Tensor-based algorithms for the model reduction of high dimensional problems: application to stochastic fluid problems M. Billaud Friess marie.billaud-friess@ec-nantes.fr Joint work with A. Nouy, O. Zahm CEMRACS Luminy, 2013
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions General context High dimensional problem in tensor spaces: Given b ∈ Y ′ , seek u ∈ X solution of L u = b . • L : X → Y ′ a linear and continuous isomorphism � s ||·|| X (resp. Y ) a tensor Hilbert space of dual X ′ (resp. Y ′ ). • X = a µ = 1 X µ • X (resp. Y ) is equipped with the norm || · || X (resp. || · || Y ) Typical problems: • Stochastic partial differential equations (SPDE) • Parametric partial differential equations • High dimensional algebraic systems in tensor format arising from discretization CEMRACS 2013 M. Billaud Friess (ECN) 2/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Application: Stochastic PDEs arising from fluids Problem: Find u : ( x , ξ ) ∈ Ω × Ξ → u ( x , ξ ) in X = L 2 ( Ξ , dP ξ ; V ) solution of L ( u ( · , ξ ); ξ ) = b ( ξ ) , a.s. . with uncertainties represented by m ∈ N random variables on ( Ξ , B , P ξ ) : ξ ∈ R m , and V an Hilbert space of functions on Ω ⊂ R d . Considered examples: • Reaction-Advection-Diffusion problem: non-symetric problem L = − ν △ + c ( ξ ) · ∇ + a ( ξ ) with X = L 2 ( Ξ , dP ξ ; H 1 0 (Ω)) • Oseen problem: non-symetric saddle point problem � − ν ( ξ ) △ + a ( ξ ) · ∇ � ∇ with X = L 2 ( Ξ , dP ξ ; H 1 0 (Ω)) × L 2 ( Ξ , dP ξ ; L 2 (Ω)) L = ∇· 0 Difficulty: Curse of dimensionality ❀ Model reduction • Reduced basis approaches [Rozza] • Low rank tensor approximation (Proper Generalized Decomposition) [Nouy] CEMRACS 2013 M. Billaud Friess (ECN) 3/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Low rank approximation ❶ Approximation in tensor subset for u ∈ X ≈ � u ∈ S X ⊂ X Rank- r canonical tensors: � � ||·|| X � r � s � s φ µ i ; φ µ R r ( X ) = i ∈ X µ with X = a X µ i = 1 µ = 1 µ = 1 Other: Tucker tensors, Tensor train tensors, Hierarchical Tucker tensors [Khoromskij] ❷ Best approximation in S X � u ∈ Π S X ( u ) = arg min v ∈ S X || v − u || u ∈ arg min � v ∈ S X || L v − b || ∗ ❀ ❸ Progressive constructions of approximations with Greedy approach [Temlyakov] Limitation of the classical approach: × Bad convergence rate for usual norm || · || ∗ (ex.: || · || 2 for non symmetric operator L ) × Weakly coercive problems CEMRACS 2013 M. Billaud Friess (ECN) 4/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Low rank approximation ❶ Approximation in tensor subset for u ∈ X ≈ � u ∈ S X ⊂ X Rank- r canonical tensors: � � � r with X = L 2 ( Ξ , dP ξ ) ⊗ V = S ⊗ V R r ( X ) = φ i ⊗ ψ i ; φ i ∈ V , ψ i ∈ S i = 1 ❀ Deterministic/Stochastic separation s = 2 Other: Tucker tensors, Tensor train tensors, Hierarchical Tucker tensors [Khoromskij] ❷ Best approximation in S X � u ∈ Π S X ( u ) = arg min v ∈ S X || v − u || u ∈ arg min � v ∈ S X || L v − b || ∗ ❀ ❸ Progressive constructions of approximations with Greedy approach [Temlyakov] Limitation of the classical approach: × Bad convergence rate for usual norm || · || ∗ (ex.: || · || 2 for non symmetric operator L ) × Weakly coercive problems CEMRACS 2013 M. Billaud Friess (ECN) 4/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Fig. Reaction-diffusion-advection problem : comparison of convergence error for || · || ∗ = || · || 2 for a R 20 approximation 10 0 Reference Approximation 10 − 1 u || 2 / || u || 2 10 − 2 || u − � 10 − 3 10 − 4 10 − 5 0 2 4 6 8 10 12 14 16 18 20 r CEMRACS 2013 M. Billaud Friess (ECN) 5/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Main goal of the talk Goal: Present an approximation strategy to solve high dimensional PDEs (ex.: stochastic) in tensor subsets relying on best approximation problem formulated using ideal norms. 1 Ideal algorithm (IA) 2 Perturbed ideal algorithm (PA) 3 Ad-Re-Di problem 4 Oseen problem 5 Conclusions CEMRACS 2013 M. Billaud Friess (ECN) 6/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Outline 1 Ideal algorithm (IA) 2 Perturbed ideal algorithm (PA) 3 Ad-Re-Di problem 4 Oseen problem 5 Conclusions CEMRACS 2013 M. Billaud Friess (ECN) 7/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Ideal norm Problem: Given b ∈ Y ′ find the solution u of L u = b • L : X → Y ′ linear operator of adjoint L ∗ : Y → X ′ , b ∈ Y ′ • Riesz operators R X : X → X ′ (resp. R Y : Y → Y ′ ) ∀ u , w ∈ X � u , w � X = � u , R X w � X , X ′ = � R X u , w � X ′ , X = � R X u , R X u � X ′ • L is continuous � Lv , w � Y ′ , Y � v � X � w � Y = β > 0 . sup sup v ∈ X w ∈ Y • L is weakly coercive � Lv , w � Y ′ , Y � v � X � w � Y = α > 0 . v ∈ X sup inf w ∈ Y • We have the stability condition for L α || Lu || Y ′ ≤ || u || X ≤ β || Lu || Y ′ ➥ Under these assumptions L is an isomorphism [Ern] . CEMRACS 2013 M. Billaud Friess (ECN) 8/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions How to choose the norms || · || X and || · || Y ′ ? [Cohen,Dahmen] || · || X = || L · || Y ′ ⇔ || · || X ′ = || L ∗ · || Y • This choice leads to a problem ideally conditioned i.e. α = β = 1 . • Possibility to choose a priori and arbitrary || · || X ❀ "Goal oriented approximations" Interpretation: Such a choice implies ∀ v , w ∈ X � v , w � X = � Lv , Lw � Y ′ = � Lv , R − 1 Y Lw � Y ′ , Y = � v , R − 1 X L ∗ R − 1 Y Lw � X X L ∗ ⇔ R X = L ∗ R − 1 ⇒ I X = R − 1 X L ∗ R − 1 Y L ⇔ R Y = LR − 1 Y L Example: algebraic system • L ∈ R n × n , u , b ∈ R n • X = Y = R n • R X = I , R Y = LL ∗ • || u || Y = || L ∗ u || 2 = || L ∗ u || X CEMRACS 2013 M. Billaud Friess (ECN) 9/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Best approximation problem ❶ Best approximation problem u ≈ � u in S X ⊂ X u − b || Y ′ = min � u ∈ Π S X ( u ) = arg min v ∈ S X || v − u || X ⇔ || L � v ∈ S X || L v − b || Y ′ ❷ Equivalent problem: v ∈ S X || L v − b || Y ′ min • Non computable norm || · || Y ′ ❸ Exact gradient type algorithm We seek { u k , y k } k ≥ 0 ⊂ S X × Y given u 0 h = 0 s.t. � Y ( L u k − b )) , R − 1 y k = Π S X ( u k − R − 1 u k + 1 X L ∗ y k ) . ∈ • This ideal gradient type algorithm converges in one iteration. • R − 1 Y ( L v − b ) not affordable in practice ! • How to compute practically Π S X ? CEMRACS 2013 M. Billaud Friess (ECN) 10/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Best approximation problem ❶ Best approximation problem u ≈ � u in S X ⊂ X u − b || Y ′ = min � u ∈ Π S X ( u ) = arg min v ∈ S X || v − u || X ⇔ || L � v ∈ S X || L v − b || Y ′ ❷ Equivalent problem: v ∈ S X || R − 1 Y ( L v − b ) || Y min • Non computable norm || · || Y ′ ❸ Exact gradient type algorithm We seek { u k , y k } k ≥ 0 ⊂ S X × Y given u 0 h = 0 s.t. � Y ( L u k − b )) , R − 1 y k = Π S X ( u k − R − 1 u k + 1 X L ∗ y k ) . ∈ • This ideal gradient type algorithm converges in one iteration. • R − 1 Y ( L v − b ) not affordable in practice ! • How to compute practically Π S X ? CEMRACS 2013 M. Billaud Friess (ECN) 10/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions Outline 1 Ideal algorithm (IA) 2 Perturbed ideal algorithm (PA) 3 Ad-Re-Di problem 4 Oseen problem 5 Conclusions CEMRACS 2013 M. Billaud Friess (ECN) 11/ 37
Introduction Ideal algorithm (IA) Perturbed ideal algorithm (PA) Ad-Re-Di problem Oseen problem Conclusions First step To compute: Find y k ∈ Y s.t. y k = R − 1 Y ( L u k − b ) • Λ δ : Y → Y is a non linear mapping s.t ∀ y ∈ { L ( S X − b ); v ∈ S X } we have || Λ δ ( y ) − y || Y ′ ≤ δ || y || Y ′ , δ ∈ ( 0 , 1 ) Y L ( u k − b ) "with" a precision δ • y k is an approximation of R − 1 How ? • Preconditionned iterative solver [Powell & al.] • Greedy construction in a fixed small low-rank subset (ex.: R 1 ) [Temlyakov] CEMRACS 2013 M. Billaud Friess (ECN) 12/ 37
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