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In a recent article the authors showed that the radiative Transfer equations with multiple frequencies and scattering can be formulated as a nonlinear integral system. In the present article, the formulation is extended to handle reflective boundary conditions. The fixed point method to solve the system is shown to be monotone. The discretization is done with a P 1 Finite Element Method. The convolution integrals are precomputed at every vertex of the mesh and stored in compressed hierarchical matrices, using Partially Pivoted Adaptive Cross-Approximation. Then the fixed point iterations involve only matrix vector products. The method is O (N N 3 ln ⁡ N) , with respect to the number of vertices, when everything is smooth. A numerical implementation is proposed and tested on two examples. As there are some analogies with ray tracing the programming is complex. • We present a method and a an algorithm using an H-matrix compression scheme to compute the temperature of a gas under electromagnetic radiations. • The method is now capable of handling reflective boundaries. • The paper improves on previously existing results and the numerical methods use state of the art computational tools. [ABSTRACT FROM AUTHOR]

Highlights • Solving the anisotropic acoustic equation without S-wave artifacts. • Correct the dispersion error and phase error of the finite-difference method. • Deal with free surface and absorbing boundary condition in a direct way. Abstract Numerical solutions of the acoustic wave equation, especially in anisotropic media, is crucial to seismic modeling, imaging and inversion as it provides efficient, practical, and stable approximate representation of the medium. However, a clean implementation (free of shear wave artifacts and dispersion) of wave propagation, especially in anisotropic media, requires an integral operator, the direct evaluation of which is extremely expensive. Recently, the low-rank method was proposed to provide a good approximation to the integral operator utilizing Fourier transforms. Thus, we propose to split the integral operator into two terms. The first term provides a differential operator that approximates that can be approximated with a standard finite-difference method. We, then, apply the low-rank approximation on the residual term of the finite-difference operator. We implement the two terms in two complementing steps, in which the spectral step corrects for any errors admitted by the finite difference step. Even though we utilize finite-difference approximations, the resulting algorithm admits spectral accuracy. Also, through the finite difference step, the method can deal approximately with the free surface and absorbing boundary conditions in a straight forward manner. Numerical examples show that the method is of high accuracy and efficiency. [ABSTRACT FROM AUTHOR]

Abstract In this paper we present an efficient and accurate integral equation method to compute the scattering resonances for a subwavelength metallic slit structure. A new boundary integral equation is derived for the scattering problem, and the computation of scattering resonances is reduced to solving the eigenvalues of the corresponding homogeneous formulation over the complex plane. The integral operators are evaluated with high-order precisions by accurate calculations of the Green's functions for the layered medium and accelerated computation of the slit Green's function. The Newton's method is employed for solving the eigenvalues of the boundary integral formulation. We propose an effective strategy for obtaining the initial guesses of scattering resonances by introducing an approximate model for the scattering problem, for which the leading orders of the resonances are derived by asymptotic analysis. Numerical experiments are provided to demonstrate the accuracy, efficiency, and robustness of the method. [ABSTRACT FROM AUTHOR]

In this work, a modified Particle Method in fluid mechanics involving irregular distribution and discontinuity issues is proposed. Due to the approximation accuracy, momentum conservation and governing equation feasibility, traditional Particle Method, Moving Particle Semi-implicit (MPS) method in this paper, cannot guarantee the accuracy and stability of computation, and the divergence calculation in Navier–Stokes equation may cause numerical error in computation domain with discontinuity. To enhance the computation accuracy of Particle Method, we modify the gradient operator with a corrected tensor emanating from minimizing the local error of the first-order Taylor Expansion approximation. Besides, inspired by Peridynamic which is mostly used in solid mechanics, a new governing equation in an integral form which maintains the momentum conservation is obtained. Combining the modified gradient and new governing equation, we obtain a new particle-based method. The numerical results verify its feasibility and illustrate a good computational performance of proposed Method in the fluid dynamics involving non-uniform particle distribution. [ABSTRACT FROM AUTHOR]

Over the recent decades, a number of fast approximate solutions of Lippmann–Schwinger equation, which are more accurate than classic Born and Rytov approximations, were proposed in the field of electromagnetic modeling. Those developments could be naturally extended to acoustic and elastic fields; however, until recently, they were almost unknown in seismology. This paper presents several solutions of this kind applied to acoustic modeling for both lossy and lossless media. We evaluated the numerical merits of those methods and provide an estimation of their numerical complexity. In our numerical realization we use the matrix-free implementation of the corresponding integral operator. We study the accuracy of those approximate solutions and demonstrate, that the quasi-analytical approximation is more accurate, than the Born approximation. Further, we apply the quasi-analytical approximation to the solution of the inverse problem. It is demonstrated that, this approach improves the estimation of the data gradient, comparing to the Born approximation. The developed inversion algorithm is based on the conjugate-gradient type optimization. Numerical model study demonstrates that the quasi-analytical solution significantly reduces computation time of the seismic full-waveform inversion. We also show how the quasi-analytical approximation can be extended to the case of elastic wavefield. [ABSTRACT FROM AUTHOR]

Scattering from large, open cavity structures is of importance in a variety of electromagnetic applications. In this paper, we propose a new well conditioned integral equation for scattering from general open cavities embedded in an infinite, perfectly conducting half-space. The integral representation permits the stable evaluation of both the electric and magnetic field, even in the low-frequency regime, using the continuity equation in a post-processing step. We establish existence and uniqueness results, and demonstrate the performance of the scheme in the cavity-of-revolution case. High-order accuracy is obtained using a Nyström discretization with generalized Gaussian quadratures. [ABSTRACT FROM AUTHOR]

We solve the 2D transverse-electrically polarized domain-integral equation in a layered background medium by applying a Gabor frame as a projection method. This algorithm employs both a spatial and a spectral discretization of the electric field and the contrast current in the direction of the layer extent. In the spectral domain we use a representation on the complex plane that avoids the poles and branchcuts found in the Green function. Because of the special choice of the complex-plane path in the spectral domain and because of the choice to use a Gabor frame to represent functions on this path, fast algorithms based on FFTs are available to transform to and from the spectral domain, yielding an O ( N log ⁡ N ) scaling in computation time. [ABSTRACT FROM AUTHOR]

To study the temperature in a gas subjected to electromagnetic radiations, one may use the Radiative Transfer equations coupled with the Navier-Stokes equations. The problem has 7 dimensions; however with minimal simplifications it is equivalent to a small number of integro-differential equations in 3 dimensions. We present the method and a numerical implementation using an H -matrix compression scheme. The result is very fast: 240K physical points, all directions of radiation and 683 frequencies require less than 35 minutes on an Apple M1 Laptop. The method is capable of handling variable absorption and scattering functions of spatial positions and frequencies. The implementation is done using htool,1 a matrix compression library interfaced with the PDE solver freefem++. Applications to the temperature in the French Chamonix valley are presented at different hours of the day with and without snow or cloud and with a variable absorption coefficient taken from the Gemini measurements. The software is precise enough to assert temperature differences due to increased absorption in the vibrational frequency subrange of greenhouse gases. [ABSTRACT FROM AUTHOR]

We apply boundary integral equations for the first time to the two-dimensional scattering of time-harmonic waves from a smooth obstacle embedded in a continuously-graded unbounded medium. In the case we solve, the square of the wavenumber (refractive index) varies linearly in one coordinate, i.e. ( Δ + E + x 2 ) u ( x 1 , x 2 ) = 0 where E is a constant; this models quantum particles of fixed energy in a uniform gravitational field, and has broader applications to stratified media in acoustics, optics and seismology. We evaluate the fundamental solution efficiently with exponential accuracy via numerical saddle-point integration, using the truncated trapezoid rule with typically 10 2 nodes, with an effort that is independent of the frequency parameter E . By combining with a high-order Nyström quadrature, we are able to solve the scattering from obstacles 50 wavelengths across to 11 digits of accuracy in under a minute on a desktop or laptop. [ABSTRACT FROM AUTHOR]

In this paper we deal with the inverse scattering problem for an impedance obstacle. The aim is to recover both the impedance function and the scatterer simultaneously. Based on boundary integral equations, our method splits the inverse problem into a well-posed direct problem followed by a smaller ill-posed problem which has advantages both in understanding the inverse problem and in the numerical reconstructions. [ABSTRACT FROM AUTHOR]

Abstract: We present a frequency-domain renormalized integral equation formulation for solving a three-dimensional visco-acoustic medium using an iterative solver. Upon applying this special renormalization, the resulting integral equation operator can be proven to have a contraction property. Hence, solving the linear-system of equations using a Krylov optimization method, will result in a good convergence rate. Furthermore since the matrix–vector multiplication can be done using a Fast-Fourier transform (FFT) technique, its operation is of the order of , where N is the size of the discretization grid. This technique also allows us to use matrix-free implementation. Hence, the memory usage is about . Numerical tests show that the computational time and memory usage of this renormalized integral equation approach can be quite competitive with the frequency-domain finite difference iterative solver. Further, the numerical examples demonstrate that it is possible to solve a problem with over 100 million unknowns using an integral equation approach. [Copyright &y& Elsevier]

Abstract: An integral equation based scheme is presented for the fast and accurate computation of effective conductivities of two-component checkerboard-like composites with complicated unit cells at very high contrast ratios. The scheme extends recent work on multi-component checkerboards at medium contrast ratios. General improvement include the simplification of a long-range preconditioner, the use of a banded solver, and a more efficient placement of quadrature points. This, together with a reduction in the number of unknowns, allows for a substantial increase in achievable accuracy as well as in tractable system size. Results, accurate to at least nine digits, are obtained for random checkerboards with over a million squares in the unit cell at contrast ratio 106. Furthermore, the scheme is flexible enough to handle complex valued conductivities and, using a homotopy method, purely negative contrast ratios. Examples of the accurate computation of resonant spectra are given. [Copyright &y& Elsevier]

Abstract: An algorithm is presented for the fast and accurate solution of the electrostatic equation on multi-component random checkerboards. It relies on a particular choice of integral equation, extended as to separate ill-conditioning due to singular fields in corners from ill-conditioning due to interaction of clusters of well-conducting squares at large distances. Two separate preconditioners take care of the two separate phenomena. In a series of numerical examples, effective conductivities are computed for random checkerboards containing up to 104 squares with conductivity ratios of up to 106. The achievable relative precision in these examples is on the order of 10−11. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider band structure calculations governed by the Helmholtz or Maxwell equations in piecewise homogeneous periodic materials. Methods based on boundary integral equations are natural in this context, since they discretize the interface alone and can achieve high order accuracy in complicated geometries. In order to handle the quasi-periodic conditions which are imposed on the unit cell, the free-space Green’s function is typically replaced by its quasi-periodic cousin. Unfortunately, the quasi-periodic Green’s function diverges for families of parameter values that correspond to resonances of the empty unit cell. Here, we bypass this problem by means of a new integral representation that relies on the free-space Green’s function alone, adding auxiliary layer potentials on the boundary of the unit cell itself. An important aspect of our method is that by carefully including a few neighboring images, the densities may be kept smooth and convergence rapid. This framework results in an integral equation of the second kind, avoids spurious resonances, and achieves spectral accuracy. Because of our image structure, inclusions which intersect the unit cell walls may be handled easily and automatically. Our approach is compatible with fast-multipole acceleration, generalizes easily to three dimensions, and avoids the complication of divergent lattice sums. [ABSTRACT FROM AUTHOR]

Abstract: Automatic Chebyshev spectral collocation methods for Fredholm and Volterra integral and integro-differential equations have been implemented as part of the chebfun software system. This system enables a symbolic syntax to be applied to numerical objects in order to pose and solve problems without explicit references to discretization. The same objects can be used in matrix-free iterative methods in linear algebra, in order to avoid very large dense matrices or allow application to problems with nonsmooth coefficients. As a further application of the ability to implement operator equations, a method of Greengard for the recasting of differential equations as integral equations is generalized to mth order boundary value and generalized eigenvalue problems. In the integral form, large condition numbers associated with differentiation matrices in high-order problems are avoided. The ability to implement the recasting process generally follows from implementation of the operator expressions in chebfun. The integral method also can be extended to first-order systems, although chebfun syntax does not currently allow easy implementation in this case. [Copyright &y& Elsevier]

Abstract: In this paper we introduce a Conjugate Gradient Fast Fourier Transform (CG-FFT) scheme for the numerical solution of the integral equation formulating three-dimensional elastic scattering problems. The formulation is in terms of the stress tensor and particle velocities as the unknown field variables. In contrast with the formulation based on particle displacements, this approach leads to integral representations that do not involve derivatives of the unknown fields, thus resulting in simplified and more stable numerics. The numerical procedure is based on suitable quadrature formulas that provide (second order) accurate approximations while retaining the convolution nature of the relevant integrals that make them amenable to efficient evaluation via FFTs. The scheme is further improved through the introduction of (approximation-based) pre-conditioners that are shown to accelerate the convergence of the CG iterations. Numerical results are presented that demonstrate the accuracy and efficiency of the proposed methodology. [Copyright &y& Elsevier]

Abstract: We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science. The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of . In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of . [Copyright &y& Elsevier]

Abstract: The conventional magnetic induction equation that governs hydromagnetic dynamo action is transformed into an equivalent integral equation system. An advantage of this approach is that the computational domain is restricted to the region occupied by the electrically conducting fluid and to its boundary. This integral equation approach is first employed to simulate kinematic dynamos excited by Beltrami-like flows in a finite cylinder. The impact of externally added layers around the cylinder on the onset of dynamo actions is investigated. Then it is applied to simulate dynamo experiments within cylindrical geometry including the “von Kármán sodium” (VKS) experiment and the Riga dynamo experiment. A modified version of this approach is utilized to investigate magnetic induction effects under the influence of externally applied magnetic fields which is also important to measure the proximity of a given dynamo facility to the self-excitation threshold. [Copyright &y& Elsevier]

Abstract: In this paper, we describe an integral equation approach for simulating diffusion problems with moving interfaces. The solutions are represented as moving layer potentials where the unknowns are only defined on the interfaces. The resulting integro-differential equation (IDE) system is solved using spectral deferred correction (SDC) techniques developed for general differential algebraic equations (DAEs), and the time dependent potentials are evaluated efficiently using fast convolution algorithms. The numerical solver is applied to the BCF model for the epitaxial step-flow growth of crystals, for which the solutions are calculated accurately instead of using quasi-static approximations. Numerical results in 1+1 dimensions are compared with available results in the literature. [Copyright &y& Elsevier]

With only a few exceptions, the numerical simulation of cosmic and laboratory hydromagnetic dynamos has been carried out in the framework of the differential equation method. However, the integral equation method is known to provide robust and accurate tools for the numerical solution of many problems in other fields of physics. The paper is intended to facilitate the use of integral equation solvers in dynamo theory. In concrete, the integral equation method is employed to solve the eigenvalue problem for a hydromagnetic dynamo model with an isotropic helical turbulence parameter α. For the case of spherical geometry, three examples of the function α(r) with steady and oscillatory solutions are considered. A convergence rate proportional to the inverse squared of the number of grid points is achieved. Based on this method, a convergence accelerating strategy is developed and the convergence rate is improved remarkably. Typically, quite accurate results can be obtained with a few tens of grid points. In order to demonstrate its suitability for the treatment of dynamos in other than spherical domains, the method is also applied to α2 dynamos in rectangular boxes. The magnetic fields and the electric potentials for the first eigenvalues are visualized. [Copyright &y& Elsevier]

An iterative approach to full vector three-dimensional inverse scattering problems, where the unknown objects can have conductivity, permittivity and permeability different from the known background medium, is discussed. Since this problem involves a large number of unknowns, it has to be solved effectively and efficiently so that the results can be obtained in timely manner. The forward modeling is based on a domain integral equation approach formulated in terms of the electric and magnetic contrast sources normalized with the characteristic impedance of the background medium. Our numerical tests indicate that this formulation is prerequisite in order to arrive at a forward solution within an acceptable number of iterations, and hence it is also of significant importance in the optimization process of the inverse problem. The inverse scattering problem is attacked using the Multiplicative Regularized Contrast Source Inversion method as known in the literature. The complexity of this inverse method is approximately equal to the complexity of two equivalent forward algorithms of the conjugate gradient type. Furthermore, this inverse method has been armed with a weighted L2-norm regularizer which has been included as a multiplicative constraint. Some representative numerical testings will be presented to illustrate the ability of the our numerical algorithms. [Copyright &y& Elsevier]

This manuscript presents a new extended linear system for integral equation based techniques for solving boundary value problems on locally perturbed geometries. The new extended linear system is similar to a previously presented technique for which the authors have constructed a fast direct solver. The key features of the work presented in this paper are that the fast direct solver is more efficient for the new extended linear system and that problems involving specialized quadrature for weakly singular kernels can be easily handled. Numerical results illustrate the improved performance of the fast direct solver for the new extended system when compared to the fast direct solver for the original extended system. • A locally perturbed geometry differs from the original in local portion of the boundary. • Local refinement in boundary discretization is also regarded as local perturbation. • Extended systems allow for linearly scaling direct solvers for the perturbed problem. • The solver built for the original problem is reused to solve the perturbed one. [ABSTRACT FROM AUTHOR]

The recently developed quadrature by expansion (QBX) technique [24] accurately evaluates the layer potentials with singular, weakly or nearly singular, or even hyper singular kernels in the integral equation reformulations of partial differential equations. The idea is to form a local complex polynomial or partial wave expansion centered at a point away from the boundary to avoid the singularity in the integrand, and then extrapolate the expansion at points near or even exactly on the boundary. In this paper, in addition to the local complex Taylor polynomial expansion, we derive new representations of the Laplace layer potentials using both the local complex polynomial and plane wave type expansions. Unlike in the QBX, the local complex polynomial expansion in the new quadrature by two expansions (QB2X) method only collects the far-field contributions and its number of expansion terms can be analyzed using tools from the classical fast multipole method (FMM). The plane wave type expansion in the QB2X method is derived by first applying the Fourier extension technique to the density and polynomial approximation of the boundary geometry, and then analytically evaluating the integral using the Residue Theorem with properly chosen complex contour. The plane wave type expansion accurately captures the high frequency properties of the layer potential that are determined (up to a prescribed accuracy) only by the local features of the density function and boundary geometry, and the nonlinear impact of the boundary on the layer potential becomes explicit. The QB2X technique allows high order numerical discretizations and can be adopted easily in existing FMM based fast integral equation solvers. We present preliminary numerical results to validate our analysis and demonstrate the accuracy and efficiency of the QB2X representations when compared with the classical QBX method. [ABSTRACT FROM AUTHOR]

• New numerical method for stochastic homogenization of diffusion equations. • Based on an embedded corrector problem introduced by the authors. • Integral equation formulation, spectral discretization using spherical harmonics, fast multipole method. • Numerical tests illustrating the performance of the approach. This contribution is the numerically oriented companion article of the work [9]. We focus here on the numerical resolution of the embedded corrector problem introduced in [8,9] in the context of homogenization of diffusion equations. Our approach consists in considering a corrector-type problem, posed on the whole space, but with a diffusion matrix which is constant outside some bounded domain. In [9] , we have shown how to define three approximate homogenized diffusion coefficients on the basis of the embedded corrector problem. We have also proved that these approximations all converge to the exact homogenized coefficients when the size of the bounded domain increases. We show here that, under the assumption that the diffusion matrix is piecewise constant, the corrector problem to solve can be recast as an integral equation. In case of spherical inclusions with isotropic materials, we explain how to efficiently discretize this integral equation using spherical harmonics, and how to use the fast multipole method (FMM) to compute the resulting matrix-vector products at a cost which scales only linearly with respect to the number of inclusions. Numerical tests illustrate the performance of our approach in various settings. [ABSTRACT FROM AUTHOR]