Related papers: The Leja method revisited: backward error analysis…
We present a publicly available software for exponential integrators that computes the $\varphi_l(z)$ functions using polynomial interpolation. The interpolation method at Leja points have recently been shown to be competitive with the…
An efficient algorithm is proposed for Bayesian model calibration, which is commonly used to estimate the model parameters of non-linear, computationally expensive models using measurement data. The approach is based on Bayesian statistics:…
Krylov-based algorithms have long been preferred to compute the matrix exponential and exponential-like functions appearing in exponential integrators. Of late, direct polynomial interpolation of the action of these exponential-like…
Deterministic interpolation and quadrature methods are often unsuitable to address Bayesian inverse problems depending on computationally expensive forward mathematical models. While interpolation may give precise posterior approximations,…
We present a novel class of methods to compute functions of matrices or their action on vectors that are suitable for parallel programming. Solving appropriate simple linear systems of equations in parallel (or computing the inverse of…
This work presents a new algorithm to compute the matrix exponential within a given tolerance. Combined with the scaling and squaring procedure, the algorithm incorporates Taylor, partitioned and classical Pad\'e methods shown to be…
Approximation and uncertainty quantification methods based on Lagrange interpolation are typically abandoned in cases where the probability distributions of one or more {system} parameters are not normal, uniform, or closely related…
Matrix functions are a central topic of linear algebra, and problems requiring their numerical approximation appear increasingly often in scientific computing. We review various limited-memory methods for the approximation of the action of…
We discuss a pointwise numerical differentiation formula on multivariate scattered data, based on the coefficients of local polynomial interpolation at Discrete Leja Points, written in Taylor's formula monomial basis. Error bounds for the…
We first give a method to get multidimensional Leja sequences by considering intertwining sequences from one-dimensional ones. An application is the existence of explicit Leja sequences for the closed unit polydisc. Next, we deal with some…
A number of applications require the computation of the trace of a matrix that is implicitly available through a function. A common example of a function is the inverse of a large, sparse matrix, which is the focus of this paper. When the…
The efficient inversion of matrix polynomials is a critical challenge in computational mathematics. We design a procedure to determine the inverse of matrices polynomial of multidimensional Laplace matrices. The method is based on…
Leja points on a compact $K \subset \mathbb{C}$ are known to provide efficient points for interpolation, but their actual implementation can be computationally challenging. So-called pseudo Leja points are a more tractable solution, yet…
The evaluation of a matrix exponential function is a classic problem of computational linear algebra. Many different methods have been employed for its numerical evaluation [Moler C and van Loan C 1978 SIAM Review 20 4], none of which…
In this paper, we consider the application of exponential integrators to problems that are advection dominated, either on the entire or on a subset of the domain. In this context, we compare Leja and Krylov based methods to compute the…
In this paper, we investigate the application of exponential integrators to advection-dominated problems. We focus on Krylov subspace and Leja interpolation methods to compute the action of exponential and related matrix functions.…
In various areas of applied numerics, the problem of calculating the logarithm of a matrix A emerges. Since series expansions of the logarithm usually do not converge well for matrices far away from the identity, the standard numerical…
The nonlinear inverse problem of exponential data fitting is separable since the fitting function is a linear combination of parameterized exponential functions, thus allowing to solve for the linear coefficients separately from the…
A novel algorithm for computing the action of a matrix exponential over a vector is proposed. The algorithm is based on a multilevel Monte Carlo method, and the vector solution is computed probabilistically generating suitable random paths…
We propose a matrix-free algorithm for evaluating linear combinations of $\varphi$-function actions, $w_i := \sum_{j=0}^{p} \alpha_i^{\,j}\,\varphi_j(t_i A)v_j$ for $i=1\colon r$, arising in exponential integrators. The method combines the…