## Applied Mathematics Colloquium: Dr. Dmitriy Leykekhman

#### University of Connecticut

Friday, February 12, 2021 · 2 PM - 3 PM

Title: Numerical Analysis of Sparse Initial Data Identification for Parabolic Problems

In this talk we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed and are very challenging computationally. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the sparsity information on data into the structure of the objective functional. In particular, the cost functional contains a penalization term of the control in a measure space, which makes the analysis of the problem rather technical. To approximate the problem, we use finite element method, namely, continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. In the case when the initial data consists of a linear combination of Dirac measures, we obtain first order convergence rates in space and optimal superconvergent rates in time. The key tool for the analysis are the new sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems. We illustrate the results with several numerical experiments.

Abstract:

In this talk we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed and are very challenging computationally. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the sparsity information on data into the structure of the objective functional. In particular, the cost functional contains a penalization term of the control in a measure space, which makes the analysis of the problem rather technical. To approximate the problem, we use finite element method, namely, continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. In the case when the initial data consists of a linear combination of Dirac measures, we obtain first order convergence rates in space and optimal superconvergent rates in time. The key tool for the analysis are the new sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems. We illustrate the results with several numerical experiments.