Numerical analysis of sparse initial data identification for parabolic problems

Abstract

In this paper 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 due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.

Linear convergence of accelerated conditional gradient algorithms in spaces of measures

“…The previous assumption can be guaranteed in several settings, and is commonly imposed for the purpose of error analysis, see, e.g., [24]. For instance, this condition holds if the operator K is injective as in, e.g., sparse initial value identification problems [45]. Furthermore, if (3.3) holds, then we note that the linear independence of (3.4) is in fact necessary for the existence of a unique sparse minimizer with nonzero coefficient functions.…”

Asymptotic linear convergence of fully-corrective generalized conditional gradient methods

“…Note that the coefficients λ † i ∈ R, the positions x † i ∈ Ω as well as the unknown number N ∈ N of points are assumed to be unknown. Taking the ill-posedness of the described inverse problem into account we follow [14,32] and consider the convex Tikhonov-regularized problem min u∈M (Ω),y…”

“…for ϕ ∈ L 2 (Ω) in the weak sense. For more details we refer to [32,16]. Note that K * is well-defined as z(0) ∈ C 0 (Ω), due to parabolic regularity estimates.…”

“…We finally remind that z is the solution to (3.17) for ϕ = y d − ȳ, ȳ = K ū and therefore by the characterization of K * it holds that z(0) = −K * ∇F (K ū). The following structural assumptions are made, see also [32,34]. Assumption 3.13.…”

Optimal Control of Semilinear Parabolic Equations with Non-smooth Pointwise-Integral Control Constraints in Time-Space

“…We point out that while the L 2 norm appearing in the cost influences the optimal solution, it does not eliminate the sparsifying effect of L 1 −terms, regardless of whether they appear in the cost or as a constraint. The literature on problems with an L 1 or measure-valued norm in the cost is quite rich, so we can only give selected references which consider evolutionary problems [1,2,3,4,5,7,8,9,11,14,15,16,18]. In all these papers, either there are no control constraints or they are box constraints.…”