Bayesian Inversion and AdaptiveLow-Rank Tensor Decomposition

AGUQ Dortmund12.3 - 14.3 2018

Martin EigelManuel Marschall

Research Group 4

Nonlinear Optimization andInverse Problems

Head:Prof. Dr. Dietmar Hömberg

4We present a Bayesian inversion method with functional representations of all quantities. Theposterior density is given in terms of a polynomial basis, based on an adaptive stochasticGalerkin discretization. The sampling-free approach, using tensor trains, alleviates the curse ofdimensionality by hierarchical subspace approximations of the respective low-rank manifolds.All computations are adjusted adaptively based on a posteriori error estimators or indicators.Convergence of the posterior can be shown with respect to the discretization parameters.

Explicit and parametric Bayesian inversionapplies in parameter identification and upscalingparametrized model operator Ξ 3 y 7→ G(y) = ∑

µ∈Λ gµ(x)Pµ(y), gµ = 〈G,Pµ〉finite measurements δ ∈ RK of indirect quantity, observed by linear operatorOprior measure π0 on parameters y and noise measureN (0,Γ) on measurement error η

Statistical inverse problem: Find y ∈ Ξ from δ s.t. δ = (O ◦G)(y) + η, η ∼ N (0,Γ)Bayes’ theorem yields existence of posterior measure πδ in functional representation [1]:

dπδdπ0

(y) = Eπδ[1]−1exp(−1

2〈δ − (O ◦G)(y),Γ−1(δ − (O ◦G)(y))〉

)=∑µ∈Λ′

αµPµ(y).

Model reduction: Tensor formatshigh dimensional problem, curse of dimensionality: O

(nM

)HT/TT allows for polynomial complexity: O(r2Mn) via

U[x1, . . . , xM ] =r∑k

M∏m=1

Um[km−1, xm, km]

Features:+ separation of variables and closedness of rank r manifold,− indirect access to tensor elements by hierarchical basisCreation by tensor recovery/reconstruction [2] or cross-approximation

145mm

B{1,2,3,4,5}

B{1,2,3} B{4,5}

B{1,2} U3 U4 U5

U1 U2

Figure: dimension partition tree

n1 n2 n3 n4 n5

U1 U2 U3 U4 U5

r1 r2 r3 r4

Figure: schematical tensor train(TT) tree of order 5 with

subspaces, dimensions, ranks

Adaptive Stochastic Galerkin FEM using Tensor Trainsrandom coefficient, parametrized and represented in functional/extended tensor train format

a(x,y) =r∑k

Na∑i=1

A0[i, k0]ϕi(x)M∏m=1

nm∑µm=1

Am[km−1, µm, km]Pµm(ym), (Pµm)µm polynomial basis

weak PDE formulation obtained using tensor train operators and system solved by preconditioned ALS

A(uN , v) := E [〈uN , v〉a] = E [〈f, v〉] , ∀v ∈ VN , uN Galerkin solution

A-posteriori adaptivity in physical mesh, stochastic polynomial space and choice of rank r, see [3]

‖u− wN‖2A . estall(wN) :=

(estdet(wN) + estparam(wN) + estdisc(wN)

)2+ estdisc(wN)2

0 200 500 1,000 1,500 2,000 2,500 3,00010−13

10−10

10−7

10−4

10−1

102

micro-iterations of ALS

erro

r

err w/o precon

error w precon

Figure: Realisation of coefficient

forward

Darcy flow model

Figure: Realisation of solution

measurement

fit to model

Figure: marginal density esimtation of parameter for various measurements

1

3

degr

ee

1 2 3 4 5 6 7 8 9 101112131415161718192021222324252627282930

5

19

stochastic dimension

rank

103 104 105100

101

102

n.d.o.f:

stoc

hast

icdi

mM

p1p2p3

103 104 105

5

10

15

20

n.d.o.f:

max

rank

r

p1p2p3

103 104 105

10−3

10−2

10−1

n.d.o.f:

ener

gyer

ror

est p1error p1est p3error p3

Sampling free Bayesian inversion using Tensor Trainsexplicit forward solver yields surrogate model in TT format

GN,M(x,y) =N∑i=1

∑µ∈ΛM

U [i,µ]ϕi(x)Pµ(y)

approximation of Bayesian potential in closed TT form by exactand anisotropic interpolation

0 5 10 15 20 25 30 3510−17

10−15

10−13

10−11

10−9

10−7

10−5

10−3

adaptive refinement step

mea

nsq

uare

erro

rofB

ayes

ian

pote

ntia

l

r = 4r = 8r = 16r = 32r = 64

Figure: Rank dependency of Bayesian potential.

exponential of TT tensor by Runge-Kutta methodconvergence in Hellinger distance

dHell(πδ, πN,Mδ,L,τ ) = E (N,M, L, σ, τ ) → 0

FEM errortruncation error

interpolation nodes

tensor approx.

Runge-Kutta time step

functional representation ofposterior density

dπN,Mδ,L,τ

dπ0(y) =

∑µ∈Λ′M

Π[µ]Pµ(y)

fast access to Q.o.I., e.g. themeanposterior as new prior

Inverse scattering: Helmholtz problemConsider two random media D1(ω), D2(ω),D1(ω) ∪D2(ω) = Rd separated by interface Γ(ω).Transmission and reflection problem for plane-waveincidence and known material parameters givenby transformed Helmholtz equation

−∇ · (a(Γ(ω), ·)∇q)− κ2(Γ(ω), ·)q = 0 in Rd

boundary conditionradiation condition

Figure: triangulation of 2D interface

Figure: reflected efficiency depending on incident angle and wave

length

Outlook and referencesAdaptive functional representation combined with hierarchical model reduction:

Statistical parameter identification for shape reconstruction in scattering applicationsReconstruction of shapes of blood-cells from measured reflection intensities (with PTB)

[1] M. EIGEL, M. MARSCHALL, R. SCHNEIDER, ”Sampling-free Bayesian inversion with adaptivehierarchical tensor representations”, Inverse Problems, Feburary 2018

[2] M. EIGEL, J. NEUMANN, R. SCHNEIDER, S. WOLF, ”Non-intrusive tensor reconstruction forhigh dimensional random PDEs”, WIAS Preprint 2444

[3] M. EIGEL, M. MARSCHALL, M. PFEFFER, R. SCHNEIDER, ”Adaptive stochastic Galerkin FEMfor lognormal coefficients in hierarchical tensor representation”, in preparation

Contact

Manuel Marschall T +49 30 20372 0E manuel.marschall@wias-berlin.de

WIASMohrenstr. 3910117 Berlin

Funding Coorperation