The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

Caliari, Marco; Kandolf, Peter; Ostermann, Alexander; Rainer, Stefan

doi:10.1137/15m1027620

Cited by 59 publications

(83 citation statements)

References 15 publications

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“…For the example shown in Figure , the Leja method outperforms Higham's approach and the Krylov subspace methods by a factor of 2 and 10, respectively. This is in agreement with the literature where Leja has been observed to perform best for normal matrices (Caliari et al, ; Kandolf, ). Therefore, we will only present results of the Leja method from now on.…”

Section: Numerical Testssupporting

confidence: 93%

“…The resulting recurrence equation reads

b_{i + 1} = L_{m, c} ((), t boldA false/ s) b_{i},

with i = 0,…, s − 1, b 0 = b . Leja interpolation L m , c reduces to the Taylor series for c = 0, but it performs better than Higham's functions for normal matrices with a value of c ≠ 0 (Caliari et al, ; Kandolf, ). In both cases, the approximate solution is derived as u = b s .…”

Section: The Paraexp Algorithmmentioning

confidence: 99%

“…Instead, efficient approximations of the action of the matrix exponential on initial condition vectors are used. Examples of such methods are the Krylov subspace-based methods as used in Gander and Güttel (2013), Higham's function (Al-Mohy & Higham, 2011), and the Leja method (Caliari et al, 2016).…”

Section: Approximation Of the Matrix Exponentialmentioning

confidence: 99%

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ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

2017

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Recently, ParaExp was proposed for the time integration of linear hyperbolic problems. It splits the time interval of interest into subintervals and computes the solution on each subinterval in parallel. The overall solution is decomposed into a particular solution defined on each subinterval with zero initial conditions and a homogeneous solution propagated by the matrix exponential applied to the initial conditions. The efficiency of the method depends on fast approximations of this matrix exponential based on recent results from numerical linear algebra. This paper deals with the application of ParaExp in combination with Leapfrog to electromagnetic wave problems in time domain. Numerical tests are carried out for a simple toy problem and a realistic spiral inductor model discretized by the Finite Integration Technique.

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Section: Numerical Testssupporting

confidence: 93%

“…The resulting recurrence equation reads

b_{i + 1} = L_{m, c} ((), t boldA false/ s) b_{i},

Section: The Paraexp Algorithmmentioning

confidence: 99%

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ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

2017

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“…It is sufficient to compute only its action on the initial solution f . Polynomial methods (see, for instance, [31,6,1], approximate the action of the exponential by a polynomial of a certain degree applied to the initial vector. They do not require to solve a linear system of equations: usually they scale the matrix and approximate the solution by an iterative procedure like u k+1 = p m k (α k τ A)u k , k = 0, 1, .…”

Section: Exact Time Discretisationmentioning

confidence: 99%

Anisotropic osmosis filtering for shadow removal in images

Parisotto¹,

Calatroni²,

Caliari³

et al. 2019

Inverse Problems

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We present an anisotropic extension of the isotropic osmosis model that has been introduced by Weickert et al. [38] for visual computing applications, and we adapt it specifically to shadow removal applications. We show that in the integrable setting, linear anisotropic osmosis minimises an energy that involves a suitable quadratic form which models local directional structures. In our shadow removal applications we estimate the local structure via a modified tensor voting approach [24] and use this information within an anisotropic diffusion inpainting that resembles edge-enhancing anisotropic diffusion inpainting [39,13]. Our numerical scheme combines the nonnegativity preserving stencil of Fehrenbach and Mirebeau [10] with an exact time stepping based on highly accurate polynomial approximations of the matrix exponential. The resulting anisotropic model is tested on several synthetic and natural images corrupted by constant shadows. We show that it outperforms isotropic osmosis, since it does not suffer from blurring artefacts at the shadow boundaries.

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“…Over the last few years many numerical methods have been proposed for computing the action of a matrix function over a vector. There are already excellent reviews in the literature describing the different numerical methods proposed so far (see [10,20,22,23], e.g. ); therefore it is not intended to go into any details here.…”

Section: Introductionmentioning

confidence: 99%

A Monte Carlo method for computing the action of a matrix exponential on a vector

Acebrón

2019

Applied Mathematics and Computation

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A Monte Carlo method for computing the action of a matrix exponential for a certain class of matrices on a vector is proposed. The method is based on generating random paths, which evolve through the indices of the matrix, governed by a given continuous-time Markov chain. The vector solution is computed probabilistically by averaging over a suitable multiplicative functional. This representation extends the existing linear algebra Monte Carlo-based methods, and was used in practice to develop an efficient algorithm capable of computing both, a single entry or the full vector solution. Finally, several relevant benchmarks were executed to assess the performance of the algorithm. A comparison with the results obtained with a Krylov-based method shows the remarkable performance of the algorithm for solving large-scale problems.

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The Leja Method Revisited: Backward Error Analysis for the Matrix Exponential

Cited by 59 publications

References 15 publications

ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

ParaExp Using Leapfrog as Integrator for High‐Frequency Electromagnetic Simulations

Anisotropic osmosis filtering for shadow removal in images

A Monte Carlo method for computing the action of a matrix exponential on a vector

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