Estimating the diagonal of matrix functions

Abstract

The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix f(A), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions f(A), where A∈double-struckRp×p is a symmetric matrix and f a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

An Uncertainty-Weighted Asynchronous ADMM Method for Parallel PDE Parameter Estimation

“…Motivated by the discussion in the previous section, we reformulate the optimization problem (10) as an equivalent weighted global variable consensus problem…”

“…where in contrast to (10), the objective function is now separable, however, the coupling is enforced by the constraints. Here, x j ∈ R n are the local variables that are brought into consensus via the global variable z ∈ R n , and W j ∈ R n×n are diagonal weight matrices.…”

A posteriori error estimate for computing tr(f(A)) by using the Lanczos method

“…Computing the trace in this manner, however, requires the computation of all the eigenvalues, which is also often prohibitively expensive. Hence, various methods proposed for approximately computing tr( f ( A )) consist of the following two ingredients: Approximate the trace of f ( A ) by using the average of unbiased samples , i =1,…, N , where the u i are independent random vectors of some nature. Approximately compute the bilinear form by using some numerical technique. …”

“…Computing the trace in this manner, however, requires the computation of all the eigenvalues, which is also often prohibitively expensive. Hence, various methods proposed for approximately computing tr( f(A)) consist of the following two ingredients 1,10,[13][14][15][16][17][18][19] The various methods differ in the random mechanism of selecting the u i and the numerical technique for computing the bilinear form. Several variants of these ingredients exist (e.g., computing deterministically tr( (A)) = ∑ n i=1 e T i (A)e i rather than using random vectors u i , or even using block vectors to replace the canonical vectors e i 20 ; or using moment extrapolation for, particularly, f(t) = t with real value 21,22 ), but they are not the focus of this work.…”

“…An outstanding problem, however, is that good extensions of the a posteriori error estimate given by Saad 25 to more general functions than the exponential are rare. The Lanczos method can be considered a polynomial approximation technique, where f is approximated by a polynomial that interpolates f on the Ritz values, but it typically converges twice as fast as the other polynomial approximation methods (e.g., Chebyshev approximation; see chapter 19 in the work of Trefethen 26 ). Such a faster convergence is owed to the Gauss quadrature interpretation that will be discussed shortly.…”