Perturbation theory for the singular value decomposition
Perturbation theory for the singular value decomposition
A fast adaptive multipole algorithm in three dimensions
Journal of Computational Physics
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Automatic Preconditioning by Limited Memory Quasi-Newton Updating
SIAM Journal on Optimization
Parallel multiscale Gauss-Newton-Krylov methods for inverse wave propagation
Proceedings of the 2002 ACM/IEEE conference on Supercomputing
A kernel-independent adaptive fast multipole algorithm in two and three dimensions
Journal of Computational Physics
Inverse Problem Theory and Methods for Model Parameter Estimation
Inverse Problem Theory and Methods for Model Parameter Estimation
Journal of Computational and Applied Mathematics
Fast Directional Multilevel Algorithms for Oscillatory Kernels
SIAM Journal on Scientific Computing
Multigrid Algorithms for Inverse Problems with Linear Parabolic PDE Constraints
SIAM Journal on Scientific Computing
A Framework for the Adaptive Finite Element Solution of Large-Scale Inverse Problems
SIAM Journal on Scientific Computing
Tensor Decompositions and Applications
SIAM Review
Applying recursion to serial and parallel QR factorization leads to better performance
IBM Journal of Research and Development
SIAM Journal on Scientific Computing
Communication-optimal Parallel and Sequential QR and LU Factorizations
SIAM Journal on Scientific Computing
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We propose an algorithm to compute an approximate singular value decomposition (SVD) of least-squares operators related to linearized inverse medium problems with multiple events. Such factorizations can be used to accelerate matrix-vector multiplications and to precondition iterative solvers. We describe the algorithm in the context of an inverse scattering problem for the low-frequency time-harmonic wave equation with broadband and multi-point illumination. This model finds many applications in science and engineering (e.g., seismic imaging, subsurface imaging, impedance tomography, non-destructive evaluation, and diffuse optical tomography). We consider small perturbations of the background medium and, by invoking the Born approximation, we obtain a linear least-squares problem. The scheme we describe in this paper constructs an approximate SVD of the Born operator (the operator in the linearized least-squares problem). The main feature of the method is that it can accelerate the application of the Born operator to a vector. If N"@w is the number of illumination frequencies, N"s the number of illumination locations, N"d the number of detectors, and N the discretization size of the medium perturbation, a dense singular value decomposition of the Born operator requires O(min(N"sN"@wN"d,N)]^2xmax(N"sN"@wN"d,N)) operations. The application of the Born operator to a vector requires O(N"@wN"s@m(N)) work, where @m(N) is the cost of solving a forward scattering problem. We propose an approximate SVD method that, under certain conditions, reduces these work estimates significantly. For example, the asymptotic cost of factorizing and applying the Born operator becomes O(@m(N)N"@w). We provide numerical results that demonstrate the scalability of the method.