Regularization by truncated total least squares

The total least squares (TLS) method is a successful method for noise reduction in linear least squares problems in a number of applications. The TLS method is suited to problems in which both the coefficient matrix and the right-hand side are not precisely known. This paper focuses on the use of TLS for solving problems with very ill-conditioned coefficient matrices whose singular values decay gradually (so-called discrete ill-posed problems), where some regularization is necessary to stabilize the computed solution.

We filter the solution by truncating the small singular values of the TLS matrix. We express our results in terms of the singular value decomposition (SVD) of the coefficient matrix rather than the augmented matrix. This leads to insight into the filtering properties of the truncated TLS method as compared to regularized least squares solutions.

In addition, we propose and test an iterative algorithm based on Lanczos bidiagonalization for computing truncated TLS solutions.; A high-order Godunov-type scheme is developed for the shock interactions in ideal magnetohydrodynamics (MHD). The scheme is based on a nonlinear Riemann solver and follows the basic procedure in the piecewise parabolic method.

The scheme takes into account all the discontinuities in ideal MHD and is in a strict conservation form. The scheme is applied to numerical examples, which include shock-tube problems in ideal MHD and various interactions between strong MHD shocks. All the waves involved in the corresponding Riemann problems are resolved and are correctly displayed in the simulation results.

The correctness of the scheme is shown by the comparison between the simulation results and the solutions of the Riemann problems. The robustness of the scheme is demonstrated through the numerical examples. It is shown that the scheme offers the principle advantages of a high-order Godunov-type scheme: robust operation in the presence of very strong waves, thin shock fronts, and thin contact and slip surface discontinuities.; An approximate method for solving the Riemann problem is needed to construct Godunov schemes for relativistic hydrodynamical equations.

Such an approximate Riemann solver is presented in this paper which treats all waves emanating from an initial discontinuity as themselves discontinuous. Therefore, jump conditions for shocks are approximately used for rarefaction waves. The solver is easy to implement in a Godunov scheme and converges rapidly for relativistic hydrodynamics.

The fast convergence of the solver indicates the potential of a higher performance of a Godunov scheme in which the solver is used.