Colloquium: Peter Binev (USC)

When:
December 16, 2015 @ 3:30 pm – 4:30 pm
2015-12-16T15:30:00-10:00
2015-12-16T16:30:00-10:00
Where:
Keller 301

Speaker: Peter Binev (USC)

Title: Data Assimilation in Reduced Modeling

Abstract: We consider the problem of optimal recovery of an element $u$ of a Hilbert space $\mathcal{H}$ from measurements of the form $\ell_j(u)$, $j=1, \dots,m$, where the $\ell_j$ are known linear functionals on $\mathcal{H}$. Problems of this type are well studied and usually are carried out under an assumption that $u$ belongs to a prescribed model class, typically a known compact subset of $\mathcal{H}$.
Motivated by reduced modeling for solving parametric partial differential equations,
we consider 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 $\mathcal{H}$ of dimension $n$,
or more generally, in the form of how well $u$ can be approximated by each of a sequence of nested subspaces $V_0\subset V_1 \cdots \subset V_n$ with each $V_k$ of dimension $k$. A recovery algorithm for the one-space formulation was proposed in
[Y. Maday, A.T. Patera, J.D. Penn and M. Yano (2015), {\em A parametrized-background data-weak approach to variational data assimilation: Formulation, analysis, and application to acoustics}, Int. J. Numer. Meth. Engng, 102: 933-965].
We prove that their algorithm is optimal and show 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. Our major contribution 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 $\mathcal{H}$. 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 $u$.

This is a joint work with Albert Cohen, Wolfgang Dahmen, Ronald DeVore, Guergana Petrova, and Przemyslaw Wojtaszczyk. The results are available at [arXiv:1506.04770].