collocation based high dimensional model representation...

Collocation based high dimensional modelrepresentation for stochastic partial differential

equations

S Adhikari1

1Swansea University, UK

ECCM 2010: IV European Conference on ComputationalMechanics, Paris

Adhikari (SU) Reduced methods for SPDE 20 May 2010 1 / 52

Nanoscale Energy Harvesting: ZnO nanowires

ZnO materials have attracted extensive attention due to theirexcellent performance in electronic, ferroelectric and piezoelectricapplications.Nano-scale ZnO is an important material for the nanoscale energyharvesting and scavenging.Investigation and understanding of the bending of ZnO nanowiresare valuable for their potential application. For example, ZnOnanowires are bend by rubbing against each other for energyscavenging.


Rubbing the right way

When ambient vibrations move a microfibre covered with zinc oxidenanowires (blue) back and forth with respect to a similar fibre that hasbeen coated with gold (orange), electrical energy is produced becauseZnO is a piezoelectric material; Nature Nanotechnology, Vol 3, March2008, pp 123.


Power shirt


Collection of ZnO

A collection of vertically grown ZnO NWs. This can be viewed as thesample space for the application of stochastic finite element method.


Stochastic nanomechanics: computational challenges

When applying the continuum stochastic mechanics at the nanoscale,the following points need to be considered:

The finite element discretization should be very small to takeaccount of nanoscale spatial resolution (large n).Due to the small length-scale, the uncertainties are relatively large(as can be seen in the SEM images (large σ).The correlation length, which governs the statistical correlationbetween two points in the space is generally very small. This isbecause the interaction between the atoms reduces significantlywith distance. This can be understood, for example, by looking atthe Lennard-Jones potential V (r) = 4ε

[( rminr

)12 −( rmin

r

)6]

(largeM).

Since the standard deviation σ, the degrees-of-freedom n and thenumber of random variables M are all expected to be large, stochasticnanomechanics is particularity challenging as the computational costcan be significantly higher.


Outline of the talk

1 IntroductionStochastic elliptic PDEs

2 Spectral decomposition in a vector spaceProjection in a finite dimensional vector-spaceProperties of the spectral functions

3 Error minimization in the Hilbert spaceThe Galerkin approachComputational method

4 Numerical illustrationZnO nanowiresResults for larger correlation lengthResults for smaller correlation length

5 Conclusions6 Acknowledgements


Introduction Stochastic elliptic PDEs

Stochastic elliptic PDE

We consider the stochastic elliptic partial differential equation(PDE)

−∇ [a(r, ω)∇u(r, ω)] = p(r); r in D (1)

with the associated boundary condition

u(r, ω) = 0; r on ∂D (2)

Here a : Rd × Ω→ R is a random field, which can be viewed as aset of random variables indexed by r ∈ Rd .We assume the random field a(r, ω) to be stationary and squareintegrable. Based on the physical problem the random field a(r, ω)can be used to model different physical quantities.



Discretized Stochastic PDE

The random process a(r, ω) can be expressed in a generalizedfourier type of series known as the Karhunen-Loeve expansion

a(r, ω) = a0(r) +∞∑

i=1

√νiξi(ω)ϕi(r) (3)

Here a0(r) is the mean function, ξi(ω) are uncorrelated standardGaussian random variables, νi and ϕi(r) are eigenvalues andeigenfunctions satisfying the integral equation∫D

Ca(r1, r2)ϕj(r1)dr1 = νjϕj(r2), ∀ j = 1,2, · · · .Truncating the series (3) upto the M-th term, substituting a(r, ω) inthe governing PDE (1) and applying the boundary conditions, thediscretized equation can be written as[

A0 +M∑

i=1

ξi(ω)Ai

]u(ω) = f (4)



Polynomial Chaos expansion

Using the Polynomial Chaos expansion, the solution (a vectorvalued function) can be expressed as

u(ω) = ui0h0 +∞∑

i1=1

ui1h1(ξi1(ω))

+∞∑

i1=1

i1∑i2=1

ui1,i2h2(ξi1(ω), ξi2(ω))

+∞∑

i1=1

i1∑i2=1

i2∑i3=1

ui1i2i3h3(ξi1(ω), ξi2(ω), ξi3(ω))

+∞∑

i1=1

i1∑i2=1

i2∑i3=1

i3∑i4=1

ui1i2i3i4 h4(ξi1(ω), ξi2(ω), ξi3(ω), ξi4(ω)) + . . . ,

Here ui1,...,ip ∈ Rn are deterministic vectors to be determined.




After the finite truncation, concisely, the polynomial chaosexpansion can be written as

u(ω) =P∑

k=1

Hk (ξ(ω))uk (5)

where Hk (ξ(ω)) are the polynomial chaoses.The value of the number of terms P depends on the number ofbasic random variables M and the order of the PC expansion r as

P =r∑

j=0

(M + j − 1)!

j!(M − 1)!(6)



Some basics of linear algebra

Definition

(Linearly independent vectors) A set of vectors φ1,φ2, . . . ,φn islinearly independent if the expression

∑nk=1 αkφk = 0 if and only if

αk = 0 for all k = 1,2, . . . ,n.

Remark

(The spanning property) Suppose φ1,φ2, . . . ,φn is a complete basisin the Hilbert space H. Then for every nonzero u ∈ H, it is possible tochoose α1, α2, . . . , αn 6= 0 uniquely such thatu = α1φ1 + α2φ2 + . . . αnφn.




We can ‘split’ the Polynomial Chaos type of expansions as

u(ω) =n∑

k=1

Hk (ξ(ω))uk +P∑

k=n+1

Hk (ξ(ω))uk (7)

According to the spanning property of a complete basis in Rn it isalways possible to project u(ω) in a finite dimensional vector basisfor any ω ∈ Ω. Therefore, in a vector polynomial chaos expansion(7), all uk for k > n must be linearly dependent.This is the motivation behind seeking a finite dimensionalexpansion.


Spectral decomposition in a vector space Projection in a finite dimensional vector-space

Projection in a finite dimensional vector-space

Theorem

There exist a finite set of functions Γk : (Rm × Ω)→ (R× Ω) and anorthonormal basis φk ∈ Rn for k = 1,2, . . . ,n such that the series

u(ω) =n∑

k=1

Γk (ξ(ω))φk (8)

converges to the exact solution of the discretized stochastic finiteelement equation (4) with probability 1.

Outline of proof: The first step is to generate a complete orthonormalbasis. We use the eigenvectors φk ∈ Rn of the matrix A0 such that

A0φk = λ0kφk ; k = 1,2, . . .n (9)




We define the matrix of eigenvalues and eigenvectors

Λ0 = diag [λ01 , λ02 , . . . , λ0n ] ∈ Rn×n;Φ = [φ1,φ2, . . . ,φn] ∈ Rn×n (10)

Eigenvalues are ordered in the ascending order: λ01 < λ02 < . . . < λ0n .Since Φ is an orthogonal matrix we have Φ−1 = ΦT so that:

ΦT A0Φ = Λ0; A0 = Φ−TΛ0Φ−1 and A−1

0 = ΦΛ−10 ΦT (11)

We also introduce the transformations

Ai = ΦT AiΦ ∈ Rn×n; i = 0,1,2, . . . ,M (12)

Note that A0 = Λ0, a diagonal matrix and

Ai = Φ−T AiΦ−1 ∈ Rn×n; i = 1,2, . . . ,M (13)




Suppose the solution of Eq. (4) is given by

u(ω) =

[A0 +

M∑i=1

ξi(ω)Ai

]−1

f (14)

Using Eqs. (10)–(13) and the orthonormality of Φ one has

u(ω) =

[Φ−TΛ0Φ

−1 +M∑

i=1

ξi(ω)Φ−T AiΦ−1

]−1

f = ΦΨ (ξ(ω))ΦT f

(15)where

Ψ (ξ(ω)) =

[Λ0 +

M∑i=1

ξi(ω)Ai

]−1

(16)

and the M-dimensional random vector

ξ(ω) = ξ1(ω), ξ2(ω), . . . , ξM(ω)T (17)




Now we separate the diagonal and off-diagonal terms of the Aimatrices as

Ai = Λi + ∆i , i = 1,2, . . . ,M (18)

Here the diagonal matrix

Λi = diag[A]

= diag[λi1 , λi2 , . . . , λin

]∈ Rn×n (19)

and ∆i = Ai − Λi is an off-diagonal only matrix.

Ψ (ξ(ω)) =

Λ0 +M∑

i=1

ξi(ω)Λi︸︷︷︸Λ(ξ(ω))

+M∑

i=1

ξi(ω)∆i︸︷︷︸∆(ξ(ω))

−1

(20)

where Λ (ξ(ω)) ∈ Rn×n is a diagonal matrix and ∆ (ξ(ω)) is anoff-diagonal only matrix.




We rewrite Eq. (20) as

Ψ (ξ(ω)) =[Λ (ξ(ω))

[In + Λ−1 (ξ(ω))∆ (ξ(ω))

]]−1(21)

The above expression can be represented using a Neumann type ofmatrix series as

Ψ (ξ(ω)) =∞∑

s=0

(−1)s[Λ−1 (ξ(ω))∆ (ξ(ω))

]sΛ−1 (ξ(ω)) (22)




Taking an arbitrary r -th element of u(ω), Eq. (15) can be rearranged tohave

ur (ω) =n∑

k=1

Φrk

n∑j=1

Ψkj (ξ(ω))(φT

j f) (23)

Defining

Γk (ξ(ω)) =n∑

j=1

Ψkj (ξ(ω))(φT

j f)

(24)

and collecting all the elements in Eq. (23) for r = 1,2, . . . ,n one has

u(ω) =n∑

k=1

Γk (ξ(ω))φk (25)


Spectral decomposition in a vector space Properties of the spectral functions

Spectral functions

DefinitionThe functions Γk (ξ(ω)) , k = 1,2, . . .n are called the spectral functionsas they are expressed in terms of the spectral properties of thecoefficient matrices of the governing discretized equation.

The main difficulty in applying this result is that each of thespectral functions Γk (ξ(ω)) contain infinite number of terms andthey are highly nonlinear functions of the random variables ξi(ω).For computational purposes, it is necessary to truncate the seriesafter certain number of terms.Different order of spectral functions can be obtained by usingtruncation in the expression of Γk (ξ(ω))



First-order spectral functions

Definition

The first-order spectral functions Γ(1)k (ξ(ω)), k = 1,2, . . . ,n are

obtained by retaining one term in the series (22).

Retaining one term in (22) we have

Ψ(1) (ξ(ω)) = Λ−1 (ξ(ω)) or Ψ(1)kj (ξ(ω)) =

δkj

λ0k +∑M

i=1 ξi(ω)λik(26)

Using the definition of the spectral function in Eq. (24), the first-orderspectral functions can be explicitly obtained as

Γ(1)k (ξ(ω)) =

n∑j=1

Ψ(1)kj (ξ(ω))

(φT

j f)

=φT

k fλ0k +

∑Mi=1 ξi(ω)λik

(27)

From this expression it is clear that Γ(1)k (ξ(ω)) are non-Gaussian

random variables even if ξi(ω) are Gaussian random variables.Adhikari (SU) Reduced methods for SPDE 20 May 2010 21 / 52


Second-order spectral functions

Definition

The second-order spectral functions Γ(2)k (ξ(ω)), k = 1,2, . . . ,n are

obtained by retaining two terms in the series (22).

Retaining two terms in (22) we have

Ψ(2) (ξ(ω)) = Λ−1 (ξ(ω))− Λ−1 (ξ(ω))∆ (ξ(ω))Λ−1 (ξ(ω)) (28)

Using the definition of the spectral function in Eq. (24), thesecond-order spectral functions can be obtained in closed-form as

Γ(2)k (ξ(ω)) =

φTk f

λ0k +∑M

i=1 ξi(ω)λik

−

n∑j=1

(φT

j f)∑M

i=1 ξi(ω)∆ikj(λ0k +

∑Mi=1 ξi(ω)λik

)(λ0j +

∑Mi=1 ξi(ω)λij

) (29)



Analysis of spectral functions

The spectral basis functions are not simple polynomials, but ratioof polynomials in ξ(ω).

We now look into the functional nature of the solution u(ω) in terms ofthe random variables ξi(ω).

Theorem

If all Ai ∈ Rn×n are matrices of rank n, then the elements of u(ω) arethe ratio of polynomials of the form

p(n−1)(ξ1(ω), ξ2(ω), . . . , ξM(ω))

p(n)(ξ1(ω), ξ2(ω), . . . , ξM(ω))(30)

where p(n)(ξ1(ω), ξ2(ω), . . . , ξM(ω)) is an n-th order completemultivariate polynomial of variables ξ1(ω), ξ2(ω), . . . , ξM(ω).




Suppose we denote

A(ω) =

[A0 +

M∑i=1

ξi(ω)Ai

]∈ Rn×n (31)

so thatu(ω) = A−1(ω)f (32)

From the definition of the matrix inverse we have

A−1 =Adj(A)

det (A)=

CTa

det (A)(33)

where Ca is the matrix of cofactors. The determinant of A contains amaximum of n number of products of Akj and their linear combinations.Note from Eq. (31) that

Akj(ω) = A0kj +M∑

i=1

ξi(ω)Aikj (34)




Since all the matrices are of full rank, the determinant contains amaximum of n number of products of linear combination ofrandom variables in Eq. (34). On the other hand, each entries ofthe matrix of cofactors, contains a maximum of (n − 1) number ofproducts of linear combination of random variables in Eq. (34).From Eqs. (32) and (33) it follows that

u(ω) =CT

a fdet (A)

(35)

Therefore, the numerator of each element of the solution vectorcontains linear combinations of the elements of the cofactormatrix, which are complete polynomials of order (n − 1).The result derived in this theorem is important because thesolution methods proposed for stochastic finite element analysisessentially aim to approximate the ratio of the polynomials given inEq. (30).




Theorem

The linear combination of the spectral functions has the samefunctional form in (ξ1(ω), ξ2(ω), . . . , ξM(ω)) as the elements of thesolution vector, that is,

ur (ω) ≡ p(n−1)r (ξ1(ω), ξ2(ω), . . . , ξM(ω))

p(n)r (ξ1(ω), ξ2(ω), . . . , ξM(ω))

, ∀r = 1,2, . . . ,n (36)

When first-order spectral functions (27) are considered, we have

u(1)r (ω) =

n∑k=1

Γ(1)k (ξ(ω))φrk =

n∑k=1

φTk f

λ0k +∑M

i=1 ξi(ω)λik

φrk (37)

All (λ0k +∑M

i=1 ξi(ω)λik ) are different for different k because it isassumed that all eigenvalues λ0k are distinct.




Carrying out the above summation one has n number of products of(λ0k +

∑Mi=1 ξi(ω)λik ) in the denominator and n sums of (n− 1) number

of products of (λ0k +∑M

i=1 ξi(ω)λik ) in the numerator, that is,

u(1)r (ω) =

∑nk=1(φT

k f)φrk∏n−1

j=16=k

(λ0j +

∑Mi=1 ξi(ω)λij

)∏n−1

k=1

(λ0j +

∑Mi=1 ξi(ω)λij

) (38)


Error minimization in the Hilbert space The Galerkin approach

The Galerkin approach

There exist a set of finite functions Γk : (Rm × Ω)→ (R× Ω), constantsck ∈ R and orthonormal vectors φk ∈ Rn for k = 1,2, . . . ,n such thatthe series

u(ω) =n∑

k=1

ck Γk (ξ(ω))φk (39)

converges to the exact solution of the discretized stochastic finiteelement equation (4) in the mean-square sense provided the vectorc = c1, c2, . . . , cnT satisfies the n × n algebraic equations S c = bwith

Sjk =M∑

i=0

Aijk Dijk ; ∀ j , k = 1,2, . . . ,n; Aijk = φTj Aiφk , (40)

Dijk = E[ξi(ω)Γj(ξ(ω))Γk (ξ(ω))

]and bj = E

[Γj(ξ(ω))

] (φT

j f).

(41)




The error vector can be obtained as

ε(ω) =

(M∑

i=0

Aiξi(ω)

)(n∑

k=1

ck Γk (ξ(ω))φk

)− f ∈ Rn (42)

The solution is viewed as a projection where

Γk (ξ(ω))φk

∈ Rn

are the basis functions and ck are the unknown constants to bedetermined.We wish to obtain the coefficients ck such that the error normχ2 = 〈ε(ω), ε(ω)〉 is minimum. This can be achieved using theGalerkin approach so that the error is made orthogonal to thebasis functions, that is, mathematically

ε(ω)⊥(

Γj(ξ(ω))φj

)or

⟨Γj(ξ(ω))φj , ε(ω)

⟩= 0 ∀ j = 1,2, . . . ,n

(43)




Imposing the orthogonality condition and using the expression ofthe error one has

E

[Γj(ξ(ω))φT

j

(M∑

i=0

Aiξi(ω)

)(n∑

k=1

ck Γk (ξ(ω))φk

)− Γj(ξ(ω))φT

j f

]= 0,∀j

(44)Interchanging the E [•] and summation operations, this can besimplified to

n∑k=1

(M∑

i=0

(φT

j Aiφk

)E[ξi(ω)Γj(ξ(ω))Γk (ξ(ω))

])ck =

E[Γj(ξ(ω))

] (φT

j f)

(45)

orn∑

k=1

(M∑

i=0

Aijk Dijk

)ck = bj (46)


Error minimization in the Hilbert space Computational method

Computational method

The mean vector can be obtained as

u = E [u(ω)] =n∑

k=1

ckE[Γk (ξ(ω))

]φk (47)

The covariance of the solution vector can be expressed as

Σu = E[(u(ω)− u) (u(ω)− u)T

]=

n∑k=1

n∑j=1

ckcjΣΓkjφkφTj (48)

where the elements of the covariance matrix of the spectralfunctions are given by

ΣΓkj = E[(

Γk (ξ(ω))− E[Γk (ξ(ω))

])(Γj(ξ(ω))− E

[Γj(ξ(ω))

])](49)



Summary of the computational method

1 Solve the eigenvalue problem associated with the mean matrix A0to generate the orthonormal basis vectors: A0Φ = Λ0Φ

2 Select a number of samples, say Nsamp. Generate the samples ofbasic random variables ξi(ω), i = 1,2, . . . ,M.

3 Calculate the spectral basis functions (for example, first-order):

Γk (ξ(ω)) =φT

k fλ0k

+∑M

i=1 ξi (ω)λik

4 Obtain the coefficient vector: c = S−1b ∈ Rn, where b = f Γ,S = Λ0 D0 +

∑Mi=1 Ai Di and

Di = E[Γ(ω)ξi(ω)ΓT (ω)

], ∀ i = 0,1,2, . . . ,M

5 Obtain the samples of the response from the spectral series:u(ω) =

∑nk=1 ck Γk (ξ(ω))φk



Computational complexity

The spectral functions Γk (ξ(ω)) are highly non-Gaussian in natureand do not in general enjoy any orthogonality properties like theHermite polynomials or any other orthogonal polynomials withrespect to the underlying probability measure.The coefficient matrix S and the vector b should be obtainednumerically using the Monte Carlo simulation or other numericalintegration technique.The simulated spectral functions can also be ‘recycled’ to obtainthe statistics and probability density function (pdf) of the solution.The main computational cost of the proposed method depends on(a) the solution of the matrix eigenvalue problem, (b) thegeneration of the coefficient matrices Di , and (c) the calculation ofthe coefficient vector by solving linear matrix equation.



Computational complexity

Both the linear matrix algebraic and the matrix eigenvalue problemscales in O(n3) in the worse case.The calculation of the coefficient matrices scales linearly with Mand n(n + 1)/2 with n. Therefore, this cost scales withO((M + 1) n(n + 1)/2). The overall cost is2O(n3) + O((M + 1) n(n + 1)/2).For large M and n, asymptotically the computational costbecomes Cs = O(Mn2) + O(n3).The important point to note here that the proposed approachscales linearly with the number of random variables M.For comparison, in the classical PC expansion one needs to solvea matrix equation of dimension Pn, which in the worse casescales with (O(Pn)3). Since P M, we haveO(P3n3) O(Mn2) + O(n3).


Numerical illustration ZnO nanowires

Collection of ZnO: Close up

Uncertainties in ZnO NWs in the close up view. The uncertainparameter include geometric parameters such as the length and thecross sectional area along the length, boundary condition and materialproperties.



ZnO nanowires

For the future nano energy scavenging devices several thousandsof ZnO NWs will be used simultaneously. This gives a naturalframework for the application of stochastic finite element methoddue a large ‘sample space’.ZnO NWs have the nano piezoelastic property so that the electriccharge generated is a function of the deformation.It is therefore vitally important to look into the ensemble behaviorof the deformation of ZnO NW for the reliable estimate ofmechanical deformation and consequently the charge generation.For the nano-scale application this is especially crucial as themargin of error is very small. Here we study the deformation of acantilevered ZnO NW with stochastic properties under the AtomicForce Microscope (AFM) tip.



ZnO nanowires

(a) The SEM image of a collection of ZnO NWshowing hexagonal cross sectional area.

(b) The atomic structure of the crosssection of a ZnO NW (the red is O2

and the grey is Zn atom)



ZnO nanowires

(c) The atomistic model of a ZnO NWgrown from a ZnO crystal in the (0, 0, 0, 1)direction.

(d) The continuum idealization of a can-tilevered ZnO NW under an AFM tip.



Problem details

We study the deflection of ZnO NW under the AFM tip consideringstochastically varying bending modulus. The variability of thedeflection is particularly important as the harvested energy fromthe bending depends on it.We assume that the bending modulus of the ZnO NW is ahomogeneous stationary Gaussian random field of the form

EI(x , ω) = EI0(1 + a(x , ω)) (50)

where x is the coordinate along the length of ZnO NW, EI0 is theestimate of the mean bending modulus, a(x , ω) is a zero meanstationary Gaussian random field.The autocorrelation function of this random field is assumed to be

Ca(x1, x2) = σ2ae−(|x1−x2|)/µa (51)

where µa is the correlation length and σa is the standard deviation.



Problem details

We consider a long nanowire where the continuum model hasbeen validated.We use the baseline parameters for the ZnO NW from Gao andWang (Nano Letters 7 (8) (2007), 2499–2505) as the lengthL = 600nm, diameter d = 50nm and the lateral point force at thetip fT = 80nN.Using these data, the baseline deflection can be obtained asδ0 = 145nm. We normalize our results with this baseline value forconvenience.Two correlation lengths are considered in the numerical studies:µa = L/3 and µa = L/10.The number of terms M in the KL expansion becomes 24 and 67(95% capture).The nanowire is divided into 50 beam elements of equal length.The number of degrees of freedom of the model n = 100(standard beam element).


Numerical illustration Results for larger correlation length

Moments: larger correlation length

(e) Mean of the normalized deflection. (f) Standard deviation of the normalizeddeflection.

Figure: The number of random variable used: M = 24. The number ofdegrees of freedom: n = 100.



Error in moments: larger correlation length

Statistics Methods σa = 0.05 σa = 0.10 σa = 0.15 σa = 0.20Mean 1st order

Galerkin0.1027 0.4240 1.0104 1.9749

2nd orderGalerkin

0.0003 0.0045 0.0283 0.1321

Standard 1st orderGalerkin

1.8693 3.0517 5.2490 11.3447

deviation 2nd orderGalerkin

0.2201 1.0425 2.7690 8.2712

Percentage error in the mean and standard deviation of the deflectionof the ZnO NW under the AFM tip when correlation length is µa = L/3.For n = 100 and M = 24, if the second-order PC was used, one wouldneed to solve a linear system of equation of size 32400. The resultsshown here are obtained by solving a linear system of equation of size100 using the proposed Galerkin approach.



Pdf: larger correlation length

(a) Probability density function for σa =0.05.

(b) Probability density function for σa =0.1.

The probability density function of the normalized deflection δ/δ0 of theZnO NW under the AFM tip (δ0 = 145nm).



Pdf: larger correlation length

(c) Probability density function for σa =0.15.

(d) Probability density function for σa =0.2.



Numerical illustration Results for smaller correlation length

Moments: smaller correlation length

(e) Mean of the normalized deflection. (f) Standard deviation of the normalizeddeflection.

Figure: The number of random variable used: M = 67. The number ofdegrees of freedom: n = 100.



Error in moments: smaller correlation length

Statistics Methods σa = 0.05 σa = 0.10 σa = 0.15 σa = 0.20Mean 1st order

Galerkin0.1761 0.7206 1.6829 3.1794

2nd orderGalerkin

0.0007 0.0113 0.0642 0.6738

Standard 1st orderGalerkin

3.9543 5.9581 9.0305 14.6568

deviation 2nd orderGalerkin

0.3222 1.8425 4.6781 8.9037

Percentage error in the mean and standard deviation of the deflectionof the ZnO NW under the AFM tip when correlation length is µa = L/3.For n = 100 and M = 67, if the second-order PC was used, one wouldneed to solve a linear system of equation of size 234,500. The resultsshown here are obtained by solving a linear system of equation of size100 using the proposed Galerkin approach.



Pdf: smaller correlation length

(a) Probability density function for σa =0.05.

(b) Probability density function for σa =0.1.




Pdf: smaller correlation length

(c) Probability density function for σa =0.15.

(d) Probability density function for σa =0.2.



Conclusions

Conclusions

1 We consider discretised stochastic elliptic partial differentialequations.

2 The solution is projected into a finite dimensional completeorthonormal vector basis and the associated coefficient functionsare obtained.

3 The coefficient functions, called as the spectral functions, areexpressed in terms of the spectral properties of the systemmatrices.

4 If n is the size of the discretized matrices and M is the number ofrandom variables, then the computational complexity grows inO(Mn2) + O(n3) for large M and n in the worse case.

5 We consider a problem with 24 and 67 random variables andn = 100 degrees of freedom. A second-order PC would requirethe solution of equations of dimension 32,400 and 234,500respectively. In comparison, the proposed Galerkin approachrequires the solution of algebraic equations of dimension n only.


Conclusions

Conclusions

(e) Basic building blocks. (f) Possible ‘solution’.

An analogy of PC based solution.Adhikari (SU) Reduced methods for SPDE 20 May 2010 50 / 52

Conclusions

Conclusions

(g) Basic building blocks. (h) Possible ‘solution’.

An analogy of the proposed spectral function based Galerkin solution.


Acknowledgements

Acknowledgements


collocation based high dimensional model representation...

Documents