Lippmann-Schwinger Integral Equation Research Articles

Four-body bound states in momentum space: the Yakubovsky approach without two-body t − matrices

This study presents a solution to the Yakubovsky equations for four-body bound states in momentum space, bypassing the common use of two-body t − matrices. Typically, such solutions are dependent on the fully-off-shell two-body t − matrices, which are obtained from the Lippmann-Schwinger integral equation for two-body subsystem energies controlled by the second and third Jacobi momenta. Instead, we use a version of the Yakubovsky equations that does not require t − matrices, facilitating the direct use of two-body interactions. This approach streamlines the programming and reduces computational time. Numerically, we found that this direct approach to the Yakubovsky equations, using 2B interactions, produces four-body binding energy results consistent with those obtained from the conventional t − matrix dependent Yakubovsky equations, for both separable (Yamaguchi and Gaussian) and non-separable (Malfliet-Tjon) interactions.

An eigenstrain-based micromechanical model for homogenization of elastic multiphase/multilayer composites

This paper presents a novel micromechanical procedure for the linear elastic homogenization of composites with periodic microstructures. The procedure is developed for composites with arbitrary number of phases and geometric shapes of the inhomogeneities, in contrast with most existing homogenization approaches. Also, no restriction is made in relation to the mismatch between the properties of the phases and volume fractions of the inhomogeneities. The proposed procedure is based on the Eshelby equivalent inclusion approach and extends a model originally derived for evaluating the effective elastic moduli of periodic two-phase composites. The procedure represents the fluctuating elastic fields within each multiphase repeating unit cell (RUC) using Fourier series, resulting in Lippmann-Schwinger integral equations governing the unknown eigenstrain fields of the inclusions. Unlike traditional iterative algorithms used in Fast Fourier Transform (FFT)-based approaches, the procedure solves the integral equations straightforwardly from a scheme of partition of the domain of each inclusion. The efficiency of the proposal procedure is demonstrated through applications to composites with different arrays of coated fibers and constituent materials.

Boundary-volume Lippmann Schwinger formulation and fast iteration schemes for numerical homogenization of conductive composites. Cases of arbitrary contrasts and Kapitza interface

In this paper, we present a boundary-volume based Lippmann Schwinger integral equation for numerical homogenization problems of heterogeneous conductive materials with arbitrary contrast ratio and imperfect interface behavior. It is shown that the interior temperature gradient within each homogeneous phase is connected to the material dissimilarity quantities at all boundaries between different phases. The Kapitza interface model can also be included in the formulation. The basic FFT iteration schemes converge fast even for infinite contrast in the case of usual interface conditions. In the case of Kapitza’s interface, Conjugate Gradient schemes based on the new formulation converge fast and yield accurate results when compared with standard Finite Element Method.

Inverse Random Potential Scattering for Elastic Waves

This paper is concerned with the inverse elastic scattering problem for a random potential in three dimensions. Interpreted as a distribution, the potential is assumed to be a microlocally isotropic Gaussian random field whose covariance operator is a classical pseudodifferential operator. Given the potential, the direct scattering problem is shown to be well-posed in the sense of distributions by studying the equivalent Lippmann–Schwinger integral equation. For the inverse scattering problem, we demonstrate that the microlocal strength of the random potential can be uniquely determined with probability one by a single realization of the high frequency limit of the averaged compressional or shear backscattered far-field pattern of the scattered wave. The analysis employs the integral operator theory, the Born approximation in the high frequency regime, the microlocal analysis for the Fourier integral operators, and the ergodicity of the wave field.

Fourier transform approach to homogenization of frequency-dependent heat transfer in porous media

Purpose The purpose of this paper is to solve the local problem involving strong contrast heterogeneous conductive material, with application to gas-filled porous media with both perfect and imperfect Kapitza boundary conditions at the bi-material interface. The effective parameters like the dynamic conductivity and the thermal permeability in the acoustics of porous media are also derived from the cell solution. Design/methodology/approach The Fourier transform method is used to solve frequency-dependent heat transfer problems. The periodic Lippmann–Schwinger integral equation in Fourier space with source term is first formulated using discrete Green operators and modified wavevectors, which can then be solved by iteration schemes. Findings Numerical examples show that the schemes converge fast and yield accurate results when compared with analytical solution for benchmark problems. Originality/value The formulation of the method is constructed using static and dynamic Green operators and can be applied to pixelized microstructure issued from tomography images.

On extension of the data driven ROM inverse scattering framework to partially nonreciprocal arrays

Data-driven reduced order models (ROMs) recently emerged as powerful tool for the solution of inverse scattering problems. The main drawback of this approach is that it was limited to measurement arrays with reciprocally collocated transmitters and receivers, that is, square symmetric matrix (data) transfer functions. To relax this limitation, we use our previous work Druskin et al (2021 Inverse Problems 37 075003), where the ROMs were combined with the Lippmann–Schwinger integral equation to produce a direct nonlinear inversion method. In this work we extend this approach to more general transfer functions, including those that are non-symmetric, e.g., obtained by adding only receivers or sources. The ROM is constructed based on the symmetric subset of the data and is used to construct all internal solutions. Remaining receivers are then used directly in the Lippmann–Schwinger equation. We demonstrate the new approach on a number of 1D and 2D examples with non-reciprocal arrays, including a single input/multiple outputs inverse problem, where the data is given by just a single-row matrix transfer function. This allows us to approach the flexibility of the Born approximation in terms of acceptable measurement arrays; at the same time significantly improving the quality of the inversion compared to the latter for strongly nonlinear scattering effects.

An accelerated, high-order accurate direct solver for the Lippmann–Schwinger equation for acoustic scattering in the plane

An efficient direct solver for solving the Lippmann–Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem with N degrees of freedom, the solver constructs an approximate inverse in $\mathcal {O}(N^{3/2})$ operations and then, given an incident field, can compute the scattered field in $\mathcal {O}(N \log N)$ operations. The solver is based on a previously published direct solver for integral equations that relies on rank-deficiencies in the off-diagonal blocks; specifically, the so-called Hierarchically Block Separable format is used. The particular solver described here has been reformulated in a way that improves numerical stability and robustness, and exploits the particular structure of the kernel in the Lippmann–Schwinger equation to accelerate the computation of an approximate inverse. The solver is coupled with a Nyström discretization on a regular square grid, using a quadrature method developed by Ran Duan and Vladimir Rokhlin that attains high-order accuracy despite the singularity in the kernel of the integral equation. A particularly efficient solver is obtained when the direct solver is run at four digits of accuracy, and is used as a preconditioner to GMRES, with each forwards application of the integral operators accelerated by the FFT. Extensive numerical experiments are presented that illustrate the high performance of the method in challenging environments. Using the 10th-order accurate version of the Duan–Rokhlin quadrature rule, the scheme is capable of solving problems on domains that are over 500 wavelengths wide to relative error below 10− 10 in a couple of hours on a workstation, using 26M degrees of freedom.

A fast rapidly convergent method for approximation of convolutions with applications to wave scattering and some other problems

In this article, we discuss an O(Nlog⁡N) rapidly convergent algorithm for the numerical approximation of the convolution integral with weakly singular kernels and compactly supported densities with possible jump discontinuities. To achieve the reduced computational complexity, we utilize the Fast Fourier Transform (FFT) on a uniform grid of size N for approximating the convolution. To facilitate this and maintain the accuracy, we primarily rely on a periodic Fourier extension of the density with a suitably large period depending on the support of the density. While the method's convergence rate improves with increasing smoothness of the periodic extension and, in fact, approximations exhibit super-algebraic convergence when the extension is infinitely differentiable, it converges only linearly when the density has jump discontinuities. In this context, we present two different procedures to enhance the convergence speed. Firstly, we utilize a certain Fourier smoothing technique to accelerate the convergence to achieve the quadratic rate in the overall approximation. Finally, to make the method truly high order, we augment the basic scheme by including a “thin” boundary grid and employing a specialized high-order boundary integrator. We validate its performance in terms of accuracy as well as computational efficiency through a variety of numerical experiments. In particular, to demonstrate the method's utility, we apply the integration scheme for the numerical solution of certain partial differential equations. Moreover, we also apply the quadrature to obtain a fast and high-order Nyström solver for the solution of the Lippmann-Schwinger integral equation.

Lippmann–Schwinger–Lanczos algorithm for inverse scattering problems

Data-driven reduced order models (ROMs) are combined with the Lippmann–Schwinger integral equation to produce a direct nonlinear inversion method. The ROM is viewed as a Galerkin projection and is sparse due to Lanczos orthogonalization. Embedding into the continuous problem, a data-driven internal solution is produced. This internal solution is then used in the Lippmann–Schwinger equation, thus making further iterative updates unnecessary. We show numerical experiments for spectral domain domain data for which our inversion is far superior to the Born inversion and works as well as when the true internal solution is known.

Direct and inverse scalar scattering problems for the Helmholtz equation in ℝ m

Abstract The boundary value problem for the Helmholtz equation in the m-dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes.

An integral equation method for the Helmholtz problem in the presence of small anisotropic inclusions

We consider the Helmholtz problem with source term in an anisotropic domain of . The aim of this paper is to investigate the interplay between the geometry and analysis of elliptic equations under small perturbation of domain. The solving of this problem, anisotropic as well as isotropic case, is based on integral equations. We exhibit the Lippmann-Schwinger integral equation in the presence of finite number of anisotropic inclusions of small diameter. We derive some results for convergence estimates.

Fourier Bases-Expansion Contraction Integral Equation for Inversion Highly Nonlinear Inverse Scattering Problem

The electromagnetic inverse scattering problems (ISPs) have counted severe multiple scattering effects, especially those for strong scatterers (the ones with high contrasts and/or electrically large dimensions). Iterative inversion methods based on the model of the Lippmann–Schwinger integral equation (LSIE) usually fail to reconstruct such highly nonlinear ISPs. In this article, in virtue of a new recently established contraction integral equation for inversion (CIE-I), we propose a Fourier bases-expansion with CIE-I (FBE-CIE-I) inversion method to handle highly nonlinear ISPs via the contrast source inversion (CSI) scheme. In the FBE-CIE-I, the unknown contrast source is spanned within the Fourier bases, and the multi-round optimization scheme is adopted by choosing the proper number of low-frequency components; not only the model of CIE-I can be stabilized well but also the computational cost can be further reduced compared with the traditional CSI. Besides, compared with the previous inversion method based on LSIE, the resolvability and robustness of the FBE-CIE-I could be significantly enhanced in handling highly nonlinear ISPs. Numerical and experimental tests verify the interests and efficiency of the proposed method.

A data-driven self-consistent clustering analysis for the progressive damage behavior of 3D braided composites

A data-driven self-consistent clustering analysis (SCA) method is applied to investigate the progressive damage behavior of 3D braided composites. The SCA-based method is split into the offline stage and the online stage. In the offline stage, the high fidelity RVE is compressed into a reduced RVE composed of several clusters. In the online stage, the mechanical responses are calculated by solving the discretized Lippmann-Schwinger integral equation. To validate the accuracy of proposed model, the SCA-based simulation is compared with the corresponding experiments and finite element analysis (FEA). The results show that the SCA method can accurately capture the stress and damage distribution, and the predictive stiffness and strength agree well with experimental data. More importantly, with the same constitutive laws and geometric model, SCA only takes a few hundred seconds, which is 1771 times faster than FEA. Because of the high efficiency, the SCA has the potential to be applied in concurrent multiscale analysis for braided composites.

Solving Phaseless Highly Nonlinear Inverse Scattering Problems With Contraction Integral Equation for Inversion

At high frequencies, phase measurement is difficult to meet the accuracy required for imaging, and it is more susceptible to noise pollution. Consequently, it is important to develop inversion models and algorithms for electromagnetic inverse scattering problems with phaseless data (PD-ISPs). Compared to the ISPs with full-data (FD-ISPs), the PD-ISPs are more nonlinear due to the lack of the phase information. Aiming at solving highly nonlinear PD-ISPs, in this article we present a contraction integral equation for inversion (CIE-I) regularized with Fourier subspace via contrast source inversion method, denoted as PD-CSI-CIE-I. Under the model of CIE-I, the nonlinearity of PD-ISPs can be effectively alleviated. In order to stabilize CIE-I, a nested optimization scheme with multi-round procedure is adopted by choosing the proper number of low-frequency components from unknown induced current spanned in the Fourier subspace. By the Fourier bases expansion of induced current, the computational cost can be further reduced. Numerical and experimental examples validate the efficiency of the proposed inversion method. Futher, it is shown that the proposed PD-CSI-CIE-I has a stronger inversion capability than the one with the Lippmann-Schwinger integral equation (LSIE) when tackling high contrast scatterers with phaseless data.

The two-step method for determining a piecewise-continuous refractive index of a 2D scatterer by near field measurements

The problem of determining an unknown piecewise-continuous refractive index of an inhomogeneous scatterer is considered. The boundary value problem is reduced to the Lippmann–Schwinger integral equation. The solution of the inverse problem is obtained in two steps. At the first step, the integral equation of the first kind is solved in the inhomogeneity region using measured values of the total field outside the region. It is proved that the solution to the integral equation of the first kind is unique in the class of piecewise constant functions. At the second step, the unknown refractive index is explicitly calculated via the found solution of the integral equation and the given total field. The proposed method was implemented and verified by solving a test inverse problem with a given refractive index. Efficiency of the two-step method was approved by comparison between the exact solution and the approximate ones.

Fast Microwave Through Wall Imaging Method With Inhomogeneous Background Based on Levenberg–Marquardt Algorithm

In this paper, a fast solution for microwave through wall imaging (TWI) with nonlinear inversion is proposed to reconstruct the unknown targets embedded in an inhomogeneous background medium. We treat inhomogeneous background, i.e., the wall around bounded in a finite domain as a known scatterer, which has the advantage of avoiding the time-consuming calculation of inhomogeneous background Green's function. Under this scheme, a new approach under the framework of difference integral equation model, i.e., difference Lippmann-Schwinger integral equation, with modified enhanced Levenberg-Marquardt algorithm is proposed. In particular, we used a hybrid regularized technique, i.e., generalized cross-validation and truncated singular value decomposition, to stabilize the inversion. It is shown that the proposed method runs fast and is stable in presence of noise. Also, it is able to alleviate the nonlinearity and reconstruct unknown scatterers of high contrast with respect to the background. Both the numerical and experimental TWI tests validate the efficiency of the proposed inversion method.

Contraction Integral Equation for Three-Dimensional Electromagnetic Inverse Scattering Problems.

Inverse scattering problems (ISPs) stand at the center of many important imaging applications, such as geophysical explorations, industrial non-destructive testing, bio-medical imaging, etc. Recently, a new type of contraction integral equation for inversion (CIE-I) has been proposed to tackle the two-dimensional electromagnetic ISPs, in which the usually employed Lippmann–Schwinger integral equation (LSIE) is transformed into a new form with a modified medium contrast via a contraction mapping. With the CIE-I, the multiple scattering effects, i.e., the physical reason for the nonlinearity in the ISPs, is substantially suppressed in estimating the modified contrast, without compromising physical modeling. In this paper, we firstly propose to implement this new CIE-I for the three-dimensional ISPs. With the help of the FFT type twofold subspace-based optimization method (TSOM), when handling the highly nonlinear problems with strong scatterers, those with higher contrast and/or larger dimensions (in terms of wavelengths), the performance of the inversions with CIE-I is much better than the ones with the LSIE, wherein inversions usually converge to local minima that may be far away from the solution. In addition, when handling the moderate scatterers (those the LSIE modeling can still handle), the convergence speed of the proposed method with CIE-I is much faster than the one with the LSIE. Secondly, we propose to relax the contraction mapping condition, i.e., different contraction mappings are used in updating contrast sources and contrast, and we find that the convergence can be further accelerated. Several numerical tests illustrate the aforementioned interests.

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

In recent years, several fast solvers for the solution of the Lippmann–Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nyström scheme (Anand et al., 2016 [22]) that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of O(Nlog⁡N) for an N-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin (2009) [13] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde (2004) [12] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nyström solver in Anand et al. (2016) [22] through a wide range of numerical experiments.

Momentum space calculations of the binding energies of argon dimer

The binding energies of argon dimer are calculated by solving the homogeneous Lippmann-Schwinger integral equation in momentum space. Our numerical analysis using two models of argon-argon interaction developed by Patkowski et al. not only confirms the eight argon dimer vibrational levels of the ground state of argon dimer (ie, for j = 0) predicted by other groups but also provides a very precise means for determining the binding energy of the ninth state which its value is a matter of discussion. Our calculations have been also extended to states with higher rotational quantum number j and we have calculated the energy of all 174 bound states for both potential models. Our numerical results for vibrational levels of the ground state of argon dimer are in excellent agreement with other theoretical calculations and available experimental data.

Quantitative Analysis of Near‐Surface Seismological Complexity Based on the Generalized Lippmann–Schwinger Matrix Equation

Near-surface complexity is characteristic of rugged topographies and strong volume heterogeneities, which significantly affects seismic data recorded at the free surface. An efficient method is presented here to quantitatively analyze the nearsurface seismological complexity. We use a boundary–volume element numerical method to discretize the generalized Lippmann–Schwinger integral equation formulated in terms of volume scattering and boundary scattering. The integral equation technique provided sufficient accuracy to model rough topographies, large-scale structures, and volume heterogeneities in a straightforward and efficient manner. Rather than solving the resultant global matrix equation at a very high computational cost, certain characteristic parameters of the scattering amplitude matrix were estimated and used as indicators to assess the near-surface seismological complexity. Applications to numerous examples validated the method’s applicability, reliability, and efficiency, whereby the resultant matrix characteristic parameters varied consistently with the near-surface complexity in geometric structures, volume heterogeneities, and computational frequencies.