arXiv subjects

Arieh Iserles

Publications and source records attributed to Arieh Iserles.

At least 19 recordsLinked to original sources

arXiv2026

On a class of modified Cayley--Magnus methods

We introduce a new class of numerical integrators for the time integration of non-autonomous linear ordinary differential equations whose coefficient matrix is sparse and evolves within a quadratic matrix Lie group. In contrast to standard Lie group integrators, the proposed methods avoid the evaluation of matrix exponentials acting on vectors and instead rely on solving a sequence of linear systems with sparse coefficient matrices. Moreover, they are well suited for problems arising from unbounded operators, as they inherently produce bounded solutions. We construct optimised schemes of orders four and six and assess their performance on a representative numerical example, demonstrating clear advantages over existing Lie-group integrators.

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arXiv2026

T-systems: a theory of orthonormal functions with a tridiagonal differentiation matrix

The starting point of this paper is that a spectral method is essentially a combination of an orthonormal basis of the underlying Hilbert space with Galerkin conditions. The choice of an orthonormal basis depends on a number of desirable features which we explore in the context of spectral methods for time-dependent partial differential equations in a single space dimension. A central role in ensuring many of the above features is played by the differentiation matrix of the underlying orthonormal system. In particular, it is beneficial if this matrix is skew-symmetric and tridiagonal. While orthonormal systems with this feature have been characterised in A. Iserles & M. Webb, ``Orthogonal systems with a skew-symmetric differentiation matrix'', Found. Comput. Maths, 19 (2019), 1191--1221, employing Fourier transforms, in this paper we provide an alternative characterisation using the differential Lanczos algorithm, which can be implemented constructively. It is valid for inner products that obey an `integration-by-parts condition', inclusive of $L_2$ and Sobolev norms on the real line. Motivated by quest for integration methods that conserve Hamiltonian energy, we conclude the paper replacing inner products by more general sesquilinear forms and presenting preliminary results. Here the Fourier transform characterisation generalises to spectral theory of Schrödinger operators and the differential Lanczos algorithm generalises to the differential Arnoldi algorithm.

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arXiv2025

On semi-separability and differentiation matrices

The theory of spectral methods for partial differential equations leads to infinite-dimensional matrices which represent the derivative operator with respect to an underlying orthonormal basis. Favourable properties of such differentiation matrices are crucial in the design of good spectral methods. It is known that bases using Laguerre and ultraspherical polynomials lead to semi-separable differentiation matrices of rank 1. In this paper we consider orthonormal bases constructed from Jacobi polynomials and prove that the underlying differentiation matrices are semi-separable of rank 2. This requires new results on semi-separable matrices which might be interesting in a wider context.

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arXiv2025

An accelerated Levin-Clenshaw-Curtis method for the evaluation of highly oscillatory integrals

The efficient approximation of highly oscillatory integrals plays an important role in a wide range of applications. Whilst traditional quadrature becomes prohibitively expensive in the high-frequency regime, Levin methods provide a way to approximate these integrals in many settings at uniform cost. In this work, we present an accelerated version of Levin methods that can be applied to a wide range of physically important oscillatory integrals, by exploiting the banded action of certain differential operators on a Chebyshev polynomial basis. Our proposed version of the Levin method can be computed essentially in just $\mathcal{O}(ν\logν)$ operations, where $ν$ is the number of quadrature points and the dependence of the cost on a number of additional parameters is made explicit in the manuscript. This presents a significant speed-up over the direct computation of the Levin method in current state-of-the-art. We outline the construction of this accelerated method for a fairly broad class of integrals and support our theoretical description with a number of illustrative numerical examples.

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arXiv2025

Fourth-order compact exponential splittings for unbounded operators

We present a derivation and error bound for the family of fourth order splittings, originally introduced by Chin and Chen, where one of the operators is unbounded and the second one bounded but time dependent, and which are dependent on a parameter. We first express the error by an iterated application of the Duhamel principle, followed by quadratures of Birkhoff-Hermite type of the underlying multivariate integrals. This leads to error estimates and bounds, derived using Peano/Sard kernels and direct estimates of the leading error term. Our analysis demonstrates that, although no single value of the parameter can minimise simultaneously all error components, an excellent compromise is the cubic Gauss--Legendre point $1/2-\sqrt{15}/10$.

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arXiv2024

Convergence of a moving window method for the Schrödinger equation with potential on $\mathbb{R}^d$

We propose a novel framework, called moving window method, for solving the linear Schrödinger equation with an external potential in $\mathbb{R}^d$. This method employs a smooth cut-off function to truncate the equation from Cauchy boundary conditions in the whole space to a bounded window of scaled torus, which is itself moving with the solution. This allows for the application of established schemes on this scaled torus to design algorithms for the whole-space problem. Rigorous analysis of the error in approximating the whole-space solution by numerical solutions on a bounded window is established. Additionally, analytical tools for periodic cases are used to rigorously estimate the error of these whole-space algorithms. By integrating the proposed framework with a classical first-order exponential integrator on the scaled torus, we demonstrate that the proposed scheme achieves first-order convergence in time and $γ/2$-order convergence in space for initial data in $H^γ(\mathbb{R}^d) \cap L^2(\mathbb{R}^d;|x|^{2γ} dx)$ with $γ\geq 2$. In the case where $γ= 1$, the numerical scheme is shown to have half-order convergence under an additional CFL condition. In practice, we can dynamically adjust the window when waves reach its boundary, allowing for continued computation beyond the initial window. Extensive numerical examples are presented to support the theoretical analysis and demonstrate the effectiveness of the proposed method.

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arXiv2024

Spectral methods on a triangle and W-systems

We present an overarching framework for stable spectral methods on a triangle, defined by a multivariate W-system and based on orthogonal polynomials on the triangle. Motivated by the Koornwinder orthogonal polynomials on the triangle, we introduce a Koornwinder W-system. Once discretised by this W-system, the resulting spatial differentiation matrix is skew symmetric, affording important advantages insofar as stability and conservation of structure are concerned. We analyse the construction of the differentiation matrix and matrix vector multiplication, demonstrating optimal computational cost. Numerical convergence is illustrated through experiments with different parameter choices. As a result, our method exhibits key characteristics of a practical spectral method, inclusive of rapid convergence, fast computation and the preservation of structure of the underlying partial differential equation.

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arXiv2024

Computation of some dispersive equations through their iterated linearisation

It is often the case that, while the numerical solution of the non-linear dispersive equation $\mathrm{i}\partial_t u(t)=\mathcal{H}(u(t),t)u(t)$ represents a formidable challenge, it is fairly easy and cheap to solve closely related linear equations of the form $\mathrm{i}\partial_t u(t)=\mathcal{H}_1(t)u(t)+\widetilde{\mathcal H}_2(t)u(t)$, where $\mathcal{H}_1(t)+\mathcal{H}_2(v,t)=\mathcal{H}(v,t)$. In that case we advocate an iterative linearisation procedure that involves fixed-point iteration of the latter equation to solve the former. A typical case is when the original problem is a nonlinear Schrödinger or Gross--Pitaevskii equation, while the `easy' equation is linear Schrödinger with time-dependent potential. We analyse in detail the iterative scheme and its practical implementation, prove that each iteration increases the order, derive upper bounds on the speed of convergence and discuss in the case of nonlinear Schrödinger equation with cubic potential the preservation of structural features of the underlying equation: the $\mathrm{L}_2$ norm, momentum and Hamiltonian energy. A key ingredient in our approach is the use of the Magnus expansion in conjunction with Hermite quadratures, which allows effective solutions of the linearised but non-autonomous equations in an iterative fashion. The resulting Magnus--Hermite methods can be combined with a wide range of numerical approximations to the matrix exponential. The paper concludes with a number of numerical experiments, demonstrating the power of the proposed approach.

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arXiv2024

Splitting methods for unbounded operators

This paper considers computational methods that split a vector field into three components in the case when both the vector field and the split components might be unbounded. We first employ classical Taylor expansion which, after some algebra, results in an expression for a second-order splitting which, strictly speaking, makes sense only for bounded operators. Next, using an alternative approach, we derive an error expression and an error bound in the same setting which are however valid in the presence of unbounded operators. While the paper itself is concerned with second-order splittings using three components, the method of proof in the presence of unboundedness remains valid (although significantly more complicated) in a more general scenario, which will be the subject of a forthcoming paper.

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arXiv2024

Mathematical foundations of spectral methods for time-dependent PDEs

The contention of this paper is that a spectral method for time-dependent PDEs is basically no more than a choice of an orthonormal basis of the underlying Hilbert space. This choice is governed by a long list of considerations: stability, speed of convergence, geometric numerical integration, fast approximation and efficient linear algebra. We subject different choices of orthonormal bases, focussing on the real line, to these considerations. While nothing is likely to improve upon a Fourier basis in the presence of periodic boundary conditions, the situation is considerably more interesting in other settings. We introduce two kinds of orthonormal bases, T-systems and W-systems, and investigate in detail their features. T-systems are designed to work with Cauchy boundary conditions, while W-systems are suited to zero Dirichlet boundary conditions.

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arXiv2024

An elementary approach to splittings of unbounded operators

Using elementary means, we derive the three most popular splittings of $e^{(A+B)}$ and their error bounds in the case when $A$ and $B$ are (possibly unbounded) operators in a Hilbert space, generating strongly continuous semigroups, $e^{tA}$, $e^{tB}$ and $e^{t(A+B)}$. The error of these splittings is bounded in terms of the norm of the commutators $[A, B]$, $[A, [A, B]]$ and $[B, [A, B]]$.

math.FA↗

arXiv2023

A framework for stable spectral methods in $d$-dimensional unit balls

The subject of this paper is the design of efficient and stable spectral methods for time-dependent partial differential equations in unit balls. We commence by sketching the desired features of a spectral method, which is defined by a choice of an orthonormal basis acting in the spatial domain. We continue by considering in detail the choice of a $W$-function basis in a disc in $\mathbb{R}^2$. This is a nontrivial issue because of a clash between two objectives: skew symmetry of the differentiation matrix (which ensures inter alia that the method is stable) and the correct behaviour at the origin. We resolve it by representing the underlying space as an affine space and splitting the underlying functions. This is generalised to any dimension $d \geq 2$ in a natural manner and the paper is concluded with numerical examples that demonstrate how our choice of basis attains the best outcome out of a number of alternatives.

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arXiv2023

A recurrence relation for generalised connection coefficients

We formulate and prove a general recurrence relation that applies to integrals involving orthogonal polynomials and similar functions. A special case are connection coefficients between two sets of orthonormal polynomials, another example is integrals of products of Legendre functions.

math.CA↗

arXiv2023

On skyburst polynomials and their zeros

We consider polynomials orthogonal on the unit circle with respect to the complex-valued measure $z^{ω-1}\mathrm{d} z$, where $ω\in\mathbb{R}\setminus\{0\}$. We derive their explicit form, a generating function and several recurrence relations. These polynomials possess an intriguing pattern of zeros which, as $ω$ varies, are reminiscent of a firework explosion. We prove this pattern in a rigorous manner.

math.CV↗

arXiv2023

Orthogonal systems for time-dependent spectral methods

This paper is concerned with orthonormal systems in real intervals, given with zero Dirichlet boundary conditions. More specifically, our interest is in systems with a skew-symmetric differentiation matrix (this excludes orthonormal polynomials). We consider a simple construction of such systems and pursue its ramifications. In general, given any $\mathrm{C}^1(a,b)$ weight function such that $w(a)=w(b)=0$, we can generate an orthonormal system with a skew-symmetric differentiation matrix. Except for the case $a=-\infty$, $b=+\infty$, only a limited number of powers of that matrix is bounded and we establish a connection between properties of the weight function and boundedness. In particular, we examine in detail two weight functions: the Laguerre weight function $x^α\mathrm{e}^{-x}$ for $x>0$ and $α>0$ and the ultraspherical weight function $(1-x^2)^α$, $x\in(-1,1)$, $α>0$, and establish their properties. Both weights share a most welcome feature of {\em separability,\/} which allows for fast computation. The quality of approximation is highly sensitive to the choice of $α$ and we discuss how to choose optimally this parameter, depending on the number of zero boundary conditions.

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arXiv2022

Sobolev-Orthogonal Systems with Tridiagonal Skew-Hermitian Differentiation Matrices

We introduce and develop a theory of orthogonality with respect to Sobolev inner products on the real line for sequences of functions with a tridiagonal, skew-Hermitian differentiation matrix. While a theory of such L2-orthogonal systems is well established, Sobolev orthogonality requires new concepts and their analysis. We characterise such systems completely as appropriately weighed Fourier transforms of orthogonal polynomials and present a number of illustrative examples, inclusive of a Sobolev-orthogonal system whose leading N coefficients can be computed in $\mathcal{O}(N \log N)$ operations.

math.CA↗

arXiv2022

Positivity-preserving methods for population models

Many important applications are modelled by differential equations with positive solutions. However, it remains an outstanding open problem to develop numerical methods that are both (i) of a high order of accuracy and (ii) capable of preserving positivity. It is known that the two main families of numerical methods, Runge-Kutta methods and multistep methods, face an order barrier: if they preserve positivity, then they are constrained to low accuracy: they cannot be better than first order. We propose novel methods that overcome this barrier: our methods are of second order, and they are guaranteed to preserve positivity. Our methods apply to a large class of differential equations that have a special graph Laplacian structure, which we elucidate. The equations need be neither linear nor autonomous and the graph Laplacian need not be symmetric. This algebraic structure arises naturally in many important applications where positivity is required. We showcase our new methods on applications where standard high order methods fail to preserve positivity, including infectious diseases, Markov processes, master equations and chemical reactions.

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arXiv2022

Approximation of wave packets on the real line

In this paper we compare three different orthogonal systems in $\mathrm{L}_2(\mathbb{R})$ which can be used in the construction of a spectral method for solving the semi-classically scaled time dependent Schrödinger equation on the real line, specifically, stretched Fourier functions, Hermite functions and Malmquist--Takenaka functions. All three have banded skew-Hermitian differentiation matrices, which greatly simplifies their implementation in a spectral method, while ensuring that the numerical solution is unitary -- this is essential in order to respect the Born interpretation in quantum mechanics and, as a byproduct, ensures numerical stability with respect to the $\mathrm{L}_2(\mathbb{R})$ norm. We derive asymptotic approximations of the coefficients for a wave packet in each of these bases, which are extremely accurate in the high frequency regime. We show that the Malmquist--Takenaka basis is superior, in a practical sense, to the more commonly used Hermite functions and stretched Fourier expansions for approximating wave packets

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