Time-parallel iterative solvers for parabolic evolution equations: an inf-sup theoretic approach Iain Smears * Department of Mathematics University College London joint work with Martin Neum¨ uller, Johannes Kepler University Linz *:Part of this work was completed whilst at Inria. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No 647134 GATIPOR).
Parallel-in-time methods Motivations for parallel-in-time: • Potential for faster total time to solution than sequential approach on parallel computers, and can complement spatial parallelism. • Some problems have forward/backward structure (e.g. control problems) that cannot be solved sequentially like initial value problems. • Many methods (parareal, space-time multigrid, PFASST, MGRIT...) Nievergelt 64, Hackbusch 84, Womble 90, Horton 92, Horton Vandewalle 95, Lions Maday & Turinici 01, Bal 05, Gander & Vandewalle 07, Emmett & Minion 12, Falgout et al. 14, Gander & Neum¨ uller 16 . . . 1/39
Parallel-in-time methods Another reason to be interested in PinT • Available theory and understanding of iterative methods for nonsymmetric systems is much less developed than for symmetric problems. • Time-global formulation of evolution problems leads to nonsymmetric systems that are not “perturbations” of symmetric ones (e.g. non-diagonalizability) 1 + τ a y 1 y 0 y ′ + ay = 0 → − 1 1 + τ a y 2 0 = . . ... . . . . • Suggests understanding of PinT methods is relevant in the broader context of iterative methods for nonsymmetric systems. Can we develop a (reasonably) systematic approach to preconditioning nonsymmetric linear systems? 2/39
Approach based on inf-sup theory Key motivation: sufficient and necessary conditions for well-posedness for linear problems (Neˇ cas 62, Babuˇ ska 72, Brezzi 74) Applications of inf-sup theory in numerical analysis of time-dependent problems are diverse: • A priori error analysis, e.g. Tantardini & Veeser ’16 • A posteriori error analysis, e.g. Ern, S. & Vohralik ’17 • Reduced basis methods, e.g. Urban & Patera ’14 In the context of iterative methods for solving discrete systems: • Andreev, SIAM J. Numer. Anal. 16: wavelet-in-time method, multigrid in space, based on continuous inf-sup stability of problem • S., IMA J. Numer. Anal. 17: high-order DG time-stepping, based on discrete inf-sup stability of the method, considered system of a single time-step, robust with respect to space, time, & poly degree. 3/39
I. Inf-sup theory 4/39
Reminder Inf-sup theorem (quoted here from Schwab 98) Let X and Y real reflexive Banach spaces with norms �·� X and �·� Y respectively. Let Y ∗ be the dual of Y . Let further B : X → Y ∗ be a bounded linear operator. Then the conditions � Bu , v � Y ∗ × Y inf sup ≥ β > 0 , ( ∗ ) � u � X � v � Y u ∈ X \{ 0 } v ∈ Y \{ 0 } sup � Bu , v � Y ∗ × Y > 0 ∀ v ∈ Y \ { 0 } , ( ∗∗ ) u ∈ X are necessary and sufficient for well-posedness: ∀ f ∈ Y ∗ , ∃ ! u ∈ X such that Bu = f and � u � X ≤ β − 1 � f � Y ∗ . Remark: can be equivalently formulated in terms of bilinear forms with b ( u , v ) = � Bu , v � Y ∗ × Y . 5/39
Inf-sup theory Inf-sup theory for an abstract parabolic problem ∂ t u + A ( t ) u = f in (0 , T ) , u (0) = u 0 ∈ H (1) → V ∗ (densely and compactly) with separable Hilbert spaces V ֒ → H ֒ and A ( t ): V → V ∗ , �A ( t ) � V→V ∗ ≤ C bounded �A ( t ) u , v � V ∗ ×V = �A ( t ) v , u � V ∗ ×V , self-adjoint α � u � 2 V ≤ �A ( t ) u , u � V ∗ ×V , coercive for all u , v ∈ V , with C and α > 0 independent of t . Suppose also that f ∈ L 2 (0 , T ; V ∗ ). 6/39
Inf-sup theory Let �· , ·� be the duality pairing on V ∗ × V from now on. Well-posed weak formulation Find u ∈ S := L 2 (0 , T ; V ) ∩ H 1 (0 , T ; V ∗ ) s.t. u (0) = u 0 and � T � T ∀ v ∈ L 2 (0 , T ; V ) , � ∂ t u + A ( t ) u , v � d t = � f , v � d t 0 0 Full details of theory in many standard references, see e.g. Wloka 87, Zeidler 90 (II/A). Extension to many nonlinear problems in Roub´ ıˇ cek 05. � T 0 �· , ·� d t is equivalent to the duality pairing on L 2 (0 , T ; V ∗ ) and Remark: L 2 (0 , T ; V ) 7/39
Inf-sup theory Key identity: For all u ∈ S := L 2 (0 , T ; V ) ∩ H 1 (0 , T ; V ∗ ) � � 2 � T 0 � ∂ t u + A ( t ) u , v � d t � u � 2 + � u (0) � 2 S = sup ( † ) H � v � A v ∈ X \{ 0 } where the norms are defined by � T � u � 2 � ∂ t u � 2 ∗ , t + � u � 2 A ( t ) d t + � u ( T ) � 2 S := H 0 � T � v � 2 � v � 2 A := A ( t ) d t 0 with �·� 2 A ( t ) = �A ( t ) · , ·� V ∗ ×V , and with �·� ∗ , t the dual-norm on V ∗ wrt �·� A ( t ) , i.e. � φ � 2 ∗ , t = � φ, A − 1 ( t ) φ � for φ ∈ V ∗ . The identity implies that inf-sup condition ( ∗ ) holds here with constant β = 1. 8/39
Proof For all u ∈ S := L 2 (0 , T ; V ) ∩ H 1 (0 , T ; V ∗ ) � � 2 � T 0 � ∂ t u + A ( t ) u , v � d t � u � 2 + � u 0 � 2 S = sup H � v � A v ∈ X \{ 0 } Proof. Let w ∗ = A − 1 ( t ) ∂ t u , then � ∂ t u + A ( t ) u , v � = �A ( t )( w ∗ + u ) , v � and � � 2 � T � T 0 �A ( t )( w ∗ + u ) , v � d t � w ∗ + u � 2 sup = A ( t ) d t (equality with v = w ∗ + u ) � v � A v ∈ L 2 (0 , T ; V ) \{ 0 } 0 � T � w ∗ � 2 A ( t ) + 2 �A ( t ) w ∗ , u � + � u � 2 = A ( t ) d t 0 � T � ∂ t u � 2 ∗ , t + 2 � ∂ t u , u � + � u � 2 = A ( t ) d t 0 � T � ∂ t u � 2 ∗ , t + � u � 2 A ( t ) d t + � u ( T ) � 2 −� u (0) � 2 = H H 0 � �� � = � u � 2 9/39 S
Proof For all u ∈ S := L 2 (0 , T ; V ) ∩ H 1 (0 , T ; V ∗ ) � � 2 � T 0 � ∂ t u + A ( t ) u , v � d t � u � 2 + � u 0 � 2 S = sup H � v � A v ∈ X \{ 0 } Proof. Let w ∗ = A − 1 ( t ) ∂ t u , then � ∂ t u + A ( t ) u , v � = �A ( t )( w ∗ + u ) , v � and � � 2 � T � T 0 �A ( t )( w ∗ + u ) , v � d t � w ∗ + u � 2 sup = A ( t ) d t (equality with v = w ∗ + u ) � v � A v ∈ L 2 (0 , T ; V ) \{ 0 } 0 � T � w ∗ � 2 A ( t ) + 2 �A ( t ) w ∗ , u � + � u � 2 = A ( t ) d t 0 � T � ∂ t u � 2 ∗ , t + 2 � ∂ t u , u � + � u � 2 = A ( t ) d t 0 � T � ∂ t u � 2 ∗ , t + � u � 2 A ( t ) d t + � u ( T ) � 2 −� u (0) � 2 = H H 0 � �� � = � u � 2 9/39 S
Discrete inf-sup theory of Implicit Euler Implicit Euler discretization of abstract time-dependent equation: find u n ∈ V M ( u n − u n − 1 ) + τ n A n u n = τ n f n , n = 1 , . . . , N where M and { A n } N n =1 are SPD matrices, and u 0 is given. No assumption on time-regularity/continuity of A n or f n . No assumption on connection between M and A n (so no assumption on τ and h 2 ) 10/39
Discrete inf-sup theory of Implicit Euler M ( u n − u n − 1 ) + τ n A n u n = τ n f n , n = 1 , . . . , N The link between analysis of continuous and discrete settings: equivalent variational formulation (DG0): piecewise-constant approximation on intervals I n = ( t n − 1 , t n ]: ∀ v τ ∈ V τ := ⊕ N Find u τ s.t. b ( u τ , v τ ) = ℓ ( v τ ) n =1 P 0 ( I n ; V ) . � � N where b ( u τ , v τ ) := ( ∂ t I u τ , v τ ) M + ( u τ , v τ ) A n d t , I n n =1 � � N ℓ ( v τ ) := ( u 0 , v 1 ) M + ( f n , v τ ) M d t , I n n =1 where I u τ is P1 interpolatory reconstruction. I u τ u τ 11/39
Discrete inf-sup theory of Implicit Euler Discrete inf-sup condition b ( u τ , v τ ) � u τ � S = sup ∀ u τ ∈ V τ (2) � v τ � A v ∈ V τ \{ 0 } where N N � � � � u τ � 2 � ∂ t I u τ � 2 M + � u τ � 2 A n d t + � u N � 2 � � u τ � n − 1 � 2 S := M + , MA − 1 M n � �� � I n n =1 n =1 jump terms N � � � v τ � 2 � v τ � 2 A := A n d t , I n n =1 Full details of proof in Neum¨ uller & S. ’18, arxiv:1802.08126. Extends to higher-order DG, see S. 17. NB: Dual norm � ( v , w ) M v ⊤ MA − 1 � v � MA − 1 M = sup = n Mv � w � A n n w ∈ V \{ 0 } 12/39
Relation to other norms Relation to maximum norm For any u ∈ S , � u � L ∞ (0 , T ; H ) ≤ � u � S . For any u τ ∈ V τ , t ∈ [0 , T ] � u τ ( t ) � M ≤ � u τ � S . max Constant is 1 for any T , any spaces V , H , and operator A ( t ) (and in discrete case any { A n } , any M , and N , . . . ) 13/39
II. Symmetric reformulations & inexact Uzawa iterations 14/39
Symmetric reformulations Matrix form ⇒ u = [ u 1 , . . . , u N ] ∈ V N := V × · · · × V , Function u τ ∈ V τ ⇐ b ( u τ , v τ ) = ℓ ( v τ ) in matrix form M + τ 1 A 1 u 1 τ 1 f 1 + Mu 0 . − M M + τ 2 A 2 . = τ 2 f 2 . ... . . . u N � �� � � �� � � �� � f u B Can write B = K ⊗ M + diag { τ n A n } N n =1 = K + A � 1 � − 1 1 ∈ R N × N where K = ... 15/39
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