High-Order Quasi Monte-Carlo Integration for Bayesian Inversion of Parametric Operator Equations Christoph Schwab Seminar f¨ ur Angewandte Mathematik ETH Z¨ urich, Switzerland Joint with J. Dick, F. Kuo and T. Le Gia (Sydney), D. Nuyens (Leuven) ICERM WS on Integration and Optimization, September 2014 ARC QE2 (J. Dick) and ARC FF (F. Kuo), ERC AdG and SNF (Ch. Schwab)
Outline • Infinite-Dimensional Parametric Operator Equations • Quasi Monte-Carlo Integration and High Order Digital Nets • Quasi Monte-Carlo Petrov-Galerkin FEM convergence rate • Bayesian Inverse Problem • Numerical Results • Conclusions • References
Infinite-Dimensional Parametric Operator Equations • Example: Linear, Affine-Parametric Operator Equation Given f ∈ Y ′ , for every y ∈ U find u ( y ) ∈ X : A ( y ) u ( y ) = f . (1) Here � ∀ y := ( y j ) j ≥ 1 ∈ U := [ − 1 / 2 , 1 / 2] N . y j A j ∈ L ( X ; Y ′ ) , A ( y ) = A 0 + (2) j ≥ 1 • Assumptions: � � � A j � p Small Fluctuations : � A j � L ( X , Y ′ ) “small” , Sparsity : ∃ 0 < p < 1 : L ( X , Y ′ ) < ∞ . j ≥ 1 j ≥ 1 (3) a ( x ) + � • Example: Karh´ unen-Loeve expansion: a ( x, y ) = ¯ j ≥ 1 y j ψ j ( x ) � X = Y = H 1 A ( y ) = −∇ x · a ( x, y ) ∇ x = −∇ x · ¯ a ( x ) ∇ x + −∇ x · ψ j ( x ) ∇ x , 0 ( D ) . � �� � � �� � j ≥ 1 A 0 A j • Goal: given G ∈ X ′ , find E [ G ( u ( · ))] with respect to y ∈ U , i.e., � I ( G ( u )) := G ( u ( y )) d y . (4) U
Infinite-Dimensional Parametric Operator Equations • Strategy: 1. Dimension Truncation : truncate (2) to s terms, 2. solve s -dimensional equation (1) by Petrov-Galerkin discretization from {X h } ⊂ X , 3. approximate s -dimensional integral using QMC integration , N − 1 1 � � � �� u h y n − 1 G , (5) s 2 N n =0 where y 0 , . . . , y N − 1 ∈ [0 , 1] s denote N points from a higher order digital net. 4. Error bounds explicit w.r. to N , h and truncation dimension s .
Infinite-Dimensional Parametric Operator Equations • Variational Formulation: a j ( · , · ) : X × Y → R via ∀ v ∈ X , w ∈ Y : a j ( v, w ) = Y � w, A j v � Y ′ , j = 0 , 1 , 2 , . . . . Assumption : The sequence { A j } j ≥ 0 satisfies: 1. the nominal operator A 0 ∈ L ( X , Y ′ ) is boundedly invertible: a 0 ( v, w ) a 0 ( v, w ) 0 � = v ∈X sup inf ≥ µ 0 > 0 , 0 � = w ∈Y sup inf ≥ µ 0 > 0 . (6) � v � X � w � Y � v � X � w � Y 0 � = w ∈Y 0 � = v ∈X 2. the fluctuation operators { A j } j ≥ 1 are small with respect to A 0 : exists 0 < κ < 2 such that � β j := � A − 1 β j ≤ κ < 2 , where 0 A j � L ( X , Y ′ ) , j = 1 , 2 , . . . . (7) j ≥ 1
Infinite-Dimensional Parametric Operator Equations Assumption is sufficient for bounded invertibility of A ( y ) uniformly w.r. to y ∈ U : for every realization y ∈ U of the parameter vector a ( y ; v, w ) := Y � w, A ( y ) v � Y ′ , (8) satisfies uniform (with respect to y ∈ U ) inf-sup conditions: with µ = (1 − κ/ 2) µ 0 > 0 , a ( y ; v, w ) a ( y ; v, w ) ∀ y ∈ U : 0 � = v ∈X sup inf ≥ µ , 0 � = w ∈Y sup inf ≥ µ . (9) � v � X � w � Y � v � X � w � Y 0 � = w ∈Y 0 � = v ∈X For every f ∈ Y ′ and for every y ∈ U , the parametric problem find u ( y ) ∈ X : a ( y ; u ( y ) , w ) = Y � w, f � Y ′ ∀ w ∈ Y (10) admits unique solution u ( y ) which satisfies the a-priori estimate � u ( y ) � X ≤ 1 µ � f � Y ′ . (11)
Infinite-Dimensional Parametric Operator Equations • Parametric regularity of solutions � ( ∂ ν y u )( y ) � X ≤ C 0 | ν | ! β ν � f � Y ′ for all ν ∈ N N ∀ y ∈ U : 0 with | ν | < ∞ , (12) where 0! := 1 , β ν := � ν j 0 A j � L ( X , X ) and | ν | = � j , with β j = � A − 1 j ≥ 1 β j ≥ 1 ν j . • Spatial regularity: scales of smoothness spaces {X t } t ≥ 0 , {Y t } t ≥ 0 , with X = X 0 ⊃ X 1 ⊃ X 2 ⊃ · · · , Y = Y 0 ⊃ Y 1 ⊃ Y 2 ⊃ · · · , and (13) X ′ = X ′ 2 ⊃ · · · , Y ′ = Y ′ 0 ⊃ X ′ 1 ⊃ X ′ 0 ⊃ Y ′ 1 ⊃ Y ′ 2 ⊃ · · · . • Uniform regularity shift (sufficient for single-level QMC-PG): for 0 < t ≤ ¯ t , f ∈ Y ′ u ( y ) = A ( y ) − 1 f ∈ X t . ∀ y ∈ U : t = ⇒ • Mixed regularity shift (necessary for multi-level QMC-PG): f ∈ Y ′ � ( ∂ ν y u )( y ) � X t ≤ C 0 | ν | ! β ν for all ν ∈ N N t = ⇒ sup t � f � Y ′ 0 with | ν | < ∞ . t y ∈ U
Infinite-Dimensional Parametric Operator Equations • Best N -term polynomial chaos approximation Under Assumption, u ( y ) : U → X can be expanded in (unconditionally convergent in L 2 ( U ; d y ) ) gpc series � ∀ y ∈ U : u ( y ) = u ν L ν ( y ) , where u ν = ( u, L ν ) L 2 ( U ; d y ) ∈ X . F 0 : | ν | < ∞} , and L ν is the ( L 2 ( U ; d y ) -normalized) tensorized Legendre polynomial Here F = { ν ∈ N N � ∀ ν ∈ F , ∀ y ∈ U : L ν ( y ) := L ν j ( y j ) ( note L 0 ≡ 1) . j ≥ 1 • [Cohen & DeVore & CS (2011), (Chkifa & Cohen & CS 2014)] Assume that β ∈ ℓ p ( N ) for some 0 < p < 1 . Then for every N ∈ N exists Λ ⊂ F such that, for q = 2 , ∞ � u − u Λ � L q ( U,d y ; X ) ≤ C ( q ) N − (1 /p − 1 /q ′ ) . #(Λ) = N and q ′ conjugate of q = 2 , ∞ , constant C > 0 independent of N and of dimension. • Proof nonconstructive. “Constructive versions” (Chkifa & Cohen & DeVore & CS 2012-2014).
Infinite-Dimensional Parametric Operator Equations • Petrov-Galerkin discretization : Let {X h } h> 0 ⊂ X and {Y h } h> 0 ⊂ Y dense families of subspaces in X and Y . t and 0 < t ′ ≤ ¯ • Approximation Properties : for 0 < t ≤ ¯ t ′ , and for 0 < h ≤ h 0 , there hold v h ∈X h � v − v h � X ≤ C t h t � v � X t , ∀ v ∈ X t : inf (14) w h ∈Y h � w − w h � Y ≤ C t ′ h t ′ � w � Y t ′ . ∀ w ∈ Y t ′ : inf ∀ 0 ≤ t ≤ ¯ � A ( y ) − 1 � L ( Y ′ t : sup t , X t ) < ∞ . (15) y ∈ U • Stability : assume that ( X h , Y h ) satisfy discr. inf-sup condition for A 0 . Then there hold uniform (with respect to y ∈ U ) discrete inf-sup conditions a ( y ; v h , w h ) ∀ y ∈ U : inf sup ≥ ¯ µ > 0 , (16) � v h � X � w h � Y 0 � = v h ∈X h 0 � = w h ∈Y h a ( y ; v h , w h ) ∀ y ∈ U : inf sup ≥ ¯ µ > 0 . (17) � v h � X � w h � Y 0 � = w h ∈Y h 0 � = v h ∈X h
Infinite-Dimensional Parametric Operator Equations • For every 0 < h ≤ h 0 and for every y ∈ U , Petrov-Galerkin approximation find u h ( y ) ∈ X h : ∀ w h ∈ Y h , a ( y ; u h ( y ) , w h ) = Y � w h , f � Y ′ (18) admits a unique solution u h ( y ) which satisfies the a-priori estimate � u h ( y ) � X ≤ 1 µ � f � Y ′ . (19) ¯ • Quasioptimality: exists a constant C > 0 such that for all y ∈ U � u ( y ) − u h ( y ) � X ≤ C 0 � = v h ∈X h � u ( y ) − v h � X . inf (20) µ ¯ t with 0 < t ≤ ¯ • Convergence Rate : Ex. C > 0 such that for every f ∈ Y ′ t as h → 0 � u ( y ) − u h ( y ) � X ≤ C h t � f � Y ′ t . (21) • Dimension-Truncation: For y ∈ U and s ∈ N , ( y 1 , y 2 , ..., y s , 0 , 0 , ... ) ∈ U , so all bounds valid for [ − 1 / 2 , 1 / 2] s uniformly w.r. to s .
Infinite-Dimensional Parametric Operator Equations • Regularity : ∃ 0 < t ′ ≤ ¯ G ( · ) ∈ X ′ t : t ′ , (22) • Adjoint Regularity : exists C t ′ > 0 such that for every y ∈ U , w ( y ) = ( A ∗ ( y )) − 1 G ∈ Y t ′ , � w ( y ) � Y t ′ ≤ C t ′ � G � X ′ ∀ y ∈ U : t ′ . (23) • Superconvergence (Aubin-Nitsche): t ′ with 0 < t ′ ≤ ¯ t with 0 < t ≤ ¯ for every f ∈ Y ′ t , for every G ( · ) ∈ X ′ t � � � ≤ C h τ � f � Y ′ � G ( u ( y )) − G ( u h ( y )) sup t � G � X ′ t ′ . (24) y ∈ U where 0 < τ := t + t ′ ≤ 2¯ t .
Dimension Truncation Assume the A j are enumerated s.t. β j := � A − 1 0 A j � X satisfy β 1 ≥ β 2 ≥ · · · ≥ β j ≥ · · · . (25) Then, for every f ∈ Y ′ , every y ∈ U and for every s ∈ N , � u ( y ) − u s ( y ) � X ≤ C � sup µ � f � Y ′ β j . (26) y ∈ U j ≥ s +1 For every G ( · ) ∈ X ′ , � � � 2 ˜ C µ � f � Y ′ � G � X ′ | I ( G ( u )) − I ( G ( u s )) | ≤ β j (27) j ≥ s +1 for ˜ C > 0 independent of s , f and G . If conditions (25) hold, for any 0 < p < 1 and for any s ∈ N holds the dimension-truncation error bound � � � � � 1 /p 1 � s − (1 /p − 1) . β p β j ≤ min 1 /p − 1 , 1 j j ≥ s +1 j ≥ 1
Quasi Monte-Carlo Integration and High Order Digital Nets • Consider general s -variate integrand F ∈ C 0 ([0 , 1] s ) . Approximate s -dimensional integral � I s ( F ) := [0 , 1] s F ( y ) d y (28) where F ( y ) = G ( u h s ( y − 1 2 )) by • N -point QMC method: an equal-weight quadrature rule N − 1 Q N,s ( F ) := 1 � F ( y n ) , (29) N n =0 with judiciously chosen points y 0 , . . . , y N − 1 ∈ [0 , 1] s .
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