Authors:
Prof. Dr. Dmitriy Leykekhman | University of Connecticut | United States
Prof. Dr. Boris Vexler | TU München | Germany
M. Sc. Daniel Walter | Technical University of Munich
In this paper we consider a problem of identification of an unknown initial data $q$ for a homogenous parabolic equation
\begin{equation}\label{eq:state}
\begin{aligned}
\pa_t u-\Delta u &= 0 &&\text{in} \quad (0,T)\times \Om,\; \\
u &= 0 &&\text{on}\quad (0,T)\times\pa\Omega, \\
u(0) &= q &&\text{in}\quad \Omega,
\end{aligned}
\end{equation}
from a given (measured) data $u_d \approx u(T)$ of the terminal state $u(T)$ for some $T>0$. In general, this problem is known to be exponentially ill-posed. We are interested in the situation, where the initial data we are looking for, is known to be sparse, i.e. to have a support of Lebesgue measure zero. The strong smoothing property of the above equation makes it difficult to identify such sparse initial data. The remedy is the incorporation of the information that the unknown $q$ should be sparse in the optimal control formulation. Following the idea for measure valued formulation of sparse control problems, we will look for the initial state $q$ in the space of regular Borel measures $\M(\Omega)$ on the domain $\Omega$, which is known to be isomorphic to the dual space of continuous functions $C_0(\Omega)^*$.
We formulate the problem as following optimal control problem
$$
\text{Minimize }\; J(q,u) = \half \norm{u(T) - u_d}^2_{L^2(\Omega)} + \alpha \mnorm{q}, \; {q \in {\mathcal M}(\Omega)}, \; \text{subject to}\;\eqref{eq:state}.
$$
where, $\Omega$ is a convex polygonal/polyhedral domain in $\mathbb{R}^N$, $N=2,3$, $I=(0,T]$ is the time interval, $u_d \in L^2(\Omega)$ is the given (desired /measured) final state, and $\alpha>0$ is the regularization parameter.
To approximate the problem, we will use a discontinuous Galerkin method dG($r$) of order $r$ for temporal and linear (conforming) finite elements for spatial discretizations of the state equation. In the paper we obtain error estimates, and in the case of linear combination of Diracs we also establish convergence rates for the source locations and the corresponding coefficients and illustrate the theoretical results with numerical experiments.