Fast Algorithms for Source Identification Problems with Elliptic PDE Constraints

Abstract

We present algorithms for the solution of a class of source identification problems for systems governed by elliptic partial differential equations (PDEs) on two-dimensional regular geometries. The state is the solution of the PDE, which is driven by an unknown source field. Given observations of the state, we seek to reconstruct the source field. We consider the cases of full domain observations and boundary observations. The problem is formulated as a least-squares PDE-constrained optimization problem. We use a reduced space approach in which we “invert” the associated Hessian using a preconditioned conjugate gradient (PCG) algorithm. Using standard Fourier analysis, we derive analytical solutions for the case in which the governing PDE has constant coefficients. Based on these solutions, we construct preconditioners that accelerate the convergence of PCG in the case of variable-coefficient elliptic PDE constraints. We performed numerical experiments to show the effectiveness of the preconditioners for variable coefficients with different contrasts and smoothness properties. We observed mesh-independent and $\beta$-independent convergence for different cases of the variable coefficients. The computational complexity of solving the source identification problem is $\mathcal{O}(N\log N)$. The construction of the preconditioner costs $\mathcal{O}(N^{3/2})$, where $N$ is the discretization size for the source.