Data Assimilation in Reduced Modeling

Peter Binev (1) , Albert Cohen (2) , Wolfgang Dahmen (3) , Ronald Devore (4) , Guergana Petrova (5) , Przemyslaw Wojtaszczyk (6)

1 Department of Mathematics [Columbia]
2 LJLL - Laboratoire Jacques-Louis Lions
3 RWTH - Rheinisch-Westfälische Technische Hochschule Aachen University
4 Texas A&M University [College Station]
5 TAMU - Department of Mathematics [Texas]
6 IMSM - Institute of Applied Mathematics [Warsaw]

This paper considers the problem of optimal recovery of an element u of a Hilbert space H from measurements of the form j (u), j = 1,. .. , m, where the j are known linear functionals on H. Problems of this type are well studied [18] and usually are carried out under an assumption that u belongs to a prescribed model class, typically a known compact subset of H. Motivated by reduced modeling for solving parametric partial differential equations, this paper considers another setting, where the additional information about u is in the form of how well u can be approximated by a certain known subspace V n of H of dimension n, or more generally, in the form of how well u can be approximated by each subspace from a sequence of nested subspaces V 0 ⊂ V 1 · · · ⊂ V n , with each V k of dimension k. A recovery algorithm for the one-space formulation was proposed in [16]. Their algorithm is proven, in the present paper, to be optimal. It is also shown how the recovery problem for the one-space problem, has a simple formulation, if certain favorable bases are chosen to represent V n and the measurements. The major contribution of the present paper is to analyze the multi-space case by exploiting additional information derived from the whole hierarchy of spaces V j rather than only from the largest space V n. It is shown that in this multi-space case, the set of all u that satisfy the given information can be described as the intersection of a family of known ellipsoids in H. It follows that a near optimal recovery algorithm in the multi-space problem is provided by identifying any point in this intersection. It is easy to see that the accuracy of recovery of u in the multi-space setting can be much better than in the one-space problems. Two iterative algorithms based on alternating projections are proposed for recovery in the multi-space problem and one of them is analyzed in detail. This analysis includes an a posteriori estimate for the performance of the iterates. These a posteriori estimates can serve both as a stopping criteria in the algorithm and also as a method to derive convergence rates. Since the limit of the algorithm is a point in the intersection of the aforementioned ellipsoids, it provides a near optimal recovery for u.

Mots clés: optimal recovery; reduced modeling; data assimilation MSC numbers: 62M45; 65D05; 68Q32; 97N50 *

Domaines: Analyse numérique [math.NA]

Peter Binev, Albert Cohen, Wolfgang Dahmen, Ronald Devore, Guergana Petrova, et al.. Data Assimilation in Reduced Modeling. SIAM/ASA Journal on Uncertainty Quantification, 2017, 5, pp.1-29. ⟨hal-01352283⟩