Updating the inverse of a matrix
SIAM Review
A Sherman-Morrison-Woodbury identity for rank augmenting matrices with application to centering
SIAM Journal on Matrix Analysis and Applications
Iterative solution methods
An efficient and robust spectral solver for nonseparable elliptic equations
Journal of Computational Physics
Computational Methods for Inverse Problems
Computational Methods for Inverse Problems
SIAM Journal on Scientific Computing
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints
SIAM Journal on Scientific Computing
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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.