Reduced Modeling in Data Assimilation

Transcription

1 Reduced Modeling in Data Assimilation Peter Binev Department of Mathematics and Interdisciplinary Mathematics Institute University of South Carolina Challenges in high dimensional analysis and computation San Servolo May 5, 2016

2 Collaborators Albert Cohen, University Paris 6 Wolfgang Dahmen, RWTH-Aachen Ronald DeVore, Texas A&M University Guergana Petrova, Texas A&M University Przemyslaw Wojtaszczyk, University of Warsaw [arxiv: ]

3 A Few Reasons for Reduced Modeling computation of the full model could be very expensive in terms of time and computational resources if the model is just an approximation, there is no reason to compute it with a precision higher than the one provided by the model in parametric modeling one often wants to have a universal basis that provides a good approximation for the entire set of parameters Using a reduced model that can be viewed as approximating a subset M in an infinite-dimensional space U with an n-dimensional space V n notice that while M is usually a compact set, the set {u : dist(u, V n ) ε n } M is not even bounded

4 Example: Parameter-Dependent PDEs L ξ u = f with a differential operator L ξ depending on a parameter ξ Ξ from a (compact) (infinite-dimensional) set Ξ corresponding energy product, ξ variational formulation u, v ξ = f, v for all v V Galerkin discretization requires a very large subspace V V investigate the set of solutions M := {u ξ : ξ Ξ} reduced basis method: find a subspace V n such that dist(v n, M) ε n greedy approach using residual error estimates to build V n with V 0 := {0} find ξ j := argmax ξ Ξ Error(ũ(ξ, V j 1 ) u ξ ) define v j := ũ(ξ j, V ) and V j := span{v k } j k=1 weak greedy property: dist(v j, V j 1 ) γ dist(m, V j 1 ), 0 < γ 1

5 Data Assimilation How to best combine available reduced models with knowledge from external observations/measurements? the data can be used not only to validate a model but also to adjust the solution want to get closer to elements described by the model better estimation of the physical state described by the model estimate the model parameters that correspond to the state or nearby states of the observed phenomenon (inverse modeling) finite (often relatively small) number of measurements Measurements can be described by a finite dimensional space W, each point w W corresponds to a specific set of values; if W is the orthogonal complement of W, then all points with same measurements are described by w + W

6 Hilbert Space Setup Hilbert space U with norm U = :=, object of interest (the manifold): a (compact) subset M U goal: find (an approximation ũ U of) u M that fits given data {d j } m j=1 of measurements l j(u) = d j linear functionals l j U with representation l j (u) = u, ω j measurement space W = span{ω 1, ω 2,..., ω m } footprint of u on W w := P W u = (d j ) m j=1 W orthogonal complement W of W u ( w + W ) reduced modeling: use V n U instead of M with dist(m, V n ) ε n, where dist(m, V n ) : max u M u P V n u U approximation: find ũ that is close to V n and w + W

7 a picture instead of words

8 a picture instead of words

9 Least Square Recovery Formulation [MPPY] The problem for a single space V n was stated and considered in [Maday, Patera, Penn, and Yano, A parametrized-background data-weak approach to variational data assimilation: Formulation, analysis, and application to acoustics, Int.J.Numer.Meth.Engng. 102: (2015)] v (w) = arg min w P W v v V n For an orthonormal basis {ω j } m j=1 u (w) = v (w) + Then for β(v, W ) := inf v V u u (w) = ( 1 + m j=1 sup w W 1 β(v n, W ) of W define ( ) w, ω j v, ω j ω j v, w v w ) inf q W V n one has inf u q z z V n

10 The One-Space Algorithm

11 Least Square Recovery Formulation v (w) = arg min w P W v v V n For an orthonormal basis {ω j } m j=1 of W define u = v (w) + m j=1 ( ) w, ω j v, ω j ω j

12 Least Square Recovery Formulation v (w) = arg min w P W v v V n For an orthonormal basis {ω j } m j=1 u = v (w) + m j=1 of W define ( ) w, ω j v, ω j ω j = w + P W v We consider the mapping F : W W (not necessarily linear) called lifting that for given w finds an element F (w) of W to define u (w) = w + F (w) that is optimally close to u. We prove that the algorithm with F (w) := P W v for v as defined above is optimal even among ones with nonlinear liftings.

13 The Idea of Lifting consider the space V n W and W and its orthogonal projections on each v V n is split into two parts v = w + η, where w = w(v) W and η = η(v) W if dim P W V n = n, then the mapping V n P W V n invertible is given w W one can find w P W V n using the orthogonal projection P PW V n, then find v V n from w = w(v) and determine η(v) this is the way to find v V n for any given w W and consequently determine u

14 The One-Space Algorithm

15 Optimal Recovery [Michelli, Rivlin, Lecture Notes in Math (1985)] Find an algorithm A : W U that for each data point w = P W u W recovers A(w) = A(P W u) U A is instance optimal if u A(P W u) U C A (w)dist(u, V n ) model knowledge: u K n := {ũ U : dist(ũ, V n ) ε n } fix w W and consider U w := w + W and u Kw n := K n U w n multi-space problem K := K k and K w := K U w k=0 Performance criteria based on best approximations (S = K n, K): E A (S) := sup u A(P W u) u S E(S) := inf A E A(S)

16 Optimality of the One-Space Algorithm β(v, W ) := inf v V sup w W v, w, µ(v, W ) := sup v w η W Theorem Assume that β(v n, W ) > 0, K w and define η P V η = 1 β(v, W ) u = u (w) := arg min u P Vn u, v = v (w) := P Vn u. Then u U w (u v ) W, V n and u v = min u U w, v V n u v.

17 Optimality of the One-Space Algorithm β(v, W ) := inf v V sup w W v, w, µ(v, W ) := sup v w η W Theorem Assume that β(v n, W ) > 0, K w and define η P V η = 1 β(v, W ) u = u (w) := arg min u P Vn u, v = v (w) := P Vn u. Then u U w (u v ) W, V n and u v = min u U w, v V n u v. Moreover, (i) rad(k w ) = µ(v n, W ) ε 2 n u (w) v (w) 2 Chebychev radius

18 Optimality of the One-Space Algorithm β(v, W ) := inf v V sup w W v, w, µ(v, W ) := sup v w η W Theorem Assume that β(v n, W ) > 0, K w and define η P V η = 1 β(v, W ) u = u (w) := arg min u P Vn u, v = v (w) := P Vn u. Then u U w (u v ) W, V n and u v = min u U w, v V n u v. Moreover, (i) rad(k w ) = µ(v n, W ) ε 2 n u (w) v (w) 2 Chebychev radius (ii) A : w u (w) is optimal for recovering K w and E A (K w ) = E(K w ) = rad(k w ) = µ(v n, W ) ε 2 n u (w) v (w) 2

19 Optimality of the One-Space Algorithm β(v, W ) := inf v V sup w W v, w, µ(v, W ) := sup v w η W Theorem Assume that β(v n, W ) > 0, K w and define η P V η = 1 β(v, W ) u = u (w) := arg min u P Vn u, v = v (w) := P Vn u. Then u U w (u v ) W, V n and u v = min u U w, v V n u v. Moreover, (i) rad(k w ) = µ(v n, W ) ε 2 n u (w) v (w) 2 Chebychev radius (ii) A : w u (w) is optimal for recovering K w and E A (K w ) = E(K w ) = rad(k w ) = µ(v n, W ) ε 2 n u (w) v (w) 2 (iii) A is also optimal for recovering K W := E A (K W ) = E(K w ) = µ(v n, W ) ε n w W K w and

20 Multi-Space Problem Can we do better if consider all subspaces V 0 V 1... V n?

21 Multi-Space Problem the spaces V j for j = 1, 2,..., n are usually defined sequentially using a greedy procedure for each j = 0, 1,..., n we get an estimate dist(m, V j ) ε j thus M is a subset of K j := {x U : dist(x, V j ) ε j } since u (w + W ) we can consider K j w := K j (w + W ) all we know about the solution is u Kw j and it is better to use the intersection of all the sets instead of just one of them K 0 w is a hypersphere and in case V n W = {0}, K j w are (infinite-dimensional) hyperellipsoids for all practical purposes one can consider the intersections K j w (V n + W )

22 Multi-Space Algorithm Favorable bases } {φ 1,..., φ j } ONB for V j, j n {ω 1,..., ω m} ONB for W ω i, φ j SVD of ( { ) m,n {φ ω i, φ j i,j=1 1,..., φ n } ONB for Vn {ω1,..., ω m} ONB for W = si δ i,j (ψ j ) j 1 ONB for V n W if s j < 1 for j = p + 1,..., n, then ψ j := φ j s j ω j 1 s 2 j ONB for P W V n

23 Multi-Space Algorithm Favorable bases } {φ 1,..., φ j } ONB for V j, j n {ω 1,..., ω m} ONB for W ω i, φ j SVD of ( { ) m,n {φ ω i, φ j i,j=1 1,..., φ n } ONB for Vn {ω1,..., ω m} ONB for W = si δ i,j (ψ j ) j 1 ONB for V n W if s j < 1 for j = p + 1,..., n, then ψ j := φ j s j ω j 1 s 2 j ONB for P W V n u = m n n w j ω j + x j ψ j + y j ψj j=1 j=p+1 j 1 K n w n j=p+1 s 2 j (x j a j ) 2 + n j 1 the solution u (w) K w := y 2 j ε 2 n m j=n+1 (1 s 2 j )w 2 n Kw j an intersection of ellipsoids j=1 j

24 Multi-Space Algorithm universal bound Favorable bases } {φ 1,..., φ j } ONB for V j, j n {ω 1,..., ω m} ONB for W SVD of ( { ) m,n {φ ω i, φ j i,j=1 1,..., φ n } ONB for Vn {ω1,..., ω m} ONB for W ω i, φ j = si δ i,j, µ(v n, W ) = 1 s n Theorem Define θ i := n φ j, φ i ε j 1, k n, δ: δθk 2 n sk 2 n + j=1 for some 0 < δ 1 and let E 2 n := ε 2 n + δθ 2 k n + n j=k n+1 n j=k n+1 θ 2 j s 2 j θ 2 j, then rad(k 0 ) E n, rad(k w ) 2E n, w W, rad(k) 2E n E n 2 min 0 k n µ(v k, W ) ε k = ε n

25 Numerical Algorithm to Find a Point Near the Intersection Restrict the search on a finite dimensional space V n + W cyclic sequential projection: u 0 := w, u k+1 := P K np K n 1...P K 1P K 0P Uw u k

26 The End Thank you!