Abstract. The nonlocal nature of the fractional integral makes the numerical treatment of fractional differential equations expensive in terms of computational effort and memory requirements. In this paper we propose a method to reduce these costs while controlling the accuracy of the scheme. This is achieved by splitting the fractional integral of a function f into a local term and a history term. Observing that the history term is a convolution of the history of f and a regular kernel, we derive a multipole approximation to the Laplace transform of the kernel. This enables the history term to be replaced by a linear combination of auxiliary variables defined as solutions to standard ordinary differential equations. We derive a priori error estimates, uniform in f , and obtain estimates on the number of auxiliary variables required to satisfy an error tolerance. The resulting formulation is discretized to produce a time stepping method. The method is applied to some test cases to illustrate the performance of the scheme.

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High-order adaptive methods for fractional differential equations are proposed. The methods rely on a kernel compression scheme for the approximation and localization of the history term. To avoid complications typical to multistep methods, we focus our study on 1-step methods and approximate the local part of the fractional integral by integral deferred correction to enable high order accuracy. We study the local truncation error of integral deferred correction schemes for Volterra equations and present numerical results obtained with both implicit and the explicit methods applied to different problems.

Wave problems in unbounded domains are often treated numerically by truncating the domain to produce a finite computational domain. The double absorbing boundary (DAB) method, which was invented recently as an alternative to methods of high-order absorbing boundary conditions and to the perfectly matched layer, is investigated here for problems in acoustics and elastodynamics. The paper offers two main contributions. The first one pertains to the well-posedness of the DAB scheme for the acoustics problem written in second-order form. The energy method is employed to obtain uniform-in-time estimates of the norm of the solution and the auxiliary functions, thus establishing the well-posedness and asymptotic stability of the DAB formulation. The second part pertains to the extension of the DAB to isotropic elastodynamics, written in first-order conservation form. Numerical experiments for an elastic wave guide demonstrate the performance of the scheme. Introduction.Efficient methods for truncating unbounded domains are key to accurate simulations in many fields of research. Two high-fidelity types of schemes are those based on a perfectly matched layer (PML) and those based on a local highorder absorbing boundary condition (ABC). See the review papers [1, 2, 3, 4, 5] and references therein. The method investigated in this paper is closer to that of highorder ABCs, although it does involve a truncating layer of a finite (small) thickness and in this sense bears similarity to a PML.High-order ABCs have been developed mainly for acoustics problems. Some such ABCs are Collino's ABC [6] (which was the first high-order ABC), the Givoli-Neta ABC [7,8], and the Hagstrom-Warburton ABC [9] and its extensions [10,11]. More recently, Hagstrom and Warburton devised the complete radiation boundary condition (CRBC) formulation [12], with the goal of combining the flexibility of local methods with the long-time accuracy of nonlocal methods. It was developed for first-order systems and was tested and optimized for acoustics [12,13,14]. It was later adapted for isotropic elastodynamics in [15]. In [16] is a study of the reflection coefficients for elasticity.

Inverse medium problems involve the reconstruction of a spatially varying unknown medium from available observations by exploring a restricted search space of possible solutions. Standard grid-based representations are very general but all too often computationally prohibitive due to the high dimension of the search space. Adaptive spectral decompositions instead expand the unknown medium in a basis of eigenfunctions of a judicious elliptic operator, which depends itself on the medium. Here the AS decomposition is combined with a standard inexact Newton-type method for the solution of time-harmonic scattering problems governed by the Helmholtz equation. By repeatedly adapting both the eigenfunction basis and its dimension, the resulting adaptive spectral inversion (ASI) method substantially reduces the dimension of the search space during the nonlinear optimization. Rigorous estimates of the AS decomposition are proved for a general piecewise constant medium. Numerical results illustrate the accuracy and efficiency of the ASI method for time-harmonic inverse scattering problems, including a salt dome model from geophysics.

A scheme for approximating the kernel w of the fractional α-integral by a linear combination of exponentials is proposed and studied. The scheme is based on the application of a composite Gauss-Jacobi quadrature rule to an integral representation of w. This results in an approximation of w in an interval [δ, T ], with 0 < δ, which converges rapidly in the number J of quadrature nodes associated with each interval of the composite rule. Using error analysis for Gauss-Jacobi quadratures for analytic functions, an estimate of the relative pointwise error is obtained. The estimate shows that the number of terms required for the approximation to satisfy a prescribed error tolerance is bounded for all α ∈ (0, 1), and that J is bounded for α ∈ (0, 1), T > 0, and δ ∈ (0, T ).

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Adaptive spectral (AS) decompositions associated with a piecewise constant function, u, yield small subspaces where the characteristic functions comprising u are well approximated. When combined with Newton-like optimization methods for the solution of inverse medium problems, AS decompositions have proved remarkably efficient in providing at each nonlinear iteration a low-dimensional search space. Here, we derive $$L^2$$ L 2 -error estimates for the AS decomposition of u, truncated after K terms, when u is piecewise constant and consists of K characteristic functions over Lipschitz domains and a background. Our estimates apply both to the continuous and the discrete Galerkin finite element setting. Numerical examples illustrate the accuracy of the AS decomposition for media that either do, or do not, satisfy the assumptions of the theory.