Alessandra Papini

Compututational Techniques for Real Logarithms of Matrices

In this work, we consider computing the real logarithm of a real matrix. We pay attention to general conditioning issues, provide careful implementation for several techniques including scaling issues, and finally test and compare the techniques on a number of problems. All things considered, our recommendation for a general purpose method goes to the Schur decomposition approach with eigenvalue grouping, followed by square roots and diagonal Padé approximants of the diagonal blocks. Nonetheless, in some cases, a well-implemented series expansion technique outperformed the other methods. We have also analyzed and implemented a novel method to estimate the Frech&eacutet derivative of the

Padé approximation for the exponential of a block triangular matrix

\log

Conditioning and Padé Approximation of the Logarithm of a Matrix

, which proved very successful for condition estimation.

Conditioning of the Exponential of a Block Triangular Matrix

Abstract In this work, we obtain improved error bounds for Pade approximations to e A when A is block triangular. As a result, improved scaling strategies ensue which avoid some common overscaling difficulties.

Continuation of eigendecompositions

In this work we (i) use the theory of piecewise analytic functions to represent the Frechet derivative of any primary matrix function, in particular of primary logarithms; (ii) propose an indicator to assess inherent difficulties to compute a logarithm; and (iii) revisit Pade approximation techniques for the principal logarithm of block triangular matrices.

Coordinate search algorithms in multilevel optimization

We propose a new measure of conditioning for the exponential of a block triangular matrix. We also show that different “condition numbers” must be used to assess the accuracy of different algorithms which implement diagonal Padé with scaling and squaring.

On real logarithms of nearby matrices and structured matrix interpolation

In this work, we consider continuation of block eigendecompositions of a matrix valued function. We give new theoretical results on reduction to Hessenberg and bidiagonal forms, introduce and implement algorithms to continue eigendecompositions, and give numerical examples.

Generating Set Search Methods for Piecewise Smooth Problems

Many optimization problems of practical interest arise from the discretization of continuous problems. Classical examples can be found in calculus of variations, optimal control and image processing. In the recent years a number of strategies have been proposed for the solution of such problems, broadly known as multilevel methods. Inspired by classical multigrid schemes for linear systems, they exploit the possibility of solving the problem on coarser discretization levels to accelerate the computation of a finest-level solution. In this paper, we study the applicability of coordinate search algorithms in a multilevel optimization paradigm. We develop a multilevel derivative-free coordinate search method, where coarse-level objective functions are defined by suitable surrogate models. We employ a recursive v-cycle correction scheme, which exhibits multigrid-like error smoothing properties. On a practical level, the algorithm is implemented in tandem with a full-multilevel initialization. A suitable strategy to manage the coordinate search stepsize on different levels is also proposed, which gives a substantial contribution to the overall speed of the algorithm. Numerical experiments on several examples show promising results. The presented algorithm can solve large problems in a reasonable time, thus overcoming size and convergence speed limitations typical of coordinate search methods.

Path following by SVD

Theoretical and algorithmic results are given for the numerical computation of real logarithms of nearby matrices. As an application, and an original motivation for this study, interpolation for sequences of invertible matrices is considered particularly for matrices with a given structure (for example, orthogonal, symplectic, or positive definite), so that the resulting interpolants share the structural properties of the data. Error analysis, implementation details and examples are provided.

Approximating Coalescing Points for Eigenvalues of Hermitian Matrices of Three Parameters

We consider a direct search approach for solving nonsmooth minimization problems where the objective function is locally Lipschitz continuous and piecewise continuously differentiable on a finite family of polyhedra. A generating set search method is proposed, which is named structured because the structure of the set of nondifferentiability near the current iterate is exploited to define the search directions at each iteration. Some numerical results are presented to validate the approach.