Rank reduction of parameterized time-dependent PDEs

Transcription

1 Rank reduction of parameterized time-dependent PDEs A. Spantini 1, L. Mathelin 2, Y. Marzouk 1 1 AeroAstro Dpt., MIT, USA 2 LIMSI-CNRS, France UNCECOMP 2015 (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

2 A common situation Design of a complex system: optimize some quantitie(s) of interest but... multiple operating conditions (Reynolds number, loading,... ), parameterized geometry, initial/boundary conditions, source terms, etc., uncertainty in some variables naturally leads to the introduction of an image probability space (Ξ, B Ξ, µ Ξ ),... one routinely faces parametric PDEs potentially high-dimensional solution space need for a good approximation method. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

3 Problem statement Let u (x, ξ) : X Ξ R, u S = S X Ξ S X S Ξ, be the solution at any given time of F possibly nonlinear. F (u; ξ) = f (x, ξ), Goal Solve and represent the solution u of F (u) = f efficiently. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

4 Problem statement Let u (x, ξ) : X Ξ R, u S = S X Ξ S X S Ξ, be the solution at any given time of F (u; ξ) = f (x, ξ), F possibly nonlinear. Approximating the solution in a finite dimensional space, and upon introduction of suitable bases for the product space S, yields u S u h = n p i=1 j=1 u ij h φ i (x) ψ j (ξ) S h, u ij h R, and S h is isomorphic to R n R p R n p. (U) ij = u ij h, U Rn p. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

5 Problem statement Let u (x, ξ) : X Ξ R, u S = S X Ξ S X S Ξ, be the solution at any given time of F (u; ξ) = f (x, ξ), F possibly nonlinear. Approximating the solution in a finite dimensional space, and upon introduction of suitable bases for the product space S, yields u S u h = n p i=1 j=1 u ij h φ i (x) ψ j (ξ) S h, u ij h R, and S h is isomorphic to R n R p R n p. (U) ij = u ij h, U Rn p. Complexity of the solution field: O (n p). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

6 Complexity of representation Alternatively, u can be written as a sum of rank one tensors. In the finite dimensional case, they can be evaluated from the SVD of U: u h KL = min[n,p] r=1 λr u Ξ r (ξ) u X r (x), [Karhunen-Loève-like], u Ξ r (ξ) R p, u X r (x) R n. R-term KL approximation: Complexity: O (R (n + p + 1)). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

7 Low rankness is good... A low rank description of the solution is desirable in terms of storage (O (R (n + p))) and computational effort. Several techniques exploit the low rankness of the solution structure in the solution process: Generalized Spectral Decomposition (GSD), [Chinesta, Nouy,... ], Dynamically Orthogonal (DO), [Sapsis & Lermusiaux, 2009], Dynamically Bi-Orthogonal Decomposition, [Cheng, Hou & Zhang, 2013], Reduced Basis Methods, [Patera, Maday,... ], Dynamical low-rank approximation [Koch & Lubich, 2007],... (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

8 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

9 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

10 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

11 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

12 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

13 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

14 Stochastic advection equation u t + V (ξ) u = 0, V (ξ) = ξ, ξ U ( 1, 1) x R(n) Progressive increase of the rank u (x, ξ; t n) w r (x) λ r (ξ), Long time integration issue. r=1 (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

15 Low rankness is good... Several techniques exploit the low rankness of the solution structure in the solution process: Generalized Spectral Decomposition (GSD), [Chinesta, Nouy,... ], Dynamically Orthogonal (DO), [Sapsis & Lermusiaux, 2009], Dynamically Bi-Orthogonal Decomposition, [Cheng, Hou & Zhang, 2013], Reduced Basis Methods, [Patera, Maday,... ], Dynamical low-rank approximation [Koch & Lubich, 2007],... (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

16 Low rankness is good... Yes, but... Several techniques exploit the low rankness of the solution structure in the solution process: Generalized Spectral Decomposition (GSD), [Chinesta, Nouy,... ], Dynamically Orthogonal (DO), [Sapsis & Lermusiaux, 2009], Dynamically Bi-Orthogonal Decomposition, [Cheng, Hou & Zhang, 2013], Reduced Basis Methods, [Patera, Maday,... ], Dynamical low-rank approximation [Koch & Lubich, 2007],... the solution is approximated in a low-dimensional hyperplane (linear manifold). There might exist a low-dimensional nonlinear manifold on which the solution is well approximated. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

17 A naive observation Let U = rank [U] = n (full rank). Complexity: O ( n 2) R n n. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

18 A naive observation Let U = rank [U] = n (full rank). Complexity: O ( n 2) R n n. But its structure is fairly simple: U can be described by a recovery strategy and associated data Û and τ : Û (U) = τ = Complexity: O (n). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

19 Core idea Setting up the stage Let U be the solution field u S projected onto a finite dimensional space. Π S : S R n p, u U = Π S [u]. Let Φ : S S be a bijective form between vector spaces and consider û = Φ (u): U = Π S [u], Û = Π S [Φ (u)], [ ] Û = Π S Φ Π 1 S U. Let Û R ε r=1 w r λt r, the transformation Φ is determined such as the ε -rank R is low, R min [n, p]. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

20 Core idea (cont d) Proposed approach The solution of the transformed problem ( F Φ 1) ( û ) = f is meant to exhibit good numerical properties. The problem formulates as a constrained minimization problem: find a map Φ and a preconditioned solution field û such that s.t. {û, Φ } arg min J ( û ), J ( û ) = ε rank [ û (x, ξ; t) ], Φ ( F Φ 1) (û) = f, Φ Φ adm. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

21 Core idea (cont d) Proposed approach The solution of the transformed problem ( F Φ 1) ( û ) = f is meant to exhibit good numerical properties. The problem formulates as a constrained minimization problem: find a map Φ and a preconditioned solution field û such that s.t. {û, Φ } arg min J ( û ), J ( û ) = ε rank [ û (x, ξ; t) ], Φ ( F Φ 1) (û) = f, Φ Φ adm. But... rank minimization is non-convex, non-continuous, NP-hard need for a proxy. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

22 Heuristics for rank minimization Nuclear norm. See for instance Recht, Fazel & Parrilo (2010) J ( û ) = Û = (σ 1 σ 2...) 1. Recall û (x, ξ; t) KL = r σ r û X r (x) û Ξ r (ξ), I ( û ) = Û Rn p s.t. (Û) = û ij. ij σ ( ) r = W T û (V ) jr, i, j, r. ij ri Note: if Û is rank-deficient, the (Û) nuclear norm is non-differentiable. sub-differential Û û of the nuclear norm, smoothing with Huber penalties. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

23 Heuristics for rank minimization Nuclear norm. See for instance Recht, Fazel & Parrilo (2010) J ( û ) = Û = (σ 1 σ 2...) 1. Recall û (x, ξ; t) KL = r σ r û X r (x) û Ξ r (ξ), I ( û ) = Û Rn p s.t. (Û) = û ij. ij σ ( ) r = W T û (V ) jr, i, j, r. ij ri Note: if Û is rank-deficient, the (Û) nuclear norm is non-differentiable. sub-differential Û û of the nuclear norm, smoothing with Huber penalties. Complementary energy minimization. J ( û ) = E n c (û) (û), En c E = σ 2 (û) r, r>n Venturi (2011) uses n = 1: E = r J ( û ) = 1 σ2 1 û 2. S σr 2 (û). (û) (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

24 Heuristics for rank minimization Nuclear norm. See for instance Recht, Fazel & Parrilo (2010) J ( û ) = Û = (σ 1 σ 2...) 1. Recall û (x, ξ; t) KL = r σ r û X r (x) û Ξ r (ξ), I ( û ) = Û Rn p s.t. (Û) = û ij. ij σ ( ) r = W T û (V ) jr, i, j, r. ij ri Note: if Û is rank-deficient, the (Û) nuclear norm is non-differentiable. sub-differential Û û of the nuclear norm, smoothing with Huber penalties. Complementary energy minimization. J ( û ) = E n c (û) (û), En c E = σ 2 (û) r, r>n Venturi (2011) uses n = 1: Boyd (2003): J ( û ) = log ( r σr ). E = r J ( û ) = 1 σ2 1 û 2. S σr 2 (û). (û) (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

25 Heuristics for rank minimization (cont d) These transformation definitions rely on solution field decomposition techniques (e.g., SVD) computationally expensive. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

26 Heuristics for rank minimization (cont d) These transformation definitions rely on solution field decomposition techniques (e.g., SVD) computationally expensive. Alternative definition More affordable heuristic: reference subspace target. Let Ũref = span (u (ξ 1 ),..., u (ξ m)) be a subspace spanned by solutions of the original problem for different values of the parameters ξ. J ( û ) = 1 2 The first variation of J ( û ) is given by û ΠŨref [û] 2 S X S Ξ. DûJ ( û ) = ΠŨ ref [û] where ΠŨ ref [û] is the projection onto the orthocomplement of the reference subspace Ũ ref. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

27 A solution method for time-dependent parameterized PDEs At any time, the solution is represented under a separated format: R(t) u (x, ξ, t) = ur X (x, t) ur Ξ (ξ, t). r=1 the rank depends on time. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

28 A solution method for time-dependent parameterized PDEs At any time, the solution is represented under a separated format: R(t) u (x, ξ, t) = ur X (x, t) ur Ξ (ξ, t). r=1 the rank depends on time. Let consider an artificial time: û (x, ξ, t) := Φ (u, x, ξ, t) = u (x, ξ, τ (x, ξ, t)). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

29 A solution method for time-dependent parameterized PDEs Space-independent transformation τ (ξ, t), receding-horizon optimal control approach: J ( û ) = 1 2 t+t t û ΠŨref [û] 2 S X S Ξ. Regularization term α T 2 1 τ 2 S ξ promotes map invertibility ( τ > 0, µ Ξ -a.s.) and reduces drift from physical time (ṫ 1) as much as possible. Myopic formulation (vanishing horizon T 0 +) the (degenerate) optimal transformation τ is the solution of a parameterized ODE. ( ) Finally, for a problem of the form F (u) = + N (u) = f : t τ t [û] = α β (1 τ) + β û ΠŨref, (f F) S x. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

30 A solution method for time-dependent parameterized PDEs Space-independent transformation τ (ξ, t), receding-horizon optimal control approach: J ( û ) = 1 2 t+t t û ΠŨref [û] 2 S X S Ξ. Coupled problem where the reformulated problem and the parameterized ODE governing the transformation are solved together. ( ) F (u) = t + N (u) = f. ( ) (û) + τ N t = τ f, τ t [û] = α β (1 τ) + β û ΠŨref, (f F) S x, (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

31 Numerical experiments (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

32 Flow past a cylinder τ (ξ, t). Flow around a circular cylinder in a channel. Two-dimensional flow, laminar regime, parameterized Reynolds number: V (ξ) = ξ, ξ U ( 1, 1). ξ-dependent Reynolds number (MIT & LIMSI-CNRS) = ξ-dependent vortex shedding frequency, vortices wavelength, boundary layer thickness, etc. = growing rank in the {x, ξ}-space over time. Rank reduction of parameterized PDEs UNCECOMP / 22

33 Flow past a cylinder τ (ξ, t). Flow around a circular cylinder in a channel. Two-dimensional flow, laminar regime, parameterized Reynolds number: V (ξ) = ξ, ξ U ( 1, 1). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

34 x-velocity fields at t = 15 Non preconditioned Fields are out-of-phase. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

35 x-velocity fields at t = 15 Preconditioned Ũ ref = span (u ( ξ )) Fields are in-phase. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

36 Flow past a cylinder Time transform Substantial reduction of the numerical rank (20 3). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

37 A progressive decomposition Let u (x, ξ, t) S : u R r u X r u Ξ r S 1 S 2. In pratice, R is usually unknown a priori and u is derived progressively. Remark The following identifications are all legitimate: S (X T Ξ) S 1 (X T ) S 2 (Ξ), S 1 (X) S 2 (T Ξ), S 1 (X Ξ) S 2 (T ). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

38 A progressive decomposition Proposed approach Let û 1 s.t. u (x, ξ, t) rk 1 ( ) û 1 x, ξ, τ 1 1 (x, ξ, t) be the rank-1 approximation of the solution of ) F 1 (û1 = f. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

39 A progressive decomposition Proposed approach Let û 1 s.t. u (x, ξ, t) rk 1 ( ) û 1 x, ξ, τ 1 1 (x, ξ, t) be the rank-1 approximation of the solution of ) F 1 (û1 = f. Introducing û 2 s.t. u (x, ξ, t) rk 2 ( ) ( ) û 1 x, ξ, τ 1 1 (x, ξ, t) + û 2 x, ξ, τ 1 2 (x, ξ, t) and deflating the problem yields the approximation of the solution of F ) 2 (û2 ; û 1 = f. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

40 A progressive decomposition Proposed approach Let û 1 s.t. u (x, ξ, t) rk 1 ( ) û 1 x, ξ, τ 1 1 (x, ξ, t) be the rank-1 approximation of the solution of ) F 1 (û1 = f. Introducing û 2 s.t. u (x, ξ, t) rk 2 ( ) ( ) û 1 x, ξ, τ 1 1 (x, ξ, t) + û 2 x, ξ, τ 1 2 (x, ξ, t) and deflating the problem yields the approximation of the solution of F ) 2 (û2 ; û 1 = f. The artificial time τ r (x, ξ, t) is derived s.t. ( τ r arg min J r û r, τ r ; { } ) r 1 û i, τ i, J i=1 r = F r f, τ r S τ s.t. Fr f span ( ) ( ) ( ) û r, ûr x, ξ, τr 1 = ur X x, τr 1 ur Ξ (ξ). Finally u (x, ξ, t) r ( ) û r x, ξ, τr 1, τ r = τr 1 (x, t, ξ). (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

41 A progressive decomposition (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

42 Concluding remarks Strategy for parameterized PDEs which may achieve significant savings both in terms of CPU and memory. Solution to the original problem is obtained via a recovery strategy. Reformulate the problem so that its solution exhibits a particular structure and efficient numerical tools can be exploited. By-product: alleviates the long-time integration issue in parameterized PDEs. The proposed approach can be linked to the action of a Lie group on the solution manifold. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22

43 Concluding remarks Strategy for parameterized PDEs which may achieve significant savings both in terms of CPU and memory. Solution to the original problem is obtained via a recovery strategy. Reformulate the problem so that its solution exhibits a particular structure and efficient numerical tools can be exploited. By-product: alleviates the long-time integration issue in parameterized PDEs. The proposed approach can be linked to the action of a Lie group on the solution manifold. Yes, but... Mathematical and physical issues associated with a space-dependent time. Impact on properties of the problem (say, ellipticity), stability of the solution numerical scheme, etc. Theoretical analysis on how much complexity can be reduced and splitted between û and Φ. Best compromise and most suitable functional forms for Φ. (MIT & LIMSI-CNRS) Rank reduction of parameterized PDEs UNCECOMP / 22