

⇤ Optimization-based transport

April 7, 2014 14

Convergence Test

t = 0 t = 2.5 t = 5

OBT⇤ Unlimitedmesh steps l

2

l1 l

2

l1

3

� 600 0.386 0.465 0.368 0.4251.5

� 1200 0.182 0.268 0.172 0.2250.75

� 2400 0.0626 0.113 0.0559 0.08430.375

� 4800 0.0167 0.0425 0.0144 0.0233

Rate 1.51 1.16 1.56 1.403 1.5 0.75 0.375

10−2

10−1

100

l2 OBT

l∞

OBT

l2 Unlim

l∞

Unlim

⇤ Optimization-based transport

April 7, 2014 14

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SL-FV:Lineartracercorrelations

41

Linear Tracer Correlation Test

Two tracers with initial distributions linearly correlated cosine bells,q

1

has min = 0.1 and max = 1.0, q2

= �0.8q1

+ 0.9.

q1 q2

Correlation t = 2.5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

q1

q 2

April 7, 2014 15

Initial tracer distributions: two linearly correlated cosine bells

q2=-0.8q1+0.9

Optimization formulation provably preserves linear tracer correlations

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Someobservations

42

• The OBM solution is a globally optimal state that also satisfies the bounds:- By definition it is the best possible solution satisfying the bounds!

• The structure of the QPs is independent from the target scheme:- We have a fast, scalable optimization algorithm!

Cells Timesteps

FCT (sec)

Van Leer SL-FV Ratio

64x64 400 4.51 4.55 4.98 1.1128x128 810 47.60 48.35 48.78 1.0256x256 1,610 390.47 399.15 405.92 1.0512x512 3,220 5802.05 5804.66 5655.00 0.9

SL-FV timings for Leveque’s combo example.

0.2 0.4 0.6 0.8 1

0.20.4

0.60.8

10

0.20.40.60.8

1

xy

Vectorized Matlab code: wall-clock times on a 3.06GHz Intel Core Duo MacBook Pro

• The SL-FV solution provably preserves linear tracer correlations.• Cost of SL-FV essentially the same as for conventional limiters:

In the case of the SL-FV scheme we also have the following:

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Let’srevisittheSL-FVscheme

43

If we switch from RK4 to a Forward Euler, solution changes:

1.7

2.1

2.0

RK4 FE

This exposes the fact (which the more accurate RK4 hides!) that the target FV scheme does not preserve constants!

CiEXACT

CiThe departure grid approximates the true Lagrangiangrid, hence it violates the property that non-divergent

Lagrangian flows preserve volumes!Why?

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TheGeometricConservationLaw(GCL)

44

Some recent work on GCL:

Use more Lagrangian points.

Lauritzen, Nair, Ullrich, A conservative semi-Lagrangian multi-tracer transport scheme on the cubed-sphere grid, JCP 229/5 (2010)

Arbogast, Huang, A fully mass and volume conserving implementation of a characteristic method for transport problems, SISC 28 (6) (2006).

Cossette, Smolarkiewicz, Charbonneau, The Monge–Ampere trajectory correction for semi-Lagrangian schemes, JCP, (2014)

Heuristic mesh adjustment procedure:

Monge-Ampere trajectory correction

Author's personal copy

2.1. Upstream integrals

The sub-domains ak‘ over which must be integrated can have many possible shapes (Fig. 3). The practical difficulty indeveloping analytical integrals that cover all possible cases is, in general, somewhat complicated but not impossible [23].Instead the problem can be greatly simplified by converting the area-integrals into line-integrals by appropriate use ofthe Gauss–Green theorem [6].

2.1.1. Lagrangian cell boundary computation (search algorithm)Suppose the trajectories for the vertices of ak are given. Finding the location of the vertices of ak‘ basically reduces to the

computation of intersections between coordinate lines (sides of A‘) and lines of arbitrary orientation (sides of ak‘). Only threeintersection scenarios are possible when marching counter-clockwise along a side of ak‘: Intersection with a horizontal coor-dinate line (Fig. 4(a)), intersection with a vertical coordinate line (Fig. 4(b)) or intersection with a vertex of A‘ (Fig. 4(c)). Thecoordinates of the crossing are simply the location of the intersection between straight lines. Let Nh be the number of verticesof ak‘. The coordinates of the vertices of the polygon ak‘ are denoted ðxk‘;h; yk‘;hÞ; h ¼ 1; . . . ;Nh, and are numbered counter-clockwise (Fig. 5). The first subscript k refers to the kth departure cell to which ak‘ belongs, ‘ refers to the fact thatðxk‘;h; yk‘;hÞ is a vertex in the grid cell A‘ and h is the local index for the numbered vertices of ak‘.

(b)(a) (c) (d)

Fig. 2. Schematic illustrations of possible approximations to the analytical departure cell boundary (solid curved line) using different levels of refinementwith piecewise straight lines. (a) The approach used in this paper connects the four vertices of the departure cell (filled circles) with straight lines. Toimprove the approximation to the departure cell one may introduce (b) one, (c) two or (d) three Lagrangian points along the cell sides (unfilled circles) andconnect these by straight line segments to converge towards the exact departure cell boundary.

(a) (b)

(c) (d)

Fig. 3. A schematic illustration of some of the possible shapes the polygons ak‘ (shaded areas) may take depending on the location of the departure points(filled circles). The number of vertices can be (a) 3, (b) 4, (c) 5, (d) 6 and even more depending on the flow and time-step.

1404 P.H. Lauritzen et al. / Journal of Computational Physics 229 (2010) 1401–1424

2010 TODD ARBOGAST AND CHIEH-SEN HUANG

! Adjusted point to remainfixed at this stage.❞ Points adjusted simulta-neously in the directionof the characteristic.

× Points adjusted “side-ways” to the flow.

>Flow

❛❛✥✥ ✥✥❛❛✥✥

❵❵✥✥✥✥ ❛❛✥✥

❵❵✥✥ ✦✦❵❵✥✥

❵❵✥✥✥✥✦✦

☎☎☎☎

❉❉❉❉☎☎

✂✂❉❉

✄✄❉❉

☎☎✆✆❉❉

❈❈☎☎

☞☞❊❊

☎☎❊❊

! ! !!! ! !!

× ❞× ❞❞

! ! ! ×❞

!!

! ❞! ❞! ! ! × ❞

Fig. 6.1. The adjustment of the trace-back regions. At each adjustment stage, the points markedwith a solid dot remain fixed. In the first step, the points marked with a circle are simultaneouslyadjusted to obtain volume balance of the entire layer. In the second step, the points marked with across are adjusted to obtain volume balance for each element.

>

Flow ×xi,j+1/2

xi,j+1

xi,j

< >

!

!

! !!×

!! !

! !

! !×

❳❳❳❳✘✘✘ ✏✏✏

❳❳❳✘✘✘✘✥✥✥

✂✂✂❇

❇❇

☞☞☞☞▲

▲▲▲

✂✂✂▲

▲▲▲

Fig. 6.2. The adjustment of the interior midpoints. The points marked with a solid dot remainfixed, and the points marked with a cross are adjusted to obtain volume balance for each element.

converging to the correct τn. This method works well since points trace toward thesources backward in time, and so τn < tn adjusts the region to have less volume whileτn > tn gives more volume. At the conclusion of this step, we have volume balancefor the layer, but not for each element.

The second step is to adjust the interior midpoints of the layer to obtain volumebalance of each element. We choose to adjust in the direction of maximal changeto the volume (so that points move minimally). As depicted in Figure 6.2, we ad-just, e.g., xi,j+1/2 = (xi, yj+1/2) along the line through its unadjusted position andperpendicular to the line adjoining vertices xi,j and xi,j+1. On an exterior no-flowboundary where u(x) · ν(x) = 0, points should not be adjusted, since they can moveonly tangentially along the boundary, and so do not change volumes.

The trace-back layers can be of several types. Around an injection well, we mayhave a closed ring, in which case we may fix one of the midpoints and adjust the restin sequence around the ring. The volume of the entire ring is correct, so adjustmentof the last point will correct the volumes of the final two elements. We may alsohave a simple layer which meets the external boundary on two ends. Even if weare not allowed to change the end trace-back midpoints (because of no-flow boundaryconditions), again we can meet the volume constraint for each element since the entirelayer has the proper volume.

Things get more complex with multiple injection wells and/or inflow boundaries,since these can produce multiple basins of attraction. In that case, care must be takenwhen multiple trace-back adjustment layers intersect. In principle this problem canbe resolved, either through human intervention or automatically. Automatic handling

• No theoretical assurance of completion.

• Requires nontrivial solution of the nonlinear MAE • Approximate: GCL ≈ accuracy of MAE scheme

• Enforces GCL approximately.

pijcorr = pij + (t − tn )∇φ; det

∂pijcorr

∂x=1

Correct departure points according to

Our scheme violates the Geometric Conservation Law (GCL), which is critical for methods involving any kind of moving grids:

d

dtdx

Ci (t )

∫ = u ⋅nds∂Ci (t )

∫

Thomas, Lombardi, AIAA 17, 1979

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Anoptimization-basedGCL

45

Statement of the volume correction problem

C(Ω) co ∈ Rm such that c0,i

Ci

∑ = Ω and c0,i ≥ 0 ∀iGiven: a source mesh and

Find: a volume compliant mesh such that:C(Ω)

a) has the same connectivity as the source meshb) The volumes of its cells match the volumes prescribed in c0

c) Every cell is valid; or convexd) Boundary points in correspond to boundary points in

Ci ∈C(Ω)C(Ω) C(Ω)

C(Ω)

• The volume correction problem may or may not have a solution!• An important setting in which solution always exist is when

C(Ω)The source mesh is transformation of another mesh such that:C(Ω)

∀Ci ∈

C(Ω) is valid, or convex and

Ci = c0,i

In this case is a trivial solution of the volume correction problemC(Ω) =C(Ω)

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Volumecorrectionasanoptimizationproblem

46

We consider quads (simplices are actually easier). We need few things:

∀Ci ∈C(Ω), Ci =1

2(xi,1 − xi,3 )(yi,2 − yi,4 )+ (xi,2 − xi,4 )(yi,3 − yi,1)( )

Ci

Ti,r ∈Ci, Ti,r =1

2xi,ar

(yi,cr− yi,br

)− xi,br(yi,ar

− yi,cr)− xi,cr

(yi,br− yi,ar

)( )

Convexity indicator for a quad cell:

Oriented volume of a quad cell:

Ti,r ∈Ci, Ti,r = (par, pbr

, pbr) (ar, br,cr ) =

(1, 2, 4) r =1(2,3, 4) r = 2(1,3, 4) r = 3(1, 2,3) r = 4

!

"##

$##

Ci is convex, if the oriented areas of all its triangles are positive: ∀Ti,r ∈Ci, Ti,r > 0

Partitioning of a quad into triangles:

Oriented volume of a triangle

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Volumecorrectionasanoptimizationproblem

47

Optimization objective:

J0 (p, p) =1

2d(C(Ω), C(Ω))2 = p− p

22

Optimization constraints:

Mesh distance

① Volume equality

② Cell convexity

③ Boundary compliance

∀Ci ∈C(Ω), Ci = c0,i

∀Ci ∈C(Ω), ∀Ti,r ∈Ci, Ti,r > 0

∀pj ∈∂Ω, γ (pj ) = 0

Nonlinear programming problem (NLP)

p∗ = argmin J0 (p, p) subject to (1), (2), and (3){ }

M. D’Elia, D. Ridzal, K. Peterson and M. Shaskov, Optimization-based mesh correction with volume and convexity constraints. Journal of Computational Physics, Vol. 313, pp. 455-477, 2016

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AsimplifiedNLPformulation

• Boundary compliance on polygonal Ω can be subsumed in the volume constraint

Specialization to simplicial cells

• Convexity can be enforced weakly by logarithmic barrier functions

p∗ = argmin J(p) subject to Ci = c0,i ∀i{ } J(p) = J0 (p)−β log Ti,r

Ti,r∈Ci

∑Ci

∑

• This leaves only the equality volume constraint and gives the simplified NLP:

Consider a polygonal domain:

A simplex is valid if and only if Ci > 0A valid simplex is always convex

Since c0,i >0, the volume equality constraint implies !∀Ci ∈C(Ω), Ci = c0,i Ci > 0

p∗ = argmin J0 (p) subject to Ci = c0,i ∀i{ }

48

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Ascalableoptimizationalgorithm

49

Based on the inexact trust region sequential quadratic programming (SQP) method of Ridzal and Heinkenschloss. Key properties of the inexact SQP approach:

• Fast local convergence, based on its relationship to Newton’s method, • Use of ‘inexact’ solvers enables an efficient solution of very large NLP.

• Key requirement in the method: design of an efficient preconditioner.

Heinkenschloss, Ridzal, A Matrix-Free Trust-Region SQP Method for Equality Constrained Optimization, SIOPT 24/3, 2014

Given an optimization iterate pk all linear systems involved are of the form

I ∇C(pk )T

∇C(pk ) 0

"

#

$$

%

&

''

v1

vv

"

#$$

%

&''=

b1

b2

"

#$$

%

&'' C(pk ) - polynomial matrix function of coordinates

π k =I 0

0 ∇C(pk )∇C(pk )T +εI( )−1

#

$

%%

&

'

((

• ε>0 small parameter ~ 10-8h

• formed explicitly

• Inverse: smoothed aggregation AMG –Trilinos

∇C(pk )∇C(pk )T +εI

Preconditioner

∇C(pk )If is full rank, preconditioned GMRES converges in 3 iterations (Golub et al SISC 21/6, 2000)

∇C(⋅)∇C(⋅)T ≈ Δ

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Applications:Lagrangian meshmotion

50

Models the evolution of the computational mesh under a non-divergent velocity

u(p, t) =sin(π x)2 sin(2π y)cos(π t / T )

−sin(π y)2 sin(2π x)cos(π t / T )

"

#

$$

%

&

''

Deformational

R. J. LeVeque, High-resolution conservative algorithms for advection in incompressible flow, SINUM 33 (1996)

Exact Source (uncorrected) Compliant (corrected)

u(p, t) =0.5− y

0.5− x

"

#$$

%

&''Rotational

Deformational

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Improvementsinmeshgeometry

51

- exact Lagrangian mesh

- source (uncorrected)

- compliant (corrected)

Cell barycenters

Invalid cell in the source mesh:

Deformational Flow

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Improvementsinvertextrajectories

52

• The shapes of the corrected cells are close to the exact Lagrangian shapes• The barycenters of the corrected cells are very close to the exact barycenters• The trajectories of the corrected points track the exact Lagrangian trajectories very closely

Cell vertex trajectories

0.4 0.5 0.6

0.4

0.5

0.6

uncorrectedcorrectedexact

0.5 0.6 0.7

0.4

0.5

0.6

0.7

uncorrectedcorrectedexact

Rotational flow Deformational flow

We observe significant improvements in the geometry of the corrected mesh:

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ImprovementsintheFVtargetscheme

53

We add a volume correction step to our FV target scheme:

Step 1: Find departure grid ( trace back arrival grid vertices) C(Ω(t))

Step 1+: Correct the departure grid to match the cell volumes of the arrival grid

Then proceed as before. Problem solved: constant preserved!

0 0.2 0.4 0.6 0.8 11.99

2

2.01

2.02

2.03

2.04

2.05

2.06

2.07

exactuncorrectedcorrected

Uncorrected Corrected Comparison

Plots of the density at time tN = 1.5 for Forward Euler simulations with ∆t = 0.006

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Anotherexample:

54

0 0.2 0.4 0.6 0.8 1

0.8

1

1.2

1.4

1.6

1.8

2

exactuncorrectedcorrected

Initial cylindrical density distribution: rotational flow

Uncorrected Corrected Comparison

Plots of the density at time tN = 1.5 for Forward Euler simulations with ∆t = 0.006

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Conclusions

55

Traditional approaches to devise stable and accurate numerical methods are reaching a point of diminishing returns for complex applications involving multiple mathematical models, requiring diverse, heterogeneous numerical methods.

The use of optimization ideas to couple hetergoeneous numerical methods and to preserve the relevant physical properties is very promising.

However, its success depends critically on the availability of efficient and scalable optimization algorithms to solve the resulting QPs and NLPs.

We’ve presented examples where such algorithms are available and optimization leads to successful heterogenous numerical methods and property preserving schemes.

Development of fast, scalable optimization algorithms is likely to play the same role for Heterogeneous Numerical Methods as the development of fast, scalable linear solvers did in the past for conventional PDE methods.

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References

56

Coupling

Splitting

Property preservation & Data transferD. Olson, M. Luskin, A. Shapeev and P. Bochev, Analysis of an optimization-based atomistic-to-continuum coupling method for point defects. ESAIM, 2015.

D. Olson, P. Bochev, M. Luskin and A. Shapeev. An optimization-based atomistic-to-continuum coupling method. SIAM. J. Num. Anal.. Vol. 52, Issue 4, pp.2183-2204

P. Bochev and D. Ridzal, An optimization-based approach for the design of robust solution algorithms. SIAM J. Num. Anal., vol. 47, No. 5, pp.3938-3955, 2009

P. Bochev and D. Ridzal, Additive Operator Decomposition and Optimization-Based Reconnection with Applications.Proceedings of LSSC 2009, Springer Lecture Notes in Computer Science, LNCS 5910, 2010

P. Bochev and D. Ridzal, Optimization-based additive operator decomposition of weakly coercive problems with applications, CAMWA, 2016

M D’Elia, M. Perego, P. Bochev and D. Littlewood, A coupling strategy for nonlocal and local diffusion models with mixed volume constraints and boundary conditions. Computers and Mathematics with Applications, 2016

M. D’Elia, D. Ridzal, K. Peterson and M. Shaskov, Optimization-based mesh correction with volume and convexity constraints. JCP, 2016

P. Bochev, S. Moe, K. Peterson, and D. Ridzal. A Conservative, Optimization-Based Semi-Lagrangian Spectral Element Method for Passive Tracer Transport. COUPLED PROBLEMS. B. Schrefler, E. Onate and M. Papadrakakis(Eds.) 2015

K. Peterson, P. Bochev and D. Ridzal. Optimization-based conservative transport on the cubed-sphere grid. I. Lirkov, S. Margenov, and J. Wanśiewski. (Eds.): LNCS 8353, Springer Verlag, pp.205-214, 2014

P. Bochev, D. Ridzal and K. Peterson. “Optimization-based remap and transport: a divide and conquer strategy for feature-preserving discretizations”. JCP, Special Issue on Physics compatible numerical methods. Vol. 257/B, pp. 1113-1139., 2014.

P. Bochev, D. Ridzal and M. Shashkov, “Fast, optimization-based conservative remap of scalar fields through aggregate mass transfer”. JCP, 246(0):37 – 57, 2013

P. Bochev, D. Ridzal, G. Scovazi, M. Shashkov, “Constrained Optimization based data transfer – a new perspective on flux correction”. D. Kuzmin, R. Lohner, and S. Turek, eds., Flux-Corrected Transport. Principles, Algorithms and Applications, Springer Verlag, Berlin, Heidelberg, 2012, 2nd edition.

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