arXiv subjects

James Bremer

Publications and source records attributed to James Bremer.

At least 19 recordsLinked to original sources

arXiv2025

An adaptive delaminating Levin method in two dimensions

We present an adaptive delaminating Levin method for evaluating bivariate oscillatory integrals over rectangular domains. Whereas previous analyses of Levin methods impose non-resonance conditions that exclude stationary and resonance points, we rigorously establish the existence of a slowly-varying, approximate solution to the Levin PDE across all frequency regimes, even when the non-resonance condition is violated. This allows us to derive error estimates for the numerical solution of the Levin PDE via the Chebyshev spectral collocation method, and for the evaluation of the corresponding oscillatory integrals, showing that high accuracy can be achieved regardless of whether or not stationary and resonance points are present. We then present a Levin method incorporating adaptive subdivision in both two and one dimensions, as well as delaminating Chebyshev spectral collocation, which is effective in both the presence and absence of stationary and resonance points. We demonstrate the effectiveness of our algorithm with a number of numerical experiments.

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arXiv2025

Airy Phase Functions

It is well known that phase function methods allow for the numerical solution of a large class of oscillatory second order linear ordinary differential equations in time independent of frequency. Unfortunately, these methods break down in the commonly-occurring case in which the equation has turning points. Here, we resolve this difficulty by introducing a generalized phase function method designed for the case of second order linear ordinary differential equations with turning points. More explicitly, we prove the existence of a slowly-varying ``Airy phase function'' that efficiently represents a basis in the space of solutions of such an equation, and describe a numerical algorithm for calculating this Airy phase function. The running time of our algorithm is independent of the magnitude of the logarithmic derivatives of the equation's solutions, which is a measure of their rate of variation that generalizes the notion of frequency to functions which are rapidly varying but not necessarily oscillatory. Once the Airy phase function has been constructed, any reasonable initial or boundary value problem for the equation can be readily solved and, unlike step methods which output the values of a rapidly-varying solution on a sparse discretization grid that is insufficient for interpolation, the output of our scheme allows for the rapid evaluation of the obtained solution at any point in its domain. We rigorously justify our approach by proving not only the existence of slowly-varying Airy phase functions, but also the convergence of our numerical method. Moreover, we present the results of extensive numerical experiments demonstrating the efficacy of our algorithm.

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arXiv2025

An accelerated frequency-independent solver for oscillatory differential equations

Oscillatory second order linear ordinary differential equations arise in many scientific calculations. Because the running times of standard solvers increase linearly with frequency when they are applied to such problems, a variety of specialized methods, most of them quite complicated, have been proposed. Here, we point out that one of the simplest approaches not only works, but yields a scheme for solving oscillatory second order linear ordinary differential equations which is significantly faster than current state-of-the-art techniques. Our method, which operates by constructing a slowly varying phase function representing a basis of solutions of the differential equation, runs in time independent of the frequency and can be applied to second order equations whose solutions are oscillatory in some regions and slowly varying in others. In the high-frequency regime, our algorithm discretizes the nonlinear Riccati equation satisfied by the derivative of the phase function via a Chebyshev spectral collocation method and applies the Newton-Kantorovich method to the resulting system of nonlinear algebraic equations. We prove that the iterates converge quadratically to a nonoscillatory solution of the Riccati equation. The quadratic convergence of the Newton-Kantorovich method and the simple form of the linearized equations ensure that this procedure is extremely efficient. Our algorithm then extends the slowly varying phase function calculated in the high-frequency regime throughout the solution domain by solving a certain third order linear ordinary differential equation related to the Riccati equation. We describe the results of numerical experiments showing that our algorithm is orders of magnitude faster than existing schemes, including the modified Magnus method [18], the current state-of-the-art approach [7] and the recently introduced ARDC method [1].

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arXiv2024

An adaptive Levin method for complicated domains

In this paper we describe an adaptive Levin method for numerically evaluating integrals of the form $\int_Ωf(\mathbf x) \exp(i g(\mathbf x)) \,dΩ$ over general domains that have been meshed by transfinite elements. On each element, we apply the multivariate Levin method over adaptively refined sub-elements, until the integral has been computed to the desired accuracy. Resonance points on the boundaries of the elements are handled by the application of the univariate adaptive Levin method. When the domain does not contain stationary points, the cost of the resulting method is essentially independent of the frequency, even in the presence of resonance points.

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arXiv2024

On the adaptive Levin method

The Levin method is a well-known technique for evaluating oscillatory integrals, which operates by solving a certain ordinary differential equation in order to construct an antiderivative of the integrand. It was long believed that this approach suffers from "low-frequency breakdown," meaning that the accuracy of the calculated value of the integral deteriorates when the integrand is only slowly oscillating. Recently presented experimental evidence, however, suggests that if a Chebyshev spectral method is used to discretize the differential equation and the resulting linear system is solved via a truncated singular value decomposition, then no low-frequency breakdown occurs. Here, we provide a proof that this is the case, and our proof applies not only when the integrand is slowly oscillating, but even in the case of stationary points. Our result puts adaptive schemes based on the Levin method on a firm theoretical foundation and accounts for their behavior in the presence of stationary points. We go on to point out that by combining an adaptive Levin scheme with phase function methods for ordinary differential equations, a large class of oscillatory integrals involving special functions, including products of such functions and the compositions of such functions with slowly-varying functions, can be easily evaluated without the need for symbolic computations. Finally, we present the results of numerical experiments which illustrate the consequences of our analysis and demonstrate the properties of the adaptive Levin method.

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arXiv2023

A solver for linear scalar ordinary differential equations whose running time is bounded independent of frequency

When the eigenvalues of the coefficient matrix for a linear scalar ordinary differential equation are of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing such solutions using standard techniques grows with the magnitudes of the eigenvalues. As a consequence, the running times of most solvers for ordinary differential equations also grow with these eigenvalues. However, a large class of scalar ordinary differential equations with slowly-varying coefficients admit slowly-varying phase functions that can be represented at a cost which is bounded independent of the magnitudes of the eigenvalues of the corresponding coefficient matrix. Here, we introduce a numerical algorithm for constructing slowly-varying phase functions which represent the solutions of a linear scalar ordinary differential equation. Our method's running time depends on the complexity of the equation's coefficients, but is bounded independent of the magnitudes of the equation's eigenvalues. Once the phase functions have been constructed, essentially any reasonable initial or boundary value problem for the scalar equation can be easily solved. We present the results of numerical experiments showing that, despite its greater generality, our algorithm is competitive with state-of-the-art methods for solving highly-oscillatory second order differential equations. We also compare our method with Magnus-type exponential integrators and find that our approach is orders of magnitude faster in the high-frequency regime.

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arXiv2023

A frequency-independent solver for systems of first order linear ordinary differential equations

When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing such solutions using standard techniques typically grows with the magnitudes of the eigenvalues. As a consequence, the running times of standard solvers for ordinary differential equations also grow with the size of these eigenvalues. The solutions of scalar equations with slowly-varying coefficients, however, can be efficiently represented via slowly-varying phase functions, regardless of the magnitudes of the eigenvalues of the corresponding coefficient matrix. Here, we couple an existing solver for scalar equations which exploits this observation with a well-known technique for transforming a system of linear ordinary differential equations into scalar form. The result is a method for solving a large class of systems of linear ordinary differential equations in time independent of the magnitudes of the eigenvalues of their coefficient matrices. We discuss the results of numerical experiments demonstrating the properties of our algorithm.

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arXiv2023

The Levin approach to the numerical calculation of phase functions

The solutions of scalar ordinary differential equations become more complex as their coefficients increase in magnitude. As a consequence, when a standard solver is applied to such an equation, its running time grows with the magnitudes of the equation's coefficients. It is well known, however, that scalar ordinary differential equations with slowly-varying coefficients admit slowly-varying phase functions whose cost to represent via standard techniques is largely independent of the magnitude of the equation's coefficients. This observation is the basis of most methods for the asymptotic approximation of the solutions of ordinary differential equations, including the WKB method. Here, we introduce two numerical algorithms for constructing phase functions for scalar ordinary differential equations inspired by the classical Levin method for the calculation of oscillatory integrals. In the case of a large class of scalar ordinary differential equations with slowly-varying coefficients, their running times are independent of the magnitude of the equation's coefficients. The results of extensive numerical experiments demonstrating the properties of our algorithms are presented.

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arXiv2022

Phase function methods for second order linear ordinary differential equations with turning points

It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the asymptotic approximation of the solutions of such equations. More recently, it was exploited by the author to develop a highly efficient solver for second order linear ordinary differential equations whose solutions are oscillatory. In many cases of interest, that algorithm achieves near optimal accuracy in time independent of the frequency of oscillation of the solutions. Here we show that, after minor modifications, it also allows for the efficient solution of second order differential equation equations which have turning points. That is, it is effective in the case of equations whose solutions are oscillatory in some regions and behave like linear combinations of increasing and decreasing exponential functions in others. We present the results of numerical experiments demonstrating the properties of our method, including some which show that it can used to evaluate many classical special functions in time independent of the parameters on which they depend.

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arXiv2022

Phase function methods for second order inhomogeneous linear ordinary differential equations

It is well known that second order homogeneous linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation underlies the Liouville-Green method and many other techniques for the asymptotic approximation of the solutions of such equations. It is also the basis of a recently developed numerical algorithm that, in many cases of interest, runs in time independent of the magnitude of the equation's coefficients and achieves accuracy on par with that predicted by its condition number. Here we point out that a large class of second order inhomogeneous linear ordinary differential equations can be efficiently and accurately solved by combining phase function methods for second order homogeneous linear ordinary differential equations with a variant of the adaptive Levin method for evaluating oscillatory integrals.

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arXiv2021

On the numerical evaluation of the prolate spheroidal wave functions of order zero

We describe a method for the numerical evaluation of the angular prolate spheroidal wave functions of the first kind of order zero. It is based on the observation that underlies the WKB method, namely that many second order differential equations admit solutions whose logarithms can be represented much more efficiently than the solutions themselves. However, rather than exploiting this fact to construct asymptotic expansions of the prolate spheroidal wave functions, our algorithm operates by numerically solving the Riccati equation satisfied by their logarithms. Its running time grows much more slowly with bandlimit and characteristic exponent than standard algorithms. We illustrate this and other properties of our algorithm with numerical experiments.

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arXiv2021

An $\mathcal{O}\left(1\right)$ algorithm for the numerical evaluation of the Sturm-Liouville eigenvalues of the spheroidal wave functions of order zero

In addition to being the eigenfunctions of the restricted Fourier operator, the angular spheroidal wave functions of the first kind of order zero and nonnegative integer characteristic exponents are the solutions of a singular self-adjoint Sturm-Liouville problem. The running time of the standard algorithm for the numerical evaluation of their Sturm-Liouville eigenvalues grows with both bandlimit and characteristic exponent. Here, we describe a new approach whose running time is bounded independent of these parameters. Although the Sturm-Liouville eigenvalues are of little interest themselves, our algorithm is a component of a fast scheme for the numerical evaluation of the prolate spheroidal wave functions developed by one of the authors. We illustrate the performance of our method with numerical experiments.

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arXiv2021

Rapid Application of the Spherical Harmonic Transform via Interpolative Decomposition Butterfly Factorization

We describe an algorithm for the application of the forward and inverse spherical harmonic transforms. It is based on a new method for rapidly computing the forward and inverse associated Legendre transforms by hierarchically applying the interpolative decomposition butterfly factorization (IDBF). Experimental evidence suggests that the total running time of our method -- including all necessary precomputations -- is $\mathcal{O}(N^2 \log^3(N))$, where $N$ is the order of the transform. This is nearly asymptotically optimal. Moreover, unlike existing algorithms which are asymptotically optimal or nearly so, the constant in the running time of our algorithm is small enough to make it competitive with state-of-the-art $\mathcal{O}\left(N^3\right)$ methods at relatively small values of $N$. Numerical results are provided to demonstrate the effectiveness and numerical stability of the new framework.

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arXiv2019

Fast Algorithms for the Multi-dimensional Jacobi Polynomial Transform

We use the well-known observation that the solutions of Jacobi's differential equation can be represented via non-oscillatory phase and amplitude functions to develop a fast algorithm for computing multi-dimensional Jacobi polynomial transforms. More explicitly, it follows from this observation that the matrix corresponding to the discrete Jacobi transform is the Hadamard product of a numerically low-rank matrix and a multi-dimensional discrete Fourier transform (DFT) matrix. The application of the Hadamard product can be carried out via $O(1)$ fast Fourier transforms (FFTs), resulting in a nearly optimal algorithm to compute the multidimensional Jacobi polynomial transform.

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arXiv2019

A quasilinear complexity algorithm for the numerical simulation of scattering from a two-dimensional radially symmetric potential

Standard solvers for the variable coefficient Helmholtz equation in two spatial dimensions have running times which grow quadratically with the wavenumber $k$. Here, we describe a solver which applies only when the scattering potential is radially symmetric but whose running time is $\mathcal{O}\left(k \log(k) \right)$ in typical cases. We also present the results of numerical experiments demonstrating the properties of our solver, the code for which is publicly available.

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arXiv2019

An O(1) Algorithm for the Numerical Evaluation of the Prolate Spheroidal Wave Functions of Order 0

The standard algorithm for the numerical evaluation of the prolate spheroidal wave function $\mathsf{Ps}\hskip.05em{}_{n}(x;\gamma^2)$ of order $0$, bandlimit $\gamma > 0$ and characteristic exponent $n$ has running time which grows with both $n$ and $\gamma$. Here, we describe an alternate approach which runs in time independent of these quantities. We present the results of numerical experiments demonstrating the properties of our scheme, and we have made our implementation of it publicly available.

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arXiv2018

Fast algorithms for Jacobi expansions via nonoscillatory phase functions

We describe a suite of fast algorithms for evaluating Jacobi polynomials, applying the corresponding discrete Sturm-Liouville eigentransforms and calculating Gauss-Jacobi quadrature rules. Our approach is based on the well-known fact that Jacobi's differential equation admits a nonoscillatory phase function which can be loosely approximated via an affine function over much of its domain. Our algorithms perform better than currently available methods in most respects. We illustrate this with several numerical experiments, the source code for which is publicly available.

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arXiv2017

An algorithm for the numerical evaluation of the associated Legendre functions that runs in time independent of degree and order

We describe a method for the numerical evaluation of normalized versions of the associated Legendre functions $P_\nu^{-\mu}$ and $Q_\nu^{-\mu}$ of degrees $0 \leq \nu \leq 1,000,000$ and orders $-\nu \leq \mu \leq \nu$ on the interval $(-1,1)$. Our algorithm, which runs in time independent of $\nu$ and $\mu$, is based on the fact that while the associated Legendre functions themselves are extremely expensive to represent via polynomial expansions, the logarithms of certain solutions of the differential equation defining them are not. We exploit this by numerically precomputing the logarithms of carefully chosen solutions of the associated Legendre differential equation and representing them via piecewise trivariate Chebyshev expansions. These precomputed expansions, which allow for the rapid evaluation of the associated Legendre functions over a large swath of parameter domain mentioned above, are supplemented with asymptotic and series expansions in order to cover it entirely. The results of numerical experiments demonstrating the efficacy of our approach are presented, and our code for evaluating the associated Legendre functions is publicly available.

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