Sparse Quadrature Algorithms for Bayesian Inverse Problems

Transcription

1 Sparse Quadrature Algorithms for Bayesian Inverse Problems Claudia Schillings, Christoph Schwab Pro*Doc Retreat Disentis 2013 Numerical Analysis and Scientific Computing Disentis - 15 August, 2013 research supported by the European Research Council under FP7 Grant AdG C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

2 Outline 1 Bayesian Inversion of Parametric Operator Equations 2 Sparsity of the Forward Solution 3 Sparsity of the Posterior Density 4 Sparse Quadrature 5 Numerical Results Model Parametric Parabolic Problem 6 Summary C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

3 Bayesian Inverse Problems (Stuart 2010) Find the unknown data u X from noisy observations δ = G(u) + η, X separable Banach space G : X X the forward map Abstract Operator Equation Given u X, find q X : A(u; q) = f with A L(X, Y ), X, Y reflexive Banach spaces, a(v, w) := Y w, Av Y v X, w Y corresponding bilinear form O : X R K bounded, linear observation operator G : X R K uncertainty-to-observation map, G = O G η R K the observational noise (η N (0, Γ)) C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

4 Bayesian Inverse Problems (Stuart 2010) Find the unknown data u X from noisy observations δ = G(u) + η, X separable Banach space G : X X the forward map O : X R K bounded, linear observation operator G : X R K uncertainty-to-observation map, G = O G η R K the observational noise (η N (0, Γ)) Least squares potential Φ : X R K R Φ(u; δ) := 1 2 ( ) (δ G(u)) Γ 1 (δ G(u)) Reformulation of the forward problem with unknown stochastic input data as an infinite dimensional, parametric deterministic problem C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

5 Bayesian Inverse Problems (Stuart 2010) Parametric representation of the unknown u u = u(y) := u + y j ψ j X j J y = (y j) j J i.i.d sequence of real-valued random variables y j U[ 1, 1] u, ψ j X J finite or countably infinite index set Prior measure on the uncertain input data (U, B) = µ 0 (dy) := j J 1 2 λ 1(dy j ). ( [ 1, 1] J, ) j J B1 [ 1, 1] measurable space C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

6 (p, ɛ) Analyticity (p, ɛ) : 1 (well-posedness) For each y U, there exists a unique realization u(y) X and a unique solution q(y) X of the forward problem. This solution satisfies the a-priori estimate where U y C 0(y) L 1 (U; µ 0). y U : q(y) X C 0(y), (p, ɛ) : 2 (analyticity) There exist 0 < p < 1 and b = (b j) j J l p (J) such that for 0 < ɛ < 1, there exist C ɛ > 0 and ρ = (ρ j) j J of poly-radii ρ j > 1 such that ρ jb j 1 ɛ, j J and U y q(y) X admits an analytic continuation to the open polyellipse E ρ := j J Eρ j C J with z E ρ : q(z) X C ɛ(y). C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

7 Sparsity of the Forward Solution Theorem (Chkifa, Cohen, DeVore and Schwab) Assume that the parametric forward solution map q(y) admits a (p, ɛ)-analytic extension to the poly-ellipse E ρ C J. The Legendre series converges unconditionally, q(y) = q P νp ν (y) in L (U, µ 0 ; X ) ν F with Legendre polynomials P k (1) = 1, P k L ( 1,1) = 1, k = 0, 1,... There exists a p-summable, monotone envelope q = {q ν } ν F, i.e. q ν := sup µ ν q P ν X with C(p, q) := q l p (F) <. and monotone Λ P N F corresponding to the N largest terms of q with sup q(y) q P νp ν (y) C(p, q)n (1/p 1). X y U ν Λ P N C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

8 (p, ɛ) Analyticity of Affine Parametric Operator Families Affine Parametric Operator Families A(y) = A 0 + y j A j L(X, Y ). j J Assumption A1 There exists µ > 0 such that inf sup 0 v X 0 w Y a 0(v, w) v X w Y µ 0, inf sup 0 w Y 0 v X a 0(v, w) v X w Y µ 0 Assumption A2 There exists a constant 0 < κ < 1 b j κ < 1, where b j := A 1 0 A j L(X,Y ) j J Assumption A3 For some 0 < p < 1 b p l p (J) = j J b p j < C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

9 (p, ɛ) Analyticity of Affine Parametric Operator Families Theorem (Cohen, DeVore and Schwab 2010) Under Assumption A1 - A3, for every realization y U of the parameters, A(y) is boundedly invertible, uniformly with respect to the parameter sequence y U. For the parametric bilinear form a(y;, ) : X Y R, there holds the uniform inf-sup conditions with µ = (1 κ)µ 0, y U : inf 0 v X sup 0 w Y a(y; v, w) v X w Y µ, inf sup 0 w Y 0 v X a(y; v, w) v X w Y µ. The forward map q : U X, q := G(u) and the uncertainty-to-observation map G : U R K are globally Lipschitz and (p, ɛ)-analytic with 0 < p < 1 as in Assumption A3. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

10 Examples Stationary Elliptic Diffusion Problem A 1 (u; q) := (u q ) = f in D, q = 0 in D with X = Y = V = H 1 0(D). Time Dependent Diffusion where ι 0 denotes the time t = 0 trace, A 2 (y) := ( t + A 1 (y), ι 0 ) X = L 2 (0, T; V) H 1 (0, T; V ), Y = L 2 (0, T; V) H. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

11 Bayesian Inverse Problem Theorem (Schwab and Stuart 2011) Assume that G(u) is bounded and continuous. u= u + j J y jψ j Then µ δ (dy), the distribution of y U given δ, is absolutely continuous with respect to µ 0 (dy), ie. dµ δ dµ 0 (y) = 1 Z Θ(y) with the parametric Bayesian posterior Θ given by Θ(y) = exp ( Φ(u; δ) ) u= u + j J y jψ j, and the normalization constant Z = U Θ(y)µ 0 (dy). C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

12 Bayesian Inverse Problem Expectation of a Quantity of Interest φ : X S E µδ [φ(u)] = Z 1 exp ( Φ(u; δ) ) φ(u) µ 0 (dy) =: Z /Z u= u + j J y jψ j U with Z = exp( ( 1 y U 2 (δ G(u)) Γ 1 (δ G(u)) ) )µ 0(dy). Reformulation of the forward problem with unknown stochastic input data as an infinite dimensional, parametric deterministic problem Parametric, deterministic representation of the derivative of the posterior measure with respect to the prior µ 0 Approximation of Z and Z to compute the expectation of QoI under the posterior given data δ Efficient algorithm to approximate the conditional expectations given the data with dimension-independent rates of convergence C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

13 Sparsity of the Posterior Density Theorem (C.S. and Ch. Schwab 2013) Assume that the forward solution map U y q(y) is (p, ɛ)-analytic for some 0 < p < 1. Then the Bayesian posterior density Θ(y) is, as a function of the parameter y, likewise (p, ɛ)-analytic, with the same p and the same ɛ. N-term Approximation Results sup Θ(y) Θ P νp ν (y) N s θ P l p X m (F), s := 1 p 1. y U ν Λ P N Adaptive Smolyak quadrature algorithm with convergence rates depending only on the summability of the parametric operator C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

14 Univariate Quadrature Univariate quadrature operators of the form n k Q k (g) = w k i g(z k i ) i=0 with g : [ 1, 1] S for some Banach space S (Q k ) k 0 sequence of univariate quadrature formulas (z k j ) n k j=0 [ 1, 1] with zk j [ 1, 1], j, k and z k 0 = 0, k quadrature points w k j, 0 j n k, k N 0 quadrature weights Assumption 1 (i) (I Q k )(g k ) = 0, g k P k = span{y j : j N 0, j k} with I(g k ) = [ 1,1] g k(y)λ 1(dy) (ii) w k j > 0, 0 j n k, k N 0. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

15 Univariate Quadrature Univariate quadrature operators of the form n k Q k (g) = w k i g(z k i ) i=0 with g : [ 1, 1] S for some Banach space S (Q k ) k 0 sequence of univariate quadrature formulas (z k j ) n k j=0 [ 1, 1] with zk j [ 1, 1], j, k and z k 0 = 0, k quadrature points w k j, 0 j n k, k N 0 quadrature weights Univariate quadrature difference operator with Q 1 = 0 and z 0 0 = 0, w 0 0 = 1 j = Q j Q j 1, j 0 C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

16 Univariate Quadrature Univariate quadrature operators of the form n k Q k (g) = w k i g(z k i ) i=0 with g : [ 1, 1] S for some Banach space S (Q k ) k 0 sequence of univariate quadrature formulas (z k j ) n k j=0 [ 1, 1] with zk j [ 1, 1], j, k and z k 0 = 0, k quadrature points w k j, 0 j n k, k N 0 quadrature weights Univariate quadrature operator rewritten as telescoping sum Q k = k j=0 j with Z k = {z k j : 0 j n k } [ 1, 1] set of points corresponding to Q k C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

17 Tensorization Tensorized multivariate operators Q ν = j 1 Q ν j, ν = j 1 νj with associated set of multivariate points Z ν = j 1 Z ν j U If ν = 0 F, then νg = Q ν g = g(z 0F ) = g(0 F) If 0 F ν F, with ˆν = (ν j) j i Q ν g = Q ν i (t j 1 Qˆν j g t), i I ν and νg = νi (t j 1 ˆνj g t), i I ν, for g Z, g t is the function defined on Z N by g t(ŷ) = g(y), y = (..., y i 1, t, y i+1,...), i > 1 and y = (t, y 2,...), i = 1 C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

18 Sparse Quadrature Operator For any finite monotone set Λ F, the quadrature operator is defined by Q Λ = ν = νj ν Λ ν Λ j 1 with associated collocation grid Z Λ = ν Λ Z ν Theorem For any monotone index set Λ N F, the sparse quadrature Q ΛN is exact for any polynomial g P ΛN, i.e. it holds Q ΛN (g) = I(g), g P ΛN, with P ΛN = span{y ν : ν Λ N } and I(g) = U g(y)µ 0(dy). C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

19 Convergence Rates for Adaptive Smolyak Integration Theorem Assume that the forward solution map U y q(y) is (p, ɛ)-analytic for some 0 < p < 1. Then there exist two sequences (Λ 1 N ) N 1, (Λ 2 N ) N 1 of monotone index sets Λ 1,2 N F such that #Λ1,2 N N and I[Θ] Q Λ 1 N [Θ] C 1 N s, with s = 1/p 1, I[Θ] = Θ(y)µ0(dy) and, U I[Ψ] Q Λ 2 N [Ψ] X C 2 N s, s = 1 p 1. with s = 1/p 1, I[Ψ] = U Ψ(y)µ0(dy), C1, C 2 > 0 independent of N. C.S. and Ch. Schwab. Sparsity in Bayesian Inversion of Parametric Operator Equations, C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

20 Convergence Rates for Adaptive Smolyak Integration Sketch of proof Relating the quadrature error with the Legendre coefficients I(Θ) Q Λ(Θ) 2 γ ν θν P ν / Λ and I(Ψ) Q Λ(Ψ) X 2 γ ν ψν P X ν / Λ for any monotone set Λ F, where γ ν := j J (1 + νj)2. (γ ν θ P ν ) ν F l p m(f) and (γ ν ψ P ν X ) ν F l p m(f). sequence (Λ N ) N 1 of monotone sets Λ N F, #Λ N N, such that the Smolyak quadrature converges with order 1/p 1. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

21 Adaptive Construction of the Monotone Index Set Successive identification of the N largest contributions ν (Θ) = j 1 νj (Θ), ν F A. Chkifa, A. Cohen and Ch. Schwab. High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs, Set of reduced neighbors N (Λ) := {ν / Λ : ν e j Λ, j I ν and ν j = 0, j > j(λ) + 1} with j(λ) = max{j : ν j > 0 for some ν Λ}, I ν = {j N : ν j 0} N C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

22 Adaptive Construction of the Monotone Index Set 1: function ASG 2: Set Λ 1 = {0}, k = 1 and compute 0 (Θ). 3: Determine the set of reduced neighbors N (Λ 1 ). 4: Compute ν (Θ), ν N (Λ 1 ). 5: while ν N (Λ k ) ν(θ) > tol do 6: Select ν N (Λ k ) with largest ν and set Λ k+1 = Λ k {ν}. 7: Determine the set of reduced neighbors N (Λ k+1 ). 8: Compute ν (Θ), ν N (Λ k+1 ). 9: Set k = k : end while 11: end function T. Gerstner and M. Griebel. Dimension-adaptive tensor-product quadrature, Computing, 2003 C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

23 Numerical Experiments Model parametric parabolic problem t q(t, x) div(u(x) q(t, x)) = 100 tx (t, x) T D, q(0, x) = 0 x D, q(t, 0) = q(t, 1) = 0 t T with 64 u(x, y) = u + y j ψ j, where u = 1 and ψ j = α j χ Dj j=1 where D j = [(j 1) 1, j ], y = (yj)j=1,...,64 and αj =, ζ = 2, 3, j ζ Finite element method using continuous, piecewise linear ansatz functions in space, backward Euler scheme in time Uniform mesh with meshwidth h T = h D = 2 11 LAPACK s DPTSV routine C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

24 Numerical Experiments Find the unknown data u for given (noisy) data δ, δ = G(u) + η, Expectation of interest Z /Z Z = Z = U U exp ( Φ(u; δ) ) φ(u) u= u + 64 j=1 y jψ j µ 0 (dy) exp ( Φ(u; δ) ) u= u + 64 j=1 y jψ j µ 0 (dy) Observation operator O consists of system responses at K observation points in T D at t i = i 2 N, i = 1,..., K,T 2N K,T 1, x j = j 2 N, k = 1,..., K,D 2N K,D 1, o k (, ) = δ( t k )δ( x k ) with K = 1, N K,D = 1, N K,T = 1, K = 3, N K,D = 2, N K,T = 1, K = 9, N K,D = 2, N K,T = 2 G : X R K, with K = 1, 3, 9, φ(u) = G(u) η = (η j ) j=1,...,k iid with η j N (0, 1), η j N (0, ) and η j N (0, ) C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

25 Numerical Experiments Quadrature points Clenshaw-Curtis (CC) z k j = cos z k 0 = 0, if n k = 1 with n 0 = 1 and n k = 2 k + 1, for k 1 ( ) πj, j = 0,..., n k 1, if n k > 1 and n k 1 R-Leja sequence (RL) C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

26 Numerical Experiments Quadrature points Clenshaw-Curtis (CC) R-Leja sequence (RL) projection on [ 1, 1] of a Leja sequence for the complex unit disk initiated at i z k 0 = 0, z k 1 = 1, z k 2 = 1, if j = 0, 1, 2 and j 1 z k j = R(ẑ), with ẑ = argmax z 1 z z k l, j = 3,..., n k, if j odd, l=1 z k j = z k j 1, j = 3,..., n k, if j even, with n k = 2 k + 1, for k 0 J.-P. Calvi and M. Phung Van. On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation Journal of Approximation Theory, J.-P. Calvi and M. Phung Van. Lagrange interpolation at real projections of Leja sequences for the unit disk Proceedings of the American Mathematical Society, A. Chkifa. On the Lebesgue constant of Leja sequences for the unit disk Journal of Approximation Theory, C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

27 Leja quadrature points Proposition Let Q RL Λ denote the sparse quadrature operator for any monotone set Λ based on the univariate quadrature formulas associated with the R-Leja sequence. If the forward solution map U y q(y) is (p, ɛ)-analytic for some 0 < p < 1 and ɛ > 0, then (γ ν θ P ν ) ν F l p m(f) and (γ ν ψ P ν S ) ν F l p m(f). Furthermore, there exist two sequences (Λ RL,1 N ) N 1, (Λ RL,2 N )) N 1 of monotone index sets Λ RL,i N F such that #Λ RL,i N N, i = 1, 2, and such that, for some C 1, C 2 > 0 independent of N, with s = 1 p 1, I[Θ] Q Λ RL,1[Θ] C 1 N s, N and I[Ψ] Q Λ RL,2[Ψ[ S C 2 N s. N C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

28 Leja quadrature points Sketch of proof Univariate polynomial interpolation operator n k IRL(g) k = g ( z k ) i l k i, i=0 with g : U S, li k (y) := n k y z i i=0,i j z j z i the Lagrange polynomials. (I Q k RL)(g k ) = (I I[I k RL])(g k ) = I(g k I k RL(g k )) = 0 g k P k = span{y j : j N 0, j k}. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

29 Leja quadrature points Sketch of proof Univariate polynomial interpolation operator n k IRL(g) k = g ( z k ) i l k i, i=0 with g : U S, li k (y) := n k y z i i=0,i j z j z i the Lagrange polynomials. (I Q k RL)(g k ) = 0, g k P k Q k Q k RL RL = sup (g) S 0 g C(U;S) g L (U;S) IRL k sup (g) L (U;S) 0 g C(U;S) g L (U;S) 3(k + 1) 2 log(k + 1) C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

30 Leja quadrature points Sketch of proof Univariate polynomial interpolation operator n k IRL(g) k = g ( z k ) i l k i, i=0 with g : U S, li k (y) := n k y z i i=0,i j z j z i the Lagrange polynomials. (I Q k RL)(g k ) = 0, g k P k Q k RL 3(k + 1) 2 log(k + 1) Relating the quadrature error with the Legendre coefficients θ P ν of g C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

31 Normalization Constant Z Z, ζ=2, η j N(0,1) Z, ζ=3, η j N(0,1) Z, ζ=4, η j N(0,1) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, 1) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

32 Normalization Constant Z Z, ζ=2, η j N(0,0.5 2 ) Z, ζ=3, η j N(0,0.5 2 ) Z, ζ=4, η j N(0,0.5 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

33 Normalization Constant Z Z, ζ=2, η j N(0,0.1 2 ) Z, ζ=3, η j N(0,0.1 2 ) Z, ζ=4, η j N(0,0.1 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence CC with with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

34 Normalization Constant Z Z, ζ=2, η j N(0,1) Z, ζ=3, η j N(0,1) Z, ζ=4, η j N(0,1) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence RL with K = 1, 3, 9, η N (0, 1) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

35 Normalization Constant Z Z, ζ=2, η j N(0,0.5 2 ) Z, ζ=3, η j N(0,0.5 2 ) Z, ζ=4, η j N(0,0.5 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence RL with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

36 Normalization Constant Z Z, ζ=2, η j N(0,0.1 2 ) Z, ζ=3, η j N(0,0.1 2 ) Z, ζ=4, η j N(0,0.1 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence RL with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

37 Quantity Z Z, ζ=2, η j N(0,1) Z, ζ=3, η j N(0,1) Z, ζ=4, η j N(0,1) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, 1) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

38 Quantity Z Z, ζ=2, η j N(0,0.5 2 ) Z, ζ=3, η j N(0,0.5 2 ) Z, ζ=4, η j N(0,0.5 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

39 Quantity Z Z, ζ=2, η j N(0,0.1 2 ) Z, ζ=3, η j N(0,0.1 2 ) Z, ζ=4, η j N(0,0.1 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

40 Quantity Z Z, ζ=2, η j N(0,1) Z, ζ=3, η j N(0,1) Z, ζ=4, η j N(0,1) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence RL with K = 1, 3, 9, η N (0, 1) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

41 Quantity Z Z, ζ=2, η j N(0,0.5 2 ) Z, ζ=3, η j N(0,0.5 2 ) Z, ζ=4, η j N(0,0.5 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence RL with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

42 Quantity Z Z, ζ=2, η j N(0,0.1 2 ) Z, ζ=3, η j N(0,0.1 2 ) Z, ζ=4, η j N(0,0.1 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence RL with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

43 Numerical Experiments Model parametric parabolic problem t q(t, x) div(u(x) q(t, x)) = 100 tx (t, x) T D, q(0, x) = 0 x D, q(t, 0) = q(t, 1) = 0 t T with 128 u(x, y) = u + y j ψ j, where u = 1 and ψ j = α j χ Dj j=1 where D j = [(j 1) 1, j ], y = (yj)j=1,...,128 and αj =, ζ = 2, 3, j ζ Finite element method using continuous, piecewise linear ansatz functions in space, backward Euler scheme in time Uniform mesh with meshwidth h T = h D = 2 11 LAPACK s DPTSV routine C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

44 Normalization Constant Z (128 parameters) Z, ζ=2, η j N(0,1) Z, ζ=3, η j N(0,1) Z, ζ=4, η j N(0,1) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, 1) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

45 Normalization Constant Z (128 parameters) Z, ζ=2, η j N(0,0.5 2 ) Z, ζ=3, η j N(0,0.5 2 ) Z, ζ=4, η j N(0,0.5 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

46 Normalization Constant Z (128 parameters) Z, ζ=2, η j N(0,0.1 2 ) Z, ζ=3, η j N(0,0.1 2 ) Z, ζ=4, η j N(0,0.1 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the normalization constant Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

47 Quantity Z (128 parameters) Z, ζ=2, η j N(0,1) Z, ζ=3, η j N(0,1) Z, ζ=4, η j N(0,1) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, 1) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

48 Quantity Z (128 parameters) Z, ζ=2, η j N(0,0.5 2 ) Z, ζ=3, η j N(0,0.5 2 ) Z, ζ=4, η j N(0,0.5 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

49 Quantity Z (128 parameters) Z, ζ=2, η j N(0,0.1 2 ) Z, ζ=3, η j N(0,0.1 2 ) Z, ζ=4, η j N(0,0.1 2 ) Figure: Comparison of the estimated error and actual error. Curves computed by the reference solution of the quantity Z with respect to the cardinality of the index set Λ N based on the sequence CC with K = 1, 3, 9, η N (0, ) and with ζ = 2 (l.), ζ = 3 (m.) and ζ = 4 (r.), h T = h D = 2 11 for the reference and the adaptively computed solution. C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

50 Conclusions and Outlook New class of sparse, adaptive quadrature methods for Bayesian inverse problems for a broad class of operator equations Dimension-independent convergence rates depending only on the summability of the parametric operator Numerical confirmation of the predicted convergence rates C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

51 Conclusions and Outlook New class of sparse, adaptive quadrature methods for Bayesian inverse problems for a broad class of operator equations Dimension-independent convergence rates depending only on the summability of the parametric operator Numerical confirmation of the predicted convergence rates Gaussian priors and lognormal coefficients Adaptive control of the discretization error of the forward problem with respect to the expected significance of its contribution to the Bayesian estimate Efficient treatment of large sets of data δ C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24

52 References Calvi J-P and Phung Van M 2011 On the Lebesgue constant of Leja sequences for the unit disk and its applications to multivariate interpolation Journal of Approximation Theory Calvi J-P and Phung Van M 2012 Lagrange interpolation at real projections of Leja sequences for the unit disk Proceedings of the American Mathematical Society CHKIFA A 2013 On the Lebesgue constant of Leja sequences for the unit disk Journal of Approximation Theory Chkifa A, Cohen A and Schwab C 2013 High-dimensional adaptive sparse polynomial interpolation and applications to parametric PDEs Foundations of Computational Mathematics 1-33 Cohen A, DeVore R and Schwab C 2011 Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDEs Analysis and Applications Gerstner T and Griebel M 2003 Dimension-adaptive tensor-product quadrature Computing Hansen M and Schwab C 2013 Sparse adaptive approximation of high dimensional parametric initial value problems Vietnam J. Math Hansen M, Schillings C and Schwab C 2012 Sparse approximation algorithms for high dimensional parametric initial value problems Proceedings 5th Conf. High Performance Computing, Hanoi Schillings C and Schwab C 2013 Sparse, adaptive Smolyak algorithms for Bayesian inverse problems Inverse Problems Schillings C and Schwab C 2013 Sparsity in Bayesian inversion of parametric operator equations SAM-Report Schwab C and Stuart A M 2012 Sparse deterministic approximation of Bayesian inverse problems Inverse Problems C. Schillings (SAM) Sparsity in Bayesian Inversion Pro*Doc - August 15, / 24