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 [email protected]
WIASMohrenstr. 3910117 Berlin
Funding Coorperation
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