Adaptive high-order splitting schemes for large-scale differential Riccati equations

Stillfjord, Tony

doi:10.1007/s11075-017-0416-8

Cited by 29 publications

(20 citation statements)

References 30 publications

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“…A convergence analysis for a splitting method in the setting of Hilbert-Schmidt operators was proposed in [14]. Moreover, different types of splitting for DREs were proposed in [37,38].…”

Section: Differential Riccati Equationsmentioning

confidence: 99%

Convergence of a Low-Rank Lie--Trotter Splitting for Stiff Matrix Differential Equations

Ostermann¹,

Piazzola²,

Walach³

2019

SIAM J. Numer. Anal.

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We propose a numerical integrator for determining low-rank approximations to solutions of large-scale matrix differential equations. The considered differential equations are semilinear and stiff. Our method consists of first splitting the differential equation into a stiff and a non-stiff part, respectively, and then following a dynamical low-rank approach. We conduct an error analysis of the proposed procedure, which is independent of the stiffness and robust with respect to possibly small singular values in the approximation matrix. Following the proposed method, we show how to obtain low-rank approximations for differential Lyapunov and for differential Riccati equations. Our theory is illustrated by numerical experiments.

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“…A convergence analysis for a splitting method in the setting of Hilbert-Schmidt operators was proposed in [14]. Moreover, different types of splitting for DREs were proposed in [37,38].…”

Section: Differential Riccati Equationsmentioning

confidence: 99%

Convergence of a Low-Rank Lie--Trotter Splitting for Stiff Matrix Differential Equations

Ostermann¹,

Piazzola²,

Walach³

2019

SIAM J. Numer. Anal.

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Section: Introductionmentioning

confidence: 98%

“…When the matrix A is nonsingular and when the computation of the products W = A −1 V is not difficult (which is the case for sparse and structured matrices), the use of the EBA is to be preferred. We notice here that such a method was used for solving large symmetric Riccati problems in the work of Guldogan et al, 5 and other new methods were also developed recently in other works 6,7 The paper is organized as follows. In Section 2, we present a first approach based on the approximation of the exponential of a matrix times a block using a Krylov projection method.…”

Section: Introductionmentioning

confidence: 99%

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

Hached

Jbilou

2018

Numerical Linear Algebra App

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Summary In the present paper, we propose Krylov‐based methods for solving large‐scale differential Sylvester matrix equations having a low‐rank constant term. We present two new approaches for solving such differential matrix equations. The first approach is based on the integral expression of the exact solution and a Krylov method for the computation of the exponential of a matrix times a block of vectors. In the second approach, we first project the initial problem onto a block (or extended block) Krylov subspace and get a low‐dimensional differential Sylvester matrix equation. The latter problem is then solved by some integration numerical methods such as the backward differentiation formula or Rosenbrock method, and the obtained solution is used to build the low‐rank approximate solution of the original problem. We give some new theoretical results such as a simple expression of the residual norm and upper bounds for the norm of the error. Some numerical experiments are given in order to compare the two approaches.

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“…Thus, we solve the DLE to approximate the variance and the related deterministic differential equation to approximate the mean. In our numerical implementation we benefit from recently proposed solvers for both large-scale matrix DLEs [10,11,12,13,14] and large-scale matrix algebraic Lyapunov equations (ALEs) [15,16,17]. Our approach is more efficient computationally for both memory storage and computing time compared to SDE and SPDE based models solved by Taylor methods and stochastic Galerkin methods, respectively.…”

Section: Introductionmentioning

confidence: 99%

An efficient SPDE approach for El Niño

Mena

Pfurtscheller

2019

Applied Mathematics and Computation

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We consider the numerical approximation of stochastic partial differential equations (SPDEs) based models for a quasi-periodic climate pattern in the tropical Pacific Ocean known as El Niño phenomenon. We show that for these models the mean and the covariance are given by a deterministic partial differential equation and by an operator differential equation, respectively. In this context we provide a numerical framework to approximate these parameters directly. We compare this method to stochastic differential equations and SPDEs based models from the literature solved by Taylor methods and stochastic Galerkin methods, respectively. Numerical results for different scenarios taking as a reference measured data of the years 2014 and 2015 (last Niño event) validate the efficiency of our approach.

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Adaptive high-order splitting schemes for large-scale differential Riccati equations

Cited by 29 publications

References 30 publications

Convergence of a Low-Rank Lie--Trotter Splitting for Stiff Matrix Differential Equations

Convergence of a Low-Rank Lie--Trotter Splitting for Stiff Matrix Differential Equations

Computational Krylov‐based methods for large‐scale differential Sylvester matrix problems

An efficient SPDE approach for El Niño

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