Uncertainty quantification for sparse solutions of random PDEs

Transcription

1 Uncertainty quantification for sparse solutions of random PDEs L. Mathelin 1 K.A. Gallivan 2 1 LIMSI - CNRS Orsay, France 2 Mathematics Dpt., Florida State University Tallahassee, FL, USA SIAM 10 July 12-16, 2010 Pittsburgh, PA

2 A waste of resources... An uncertain quantity f L 2 (Ω ξ, p ξ ; R) is usually represented as f (ξ) = k f k Ψ k (ξ) P f k Ψ k (ξ) = f (ξ), P N, ξ Ω ξ R d. k Given an approximation basis {Ψ k }, f is entirely determined from the P coefficients f k of the q-th order ( polynomials. ) q + d When q and/or d grows, P = gets large retrieving the q coefficients becomes a computational challenge. Standard approach: determine all P coefficients and possibly retain only the K most significant to speed-up subsequent treatments Waste of resources More sounded approach: determine only the K most significant coefficients.

3 Core philosophy of the method Let u be a signal lying in the time and frequency space. From the Weyl-Heisenberg principle, one knows that the signal can not be concentrated in a time and frequency support such as supp u supp û 1. where û is the Fourier transform of u.

4 Core philosophy of the method (cnt d) In the same line of reasoning, for u R N : T W N. (1) Let K be the cardinality of u in its time approximation basis {Φ m } and let one takes M measurements of û. If 2K W < N, holds, the sparsest u matching the M measures is unique and exactly recovers the non-zero coefficients. In a nutshell, if another solution u existed, δ = u u would lie in the nullspace of {Φ m} so that supp δ W. Since supp δ 2K, supp δ supp δ 2K W so that Eq. (1) is violated, establishing uniqueness of u. stable recovery of u from M N is possible with overwhelming probability.

5 Compressed sensing Formalized by Donoho, Candès, Romberg, etc. Allows to retrieve the exact compressible solution of a vastly undetermined linear system AX = Y + ω, A R M P, M P, X R P, supp (X) = K < M P. The solution X is determined from X = arg min X 0 s.t. Y AX 2 ε. X R P L 0 -norm L 1 -norm the problem becomes convex while the solution X remains the same (but M M log P).

6 Compressed sensing (cnt d) Recovery performance depends on the properties of A. Let δ K arg max 1 i K 1 λ i, with λ an eigenvalue of A T K A K and A K any submatrix A K R M K from A. If δ 2K < 2 1 holds, then [Candès (2009)], X X 2 c 1 X X K 1 / K + c 2 ε, the recovery is almost as good as the K -best term approximation X K. Evaluation of the Restricted Isometry Property constant δ K is NP-hard intractable. But : upper-bounded by the mutual incoherence µ between the set of linear information operators {Φ} and {Ψ} defining the frame in which f is approximated: δ K µ (Φ, Ψ) max m,k Φ m Ψ k Φ m 1 Ωξ Ψ k, Ψ k 1/2. One wants to use a set {Φ} as incoherent with {Ψ} as possible.

7 CS-like UQ? Let {Ψ} a d-dimensional, q-th total order, Polynomial Chaos basis of cardinality P = (d + q)!/ (d! q!). One wants to determine the significant modes of a random quantity f assumed compressible in {Ψ}. Let f be approximated as f (ξ) f (ξ) = P X k Ψ k (ξ). (2) Let {Φ} be a set incoherent with {Ψ}. Upon application on Eq. (2), look for sparsest X such that P Φ mf Φ m (Xk Ψ k ) ε, of the form k k 2 X = min X R P X 1 s.t. Y AX 2 ε.

8 Choice of the set of information operators {Φ} should lead to maximum incoherence when applied to {Ψ} (Legendre polynomials), should allow for a tractable estimation of Φ m f. The set {Φ 1... Φ M } is chosen at random from a linear combinaison of point-wise estimators in L 2 (Ω ξ, p ξ ; R): straightforward evaluation of the constraint vector Y : Y m = Φ m f = i I α i m f (ξ i ), 1 m M.

9 Choice of the measurement set {Φ} (cnt d) Rk1 If Φ m P δξm Doostan & Owhadi. Closely related to the solution from the statistical identification LASSO method, e.g. Blatman & Sudret for UQ. Rk2 If Φ m P Ψm Least-squares solution. The incoherence is minimal can not make a worse choice for {Φ}! Requires M = P samples!

10 Shallow Water Equations test problem Solve the SWE with several independent sources of uncertainty (source term location, strength, shape, ocean depth field, etc.) we want to quantify uncertainty associated with the level of water at a given location and time. Solution method: smoothed L 1 -norm around the origin C 1, weighted L 1 -norm (iterative procedure) to mimic L 0 -norm: 1 w k = better recovery properties, X k + ε memory-limited second-order quasi-newton approach for solving a reformulated problem [convex unconstrained problem]: X = arg min X R P X 1 + α Φ mf P Φ m (X k Ψ k ). k 2

11 Some results in 2D convergence rate Compressible case d = 2, q = 12, P = 91. Denser case (but still compressible) allows to very significantly reduce # of required deterministic solutions M to reach a given accuracy. ɛ f f. 2

12 Some results in 2D (cnt d) solution spectrum Compressible case Denser case M = 65 < 257 the most significant modes are effectively retrieved.

13 Excellent performance. About 2000 samples suffice to get maximal representation accuracy with 8-th order PC. 8D case Polynomial Chaos d = 8, q = 8, P =

14 Concluding remarks Efficient non-adaptive strategy to retrieve the most dominants modes and only them large computational resources savings, most quantities of physical interest are compressible in standard bases such as PC, nice features of a non-intrusive approach are retained.

15 Concluding remarks Efficient non-adaptive strategy to retrieve the most dominants modes and only them large computational resources savings, most quantities of physical interest are compressible in standard bases such as PC, nice features of a non-intrusive approach are retained. Future directions include better sensing operators for further improved recovery properties, adaptive redundant dictionary, application to model equations instead of response surfaces is straightforward,...

16 Appendix

17 8D case Stochastic collocation d = 8, q = 8, P = Reasonably good performance. Up to 1 order of magnitude less sampling for a given accuracy. f is only weakly compressible in the normalized Lagrange basis.