Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

Angelova, Vera; Hached, Mustapha; Jbilou, Khalide

doi:10.1002/nla.2272

Cited by 7 publications

(3 citation statements)

References 36 publications

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“…Next, we apply the preceding results to estimate the approximation error E m = X(t) − X m (t) of the approximate solution X m (t) to the projected low-dimensional NDRE versus the exact solution to (1) from Theorem (4), which was already established by the authors in [19].…”

Section: Theorem 2 Let Us Denotementioning

confidence: 99%

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Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

2021

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Nonsymmetric differential matrix Riccati equations arise in many problems related to science and engineering. This work is focusing on the sensitivity of the solution to perturbations in the matrix coefficients and the initial condition. Two approaches of nonlocal perturbation analysis of the symmetric differential Riccati equation are extended to the nonsymmetric case. Applying the techniques of Fréchet derivatives, Lyapunov majorants and fixed-point principle, two perturbation bounds are derived: the first one is based on the integral form of the solution and the second one considers the equivalent solution to the initial value problem of the associated differential system. The first bound is derived for the nonsymmetric differential Riccati equation in its general form. The perturbation bound based on the sensitivity analysis of the associated linear differential system is formulated for the low-dimensional approximate solution to the large-scale nonsymmetric differential Riccati equation. The two bounds exploit the existing sensitivity estimates for the matrix exponential and are alternative.

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Section: Theorem 2 Let Us Denotementioning

confidence: 99%

“…The experimental tests are performed with Matlab R2020a on an Intel processor laptop equipped with 16GB of RAM. The reference solutions X(t) to the NDRE (1) and X(t) + ∆X(t) to the perturbed NDRE (3) are computed by the backward differential formula -BDF1-Newton method, see [19] for more details.…”

Section: Numerical Examplesmentioning

confidence: 99%

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

2021

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“…Many important equations such as differential Lyapunov equations (DLEs) and differential Riccati equations (DREs) can be put in the form. In the literature, there has been an enormous approaches to compute the solution of MDEs (7), see, e.g., [4,7,8,11,14,26,27,32,43]. Exponential integrators constitute an interesting class of numerical methods for the time integration of stiff systems of differential equations.…”

Section: Introductionmentioning

confidence: 99%

Computing the Lyapunov operator φ-functions, with an application to matrix-valued exponential integrators

Li¹,

Zhang²,

Zhang³

2022

Preprint

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In this paper, we develop efficient and accurate evaluation for the Lyapunov operator function ϕ l (L A ) [Q], where ϕ l (•) is the function related to the exponential, L A is a Lyapunov operator and Q is a symmetric and full-rank matrix. An important application of the algorithm is to the matrix-valued exponential integrators for matrix differential equations such as differential Lyapunov equations and differential Riccati equations. The method is exploited by using the modified scaling and squaring procedure combined with the truncated Taylor series. A quasi-backward error analysis is presented to determine the value of the scaling parameter and the degree of the Taylor approximation. Numerical experiments show that the algorithm performs well in both accuracy and efficiency.

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Backward differentiation formula method and random forest method to solve continuous‐time differential Riccati equations

Zhang,

Zou,

Sui

2024

Asian Journal of Control

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In this paper, we explore the utilization of machine learning techniques for solving the numerical solutions of continuous‐time differential Riccati equations. Specifically, we focus on generating a reduction matrix capable of transforming a high‐order matrix into a low‐order matrix. Additionally, we address the issue of differential terms in the continuous‐time differential Riccati equation and incorporate the backward differentiation formula of the matrix to improve stability and accuracy. Finally, by training samples through neural networks and machine learning methods, we could predict the solutions for high‐order matrix equations.

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Approximate solutions to large nonsymmetric differential Riccati problems with applications to transport theory

Cited by 7 publications

References 36 publications

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

Sensitivity of the Solution to Nonsymmetric Differential Matrix Riccati Equation

Computing the Lyapunov operator φ-functions, with an application to matrix-valued exponential integrators

Backward differentiation formula method and random forest method to solve continuous‐time differential Riccati equations

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