Application of Multigrid to an Initial Control Problem

Michael Czajkowski University Düsseldorf, Germany

April 17, 1996

We consider the following initial control problem (icp) for the wave equation: for given measured data d_{j} ∈ ℝ at discrete points in space and time (x_{j}, y_{j}, t_{j}), j = 1, …, j_{max}, we are looking for a data fitting initial function u_{0}, that is, the resulting solution u = Mu_{0} of the wave equation has to satisfy

(Mu_{0})(x_{j}, y_{j}, t_{j}) = u(x_{j}, y_{j}, t_{j}) = d_{j}, j = 1, …, j_{max}

Since this problem is ill-posed we make a regularization ansatz that guarantees a unique solution

minimize J_{α}(u_{0}) = α ∫_{Ω} (\frac{∂}{∂x}u_{0})^{2}+(\frac{∂}{∂y} u_{0})^{2}dΩ + ∑_{j=1}^{j_{max}} ((Mu_{0})(x_{j}, y_{j}, t_{j}) − d_{j})^{2}

with a regularization parameter α ∈ ℝ^{+}. After discretization we discuss the equation characterizing the minimizing element of J_{α}. For efficiency reasons, only defect-oriented iterative solvers can be used. We consider gradient methods, cg and multigrid. In the case of complete time layers as measured data, a convergence analysis of Richardson iteration and multigrid can be done by means of discrete Fourier analysis. Finally, we consider some numerical examples including exact and perturbed data and present the nested iteration technique as an efficient solver for the icp. It turns out that a proper approximation to the icp can be obtained in practice by the same order of computational amount as needed for the solution of the direct problem.