A Reduced-Basis Approach to the Reduction of Computations in Multiscale Models & Uncertainty Quantification Sébastien Boyaval 1 , 2 , 3 1 Univ. Paris Est , Laboratoire d’hydraulique Saint-Venant (EDF R&D – Ecole des Ponts Paristech – CETMEF), Chatou, France 2 INRIA , MICMAC team–project, Rocquencourt, France RICAM workshop on Numerical Analysis of Multiscale Problems & Stochastic Modelling, Linz, December 2011
Outline Introduction: why do we need to reduce computations ? 1 A multiscale paradigm: 2-scale homogenization 2 3 A standard use of the Reduced-Basis (RB) method A paradigm for uncertainty quantification: PDEs with 4 random input 5 RB Variance Reduction in parametrized Monte-Carlo Another multiscale paradigm: complex fluids 6 Conclusion: when shall one use RB ? 7 S. Boyaval 2 / 43 Reduced-Basis and stochastics
Taking perturbations into account is demanding Micro-Macro models (multiscale) couple algorithms ( S / P / D ) Equation [ α ]( u ) = 0 in Ω , ∂ Ω (1) α ( x ) = F ( w ( x )) ∀ x ∈ Ω (2) ∀ x ∈ Ω ( s / p / d ) equation [ x ]( w ) = 0 in Y [ x ] (3) (using mathematical frameworks: homogenization, non-equilibrium molecular dynamics. . . ) Uncertainty quantification (UQ) cares about fine statistics of u under the law generated by α A Reduced-Basis approach to computational reductions ? [2] B., SIAM MMS 7(1), 2008. [3] B., Le Bris, Maday, Nguyen, Patera, CMAME, 2009. [4] B., Lelièvre, CMS 8, 2010. S. Boyaval 3 / 43 Reduced-Basis and stochastics
Outline Introduction: why do we need to reduce computations ? 1 A multiscale paradigm: 2-scale homogenization 2 3 A standard use of the Reduced-Basis (RB) method A paradigm for uncertainty quantification: PDEs with 4 random input 5 RB Variance Reduction in parametrized Monte-Carlo Another multiscale paradigm: complex fluids 6 Conclusion: when shall one use RB ? 7 S. Boyaval 4 / 43 Reduced-Basis and stochastics
Ideal composites with 2-scale heterogeneity Scanning Electron Microscope Synthetic material (Google entry “microstructure”) (numerical modelling) S. Boyaval 5 / 43 Reduced-Basis and stochastics
Problem: fast-oscillating diffusion Numerical problem: Compute u ǫ ( x ) solution in D ⊂ R d to − div ( A ǫ ( x ) ∇ u ǫ ( x )) = f ( x ) (4) + Boundary Conditions (BC) for u ǫ on ∂ D , but u ǫ is expected to vary “fast” ! like A ǫ , where “fast” = at scale ǫ ≪ |D| Finite-Element (FE) discretization using simplices of size |D| / h d � |D| /ǫ d � � � h ≤ ǫ for D ⇒ O ≥ O ≫ 1 d.o.f required ! Discretization of a transformed problem → homogenization S. Boyaval 6 / 43 Reduced-Basis and stochastics
Sketch of homogenization procedure I. Problem transformed to a new problem: For some ( A ǫ ) ǫ> 0 , find u ⋆ ← u ǫ solution to − div ( A ⋆ ∇ u ⋆ ) = f H − A ǫ [Murat-Tartar] as ǫ → 0. +BC where A ⋆ ← Rm1: ( A ǫ ) ǫ> 0 is not given in practice ; weak hypotheses suffice, but computing oscillating functions ( z ǫ j ) ∈ H 1 ( D ) such that ǫ → 0 A ⋆ ( x ) e j · e i L 2 ( D ) − weak A ǫ ( x ) ∇ z ǫ ← − − − − − − − − j ( x ) · e i may be hard ! II. How to make computational use of the new problem ? Can we compute u ⋆ faster than FEM and evaluate the error ? Rm2: A ǫ ( x ) = A ( x , x ǫ − [ x ǫ ]) 2-scale periodic, z ǫ x , x � � j = x j + ǫ w j ǫ then computable through − div y ( A ( x , y ) · [ e j + ∇ y w j ( x , y )]) = 0 and � u ǫ − u ⋆ � L 2 (Ω) = O ( ǫ ) holds + “corrected” H 1 approximation S. Boyaval 7 / 43 Reduced-Basis and stochastics
Efficient multiscale computations The transformed problem is still a numerical challenge: A ⋆ ( x ) at many quadrature points requires to solve at many x ∈ D − div y ( A ( x , y ) · [ e j + ∇ y w j ( x , y )]) = 0 A generic difficulty in all numerical homogenization procedures (MsFEM,HMM,...: many micro computations) and a paradigm for many micro-macro models. A posteriori error control can moderate the problem by choos- ing good quadratures in D [Ohlberger ; Abdulle . . . ], yet some gain is still possible using a Reduced-Basis approximation (in- voking accurate solutions only at a few well-chosen points x n , n = 1 , . . . , N ) N � w i ( x , · ) ≃ w i , N ( x , · ) = α n ( x ) w i ( x n , · ) n = 1 S. Boyaval 8 / 43 Reduced-Basis and stochastics
Outline of the talk 1 Introduction: why do we need to reduce computations ? 2 A multiscale paradigm: 2-scale homogenization A standard use of the Reduced-Basis (RB) method 3 4 A paradigm for uncertainty quantification: PDEs with random input RB Variance Reduction in parametrized Monte-Carlo 5 Another multiscale paradigm: complex fluids 6 Conclusion: when shall one use RB ? 7 S. Boyaval 9 / 43 Reduced-Basis and stochastics
Standard (elliptic) RB framework Assume one can parametrize { w i ( x ) , x ∈ D} with µ ∈ Λ ⊂ R P . Problem: compute w i ( µ ) ∈ X = H 1 ♯ ( Y ) for many µ ∈ Λ ⊂ R P solution to a ( w i , v ; µ ) = f i ( v ; µ ) ∀ v ∈ X where � a ( u , v ; µ ( x )) = Y A ( x , y ) ∇ u ( y ) · ∇ v ( y ) dy ∀ ( u , v ) ∈ X × X � f i ( v ; µ ( x )) = − Y A ( x , y ) e i · ∇ v ( y ) dy ∀ v ∈ X , 1 ≤ i ≤ n + outputs A ⋆ i , j ( x ) at quadrature points x � s ij ( µ ) = − f j ( w i ; µ ( x )) = Y A ( x , y ) ∇ w i ( x , y ) · e j dy RB method: w ( µ ) ≃ w N ( µ ) ∈ X N spanned by “snapshots” � w ( µ N � 1 select X N = Span n ) , n = 1 , . . . , N ⊂ X (hopefully { w ( µ ) , µ ∈ Λ } is close to small-dimensional) 2 compute fast many w N ( µ ) ∈ arginf {� w ( µ ) − v � µ, X , v ∈ X N } (reduction hopefully efficient at given fixed accuracy ε ) S. Boyaval 10 / 43 Reduced-Basis and stochastics
RB principles to be put in practice A µ -parametrized elliptic pb. a ( w ( µ ) , v ; µ ) = l ( v ) , ∀ v ∈ X Given µ , find w ( µ ) ∈ X solution to − div ( A ( µ ) ∇ w ( µ )) = f + BC � w N best approx. in energy (Galerkin): � · � µ, X = a ( · , · ; µ ) X N N -linear space minimizing L ∞ : sup � w ( µ ) − w N ( µ ) � µ, X µ goal-oriented cases (like homogenization ) → RB also for adjoint eq. [Porsching . . . Maday, Patera, Turinici, Prud’homme, Rozza, Haasdonk, Ohlberger . . . ] Amounts to numerical approximation of “best N -linear space”: � � � w ( µ ) − w N ( µ ) � µ, X ≤ ǫ X N is a minimizer of inf sup µ 1 ,...,µ N µ [Maday, Patera, Turinici, Prud’homme, Buffa, Binev, Cohen, Dahmen, Devore . . . ] S. Boyaval 11 / 43 Reduced-Basis and stochastics
1) Constructive approximation of X N µ discretization + a posteriori esimator + greedy algorithm Provided a good discretization X N of X (d.o.f. N ≫ 1), � w N ( µ N n ) ≈ w ( µ N � build X N = Span n ); n = 1 , . . . , N using 1 a posteriori estimators ∆ N , N ( µ ) ≥ � w N ( µ ) − w N ( µ ) � µ, X , 2 a greedy algorithm selecting iteratively ( n = 1 . . . ) µ N n = µ n in a training sample while sup { ∆ N , N ( µ ) , µ ∈ sample } ≥ ε � � µ 1 = rand (); µ n + 1 ∈ , n = 1 , . . . , N − 1 sup ∆ N , n ( µ ) µ ∈ sample and use X N and precomputations for all queried µ to compute fast approximations w N ( µ ) that can be certified ∆ N , N ( µ ) ≤ ǫ ! Technical issue (application): cheap estimator, precomputation S. Boyaval 12 / 43 Reduced-Basis and stochastics
A case-dependent step in RB methodology ! Residual error estimation (with α A > 0 coercivity constant) � w i ( x , · ) − w i , N ( x , · ) � X ≤ ∆ N ( w i ( x , · )) = � a ( w i ( x , · ) − w i , N ( x , · ) , · ; x ) � X ′ α A ( x ) | s ij ( x ) − s ij , N ( x ) | ≤ ∆ s ij , N ( x ) = α A ( x ) ∆ N ( w i ( x , · )) ∆ N ( w j ( x , · )) controls RB error in outputs (w.r.t N -dimensional FE space X N ) � s ij , N ( x ) = A ( x , y ) ∇ w i , N ( x , y ) · e j dy Y � A ⋆ A ( x , y )[ e i + ∇ y w i , N ( x , y )] · e j dy N ( x ) i , j = Y S. Boyaval 13 / 43 Reduced-Basis and stochastics
2) Assembling fast the problem at any µ Typical parametrization for homogenization: moving inclusion with varying contrast y 2 ✻ 1 c 2 A ( x , · ) b 2 ✲ 0 0 b 1 c 1 1 y 1 Need to rebuild FE rigidity matrix at each x ∈ D ! � a ( u , v ; x ) = A ( x , y ) ∇ u ( y ) · ∇ v ( y ) dy ∀ ( u , v ) ∈ X N × X N Y S. Boyaval 14 / 43 Reduced-Basis and stochastics
Preprocessing trick: affine parametrization ∀ x ∈ D map Y ⊂ � d k = 1 Y k ( x ) partitioned into d nonoverlapping subsets Y k ( x ) to Y ⊂ � d k = 1 Y 0 k using . . . affine homeomorphisms Φ k ( x , · ) : Y 0 k → Y k ( x ) , 1 ≤ k ≤ d , affine parametrization for ( A ( x , Φ( x , · ))) x ∈D on each Y 0 k , Z � ∀ y ∈ Y 0 A ( x , Φ k ( x , y )) = A 0 ( y ) + Θ q ( x ) A q ( y ) (5) k q = 1 ⇒ matrix easily computed after variable change y ′ = Φ − 1 ( x , y ) (but new affine-equivalent mesh has spline-distorted shape functions φ i , beware aspect ratio !) � A q ( y ′ ) ∇ φ i ( y ′ ) · ∇ φ j ( y ′ ) | det ( ∇ y Φ k ( x , y ′ )) | dy ′ Θ q ( x ) Y S. Boyaval 15 / 43 Reduced-Basis and stochastics
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