Data Assimilation in Reduced Modeling

Abstract

This paper considers the problem of optimal recovery of an element u of a Hilbert space H from measurements of the form ℓ j (u), j = 1, . . . , m, where the ℓ 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 of 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. 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.

Model reduction from partial observations

“…Since M ⊆ M relax , one can thus construct a prior of the form (15) satisfying (2), from any combination of the subspaces (17) and scalars (18). We note that this procedure applies even if M relax is not defined via a relaxed set of parameters Θ relax as in (16).…”

“…choices of M prior [18]. When M prior is defined as the intersection of degenerate ellipsoids as in (15), the authors of [18] showed thatĥ corresponds to some specific point of M prior ∩ H h (namely the center of the Chebyshev ball of M prior ∩ H h ).…”

“…This paradigm has recently been explored in several papers [11,[16][17][18]. In [16], the authors proposed a reduced version of a Kalman filter and showed that the error on the estimate delivered by the latter can be bounded by a function of the data residual.…”

“…[11]), Maday et al considered a data assimilation problem from noiseless (resp. noisy) linear observations, where the prior model is defined as a low-dimensional approximation subspace of M. The same setup was discussed in [18] by Binev et al and extended to priors defined as an intersection of degenerate ellipsoids. Some of the theoretical considerations exposed in the present work are built upon the arguments derived in that paper.…”

“…We do not consider the case where M prior is defined as in (15) since, in this case, problem (27) does not have any simple analytical solution (the problem is actually NP-hard [18]). …”