Problems in least squares

Call number: LD2668 .R4 1965 A72

Iterative methods for solving nonlinear least squares problems

Iterative methods for solving nonlinear least squares problem

LSMR: An iterative algorithm for sparse least-squares problems

An iterative method LSMR is presented for solving linear systems $Ax=b$ and least-squares problem \min \norm{Ax-b}_2, with $A$ being sparse or a fast linear operator. LSMR is based on the Golub-Kahan bidiagonalization process. It is analytically equivalent to the MINRES method applied to the normal equation A\T Ax = A\T b, so that the quantities \norm{A\T r_k} are monotonically decreasing (where $r_k = b - Ax_k$ is the residual for the current iterate $x_k$). In practice we observe that \norm{r_k} also decreases monotonically. Compared to LSQR, for which only \norm{r_k} is monotonic, it is safer to terminate LSMR early. Improvements for the new iterative method in the presence of extra available memory are also explored.Comment: 21 page

Solving physics-driven inverse problems via structured least squares

Numerous physical phenomena are well modeled by partial differential equations (PDEs); they describe a wide range of phenomena across many application domains, from model- ing EEG signals in electroencephalography to, modeling the release and propagation of toxic substances in environmental monitoring. In these applications it is often of interest to find the sources of the resulting phenomena, given some sparse sensor measurements of it. This will be the main task of this work. Specifically, we will show that finding the sources of such PDE-driven fields can be turned into solving a class of well-known multi-dimensional structured least squares prob- lems. This link is achieved by leveraging from recent results in modern sampling theory – in particular, the approximate Strang-Fix theory. Subsequently, numerical simulation re- sults are provided in order to demonstrate the validity and robustness of the proposed framework