A Lanczos-like method for non-autonomous linear ordinary differential equations

Giscard, Pierre-Louis; Pozza, Stefano

doi:10.1007/s40574-022-00328-6

Cited by 7 publications

(7 citation statements)

References 45 publications

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“…The new method was numerically analyzed in the scalar case. The scalar analysis is a fundamental step toward understanding the more general case of systems of non-autonomous linear ODEs [12,19]. In fact, the authors are developing a method for this more general case whose analysis and understanding will be built on the crucial results presented here.…”

Section: Discussionmentioning

confidence: 99%

“…While such large problems will not be considered in this paper, the numerical approach presented here has the ambition to provide a new framework for tackling these challenging problems. This numerical framework arises from a recently developed analytical framework [9,10,12] in which the solution of (1.1) is given by a simple expression. When Ã(τ 1 ) Ã(τ 2 ) = Ã(τ 2 ) Ã(τ 1 ) for every τ 1 , τ 2 ∈ [−1, 1], Ũ (t) can be expressed in the explicit form: Ũ (t) = exp t s Ã(τ ) dτ .…”

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confidence: 99%

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A new Legendre polynomial-based approach for non-autonomous linear ODEs

Pozza¹,

Buggenhout²

2023

Preprint

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We introduce a new method with spectral accuracy to solve linear non-autonomous ordinary differential equations (ODEs) of the kind d dt ũ(t) = f (t)ũ(t), ũ(−1) = 1, with f (t) an analytic function. The method is based on a new expression for the solution ũ(t) given in terms of a convolution-like operation, the ⋆-product. This expression is represented in a finite Legendre polynomial basis translating the initial problem into a matrix problem. An efficient procedure is proposed to approximate the Legendre coefficients of ũ(t) and its truncation error is analyzed. We show the effectiveness of the proposed procedure through some numerical experiments. The method can be easily generalized to solve systems of linear ODEs.

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Section: Discussionmentioning

confidence: 99%

mentioning

confidence: 99%

A new Legendre polynomial-based approach for non-autonomous linear ODEs

Pozza¹,

Buggenhout²

2023

Preprint

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“…The matrix-valued function U (t, s) in ( 2) is composed of elements from C ∞ Θ (I). Therefore, we can define the related coefficient matrix U M as in (8). Then, expression (4) can be approximated by…”

Section: Solution Of An Ode By the ⋆-Productmentioning

confidence: 99%

“…At the heart of the new method for solving (1) is a non-commutative convolution-like product, denoted by ⋆, defined between certain distributions [3]. Thanks to this product, the solution of (1) can be expressed through the ⋆-product inverse and its formulation as a sequence of integrals and differential equations; see [4][5][6][7][8]. In [2], we illustrated that, by discretizing the ⋆-product with orthogonal functions, the solution of a scalar ODE is accessible by solving a linear system.…”

Section: Introductionmentioning

confidence: 99%

A new matrix equation expression for the solution of non‐autonomous linear systems of ODEs

Pozza

Buggenhout

2023

Proc Appl Math and Mech

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The solution of systems of non‐autonomous linear ordinary differential equations is crucial in a variety of applications, such us nuclear magnetic resonance spectroscopy. A new method with spectral accuracy has been recently introduced in the scalar case. The method is based on a product that generalizes the convolution. In this work, we show that it is possible to extend the method to solve systems of non‐autonomous linear ordinary differential equations (ODEs). In this new approach, the ODE solution can be expressed through a linear system that can be equivalently rewritten as a matrix equation. Numerical examples illustrate the method's efficacy and the low‐rank property of the matrix equation solution.

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“…However, this requires solving an NP-hard problem. In [7][8][9], the NP-hard problem is overcome by introducing the -Lanczos method, a constructive method able to tridiagonalize Ã(t). At the heart of both the path-sum and -Lanczos method is a non-commutative convolution-like product, denoted by , defined between certain distributions [12].…”

Section: Introductionmentioning

confidence: 99%

The *-product approach for linear ODEs: A numerical study of the scalar case

Pozza¹,

Buggenhout²

2023

Programs and Algorithms of Numerical Mathematics 21

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Solving systems of non-autonomous ordinary differential equations (ODE) is a crucial and often challenging problem. Recently a new approach was introduced based on a generalization of the Volterra composition. In this work, we explain the main ideas at the core of this approach in the simpler setting of a scalar ODE. Understanding the scalar case is fundamental since the method can be straightforwardly extended to the more challenging problem of systems of ODEs. Numerical examples illustrate the method's efficacy and properties in the scalar case.

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A Lanczos-like method for non-autonomous linear ordinary differential equations

Cited by 7 publications

References 45 publications

A new Legendre polynomial-based approach for non-autonomous linear ODEs

A new Legendre polynomial-based approach for non-autonomous linear ODEs

A new matrix equation expression for the solution of non‐autonomous linear systems of ODEs

The *-product approach for linear ODEs: A numerical study of the scalar case

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