We introduce a numerical method for the solution of the time-dependent Schrodinger equation with a smooth potential, based on its reformulation as a Volterra integral equation. We present versions of the method both for periodic boundary conditions, and for free space problems with compactly supported initial data and potential. A spatially uniform electric field may be included, making the solver applicable to simulations of light-matter interaction.
The primary computational challenge in using the Volterra formulation is the application of a space-time history dependent integral operator. This may be accomplished by projecting the solution onto a set of Fourier modes, and updating their coefficients from one time step to the next by a simple recurrence. In the periodic case, the modes are those of the usual Fourier series, and the fast Fourier transform (FFT) is used to alternate between physical and frequency domain grids. In the free space case, the oscillatory behavior of the spectral Green's function leads us to use a set of complex-frequency Fourier modes obtained by discretizing a contour deformation of the inverse Fourier transform, and we develop a corresponding fast transform based on the FFT.
Our approach is related to pseudo-spectral methods, but applied to an integral rather than the usual differential formulation. This has several advantages: it avoids the need for artificial boundary conditions, admits simple, inexpensive high-order implicit time marching schemes, and naturally includes time-dependent potentials. We present examples in one and two dimensions showing spectral accuracy in space and eighth-order accuracy in time for both periodic and free space problems.
Based on the perfectly matched layer (PML) technique, this paper develops a high-accuracy boundary integral equation (BIE) solver for acoustic scattering problems in locally defected layered media in both two and three dimensions. The original scattering problem is truncated onto a bounded domain by the PML. Assuming the vanishing of the scattered field on the PML boundary, we derive BIEs on local defects only in terms of using PML-transformed free-space Green's function, and the four standard integral operators: single-layer, double-layer, transpose of double-layer, and hyper-singular boundar…
This paper focuses on fast direct solvers for integral equations in the low-to-moderate-frequency regime obtained by leveraging preconditioned first kind or second kind operators regularized with Laplacian filters. The spectral errors arising from boundary element discretizations are properly handled by filtering that, in addition, allows for the use of low-rank representations for the compact perturbations of all operators involved. Numerical results show the effectiveness of the approaches and their effectiveness in the direct solution of integral equations.
We present a framework using the Quantized Tensor Train (QTT) decomposition to accurately and efficiently solve volume and boundary integral equations in three dimensions. We describe how the QTT decomposition can be used as a hierarchical compression and inversion scheme for matrices arising from the discretization of integral equations. For a broad range of problems, computational and storage costs of the inversion scheme are extremely modest $O(\log N)$ and once the inverse is computed, it can be applied in $O(N \log N)$. We analyze the QTT ranks for hierarchically low rank matrices and dis…
International Center for Theoretical Sciences, Tata Institute of Fundamental Research（塔塔基础研究院国际理论科学中心）；Florida Institute of Technology（佛罗里达理工学院）；University of Florida（佛罗里达大学）
We use a Schrödinger wave equation formalism to solve the eikonal equation. In our framework, a solution to the eikonal equation is obtained in the limit as Planck's constant $\hbar$ (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrödinger equation. The Schrödinger equation corresponding to the eikonal turns out to be a \emph{generalized, screened Poisson equation}. Despite being linear, it does not have a closed-form solution for arbitrary forcing functions. We present two different techniques to solve the screened Poisson equation. In the first appr…
An O(N) Direct Solver for Integral Equations on the Plane
Courant Institute of Mathematical Sciences, New York University（纽约大学柯朗数学科学研究所）；Department of Applied Mathematics, University of Colorado at Boulder（科罗拉多大学博尔德分校应用数学系）
Fast and high-order accurate algorithms for three dimensional elastic scattering are of great importance when modeling physical phenomena in mechanics, seismic imaging, and many other fields of applied science. In this paper, we develop a novel boundary integral formulation for the three dimensional elastic scattering based on the Helmholtz decomposition of elastic fields, which converts the Navier equation to a coupled system consisted of Helmholtz and Maxwell equations. An FFT-accelerated separation of variables solver is proposed to efficiently invert boundary integral formulations of the c…
In transient simulations of particulate Stokes flow, to accurately capture the interaction between the constituent particles and the confining wall, the discretization of the wall often needs to be locally refined in the region approached by the particles. Consequently, standard fast direct solvers lose their efficiency since the linear system changes at each time step. This manuscript presents a new computational approach that avoids this issue by pre-constructing a fast direct solver for the wall ahead of time, computing a low-rank factorization to capture the changes due to the refinement, …
A Butterfly-Based Direct Integral Equation Solver Using Hierarchical LU Factorization for Analyzing Scattering from Electrically Large Conducting Objects
Department of Electrical Engineering and Computer Science, University of Michigan（密歇根大学电气与计算机工程系）；University of Michigan（密歇根大学）；Department of Microwave Engineering, University of Electronic Science and Technology of China（电子科技大学微波工程系）；University of Electronic Science and Technology of China（电子科技大学）
This paper introduces a new method for discretizing and solving integral equation formulations of Maxwell's equations which achieves spectral accuracy for smooth surfaces. The approach is based on a hybrid Nyström-collocation method using Chebyshev polynomials to expand the unknown current densities over curvilinear quadrilateral surface patches. As an example, the proposed strategy is applied the to Magnetic Field Integral Equation (MFIE) and the N-Müller formulation for scattering from metallic and dielectric objects, respectively. The convergence is studied for several different geometries,…
A butterfly-accelerated volume integral equation (VIE) solver is proposed for fast and accurate electromagnetic (EM) analysis of scattering from heterogeneous objects. The proposed solver leverages the hierarchical off-diagonal butterfly (HOD-BF) scheme to construct the system matrix and obtain its approximate inverse, used as a preconditioner. Complexity analysis and numerical experiments validate the $O(N\log^2N)$ construction cost of the HOD-BF-compressed system matrix and $O(N^{1.5}\log N)$ inversion cost for the preconditioner, where $N$ is the number of unknowns in the high-frequency EM …
A boundary integral equation method for the 3-D Helmholtz equation in multilayered media with many quasi-periodic layers is presented. Compared with conventional quasi-periodic Green's function method, the new method is robust at all scattering parameters. A periodizing scheme is used to decompose the solution into near- and far-field contributions. The near-field contribution uses the free-space Green's function in an integral equation on the interface in the unit cell and its immediate eight neighbors; the far-field contribution uses proxy point sources that enclose the unit cell. A speciali…
On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions
Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（南锡-大东部国家研究与创新研究所）；Institut Elie Cartan de Lorraine（洛林艾利嘉当研究所）；Laboratoire de Mathématiques Raphaël Salem（拉斐尔萨莱姆数学实验室）；Université de Rouen（鲁昂大学）；Université de Rennes 1（雷恩第一大学）；IRMAR（雷恩数学与应用研究混合研究中心）；University Wien（维也纳大学）；Wolfgang Pauli Institute（沃尔夫冈·泡利研究所）
High-performance Parallel Solver for Integral Equations of Electromagnetics Based on Galerkin Method
SUNY New Paltz（纽约州立大学新帕尔茨分校）；Faculty of Computational Mathematics and Cybernetics, Lomonosov MSU（莫斯科国立大学计算数学与控制论系）；Institute of Geophysics, ETH Zurich（苏黎世联邦理工学院地球物理研究所）
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…
In this paper, we present a GPU-accelerated direct-sum boundary integral method to solve the linear Poisson-Boltzmann (PB) equation. In our method, a well-posed boundary integral formulation is used to ensure the fast convergence of Krylov subspace based linear algebraic solver such as the GMRES. The molecular surfaces are discretized with flat triangles and centroid collocation. To speed up our method, we take advantage of the parallel nature of the boundary integral formulation and parallelize the schemes within CUDA shared memory architecture on GPU. The schemes use only $11N+6N_c$ size-of-…
We present an accelerated and hardware parallelized integral-equation solver for the problem of acoustic scattering by a two-dimensional surface in three-dimensional space. The approach is based, in part, on the novel Interpolated Factored Green Function acceleration method (IFGF) that, without recourse to the Fast Fourier Transform (FFT), evaluates the action of Green function-based integral operators for an $N$-point surface discretization at a complexity of $\Ord(N\log N)$ operations instead of the $\Ord(N^2)$ cost associated with nonaccelerated methods. The IFGF algorithm exploits the slow…
We present a fast direct solver for boundary integral equations on complex surfaces in three dimensions using an extension of the recently introduced recursive strong skeletonization scheme. For problems that are not highly oscillatory, our algorithm computes an ${LU}$-like hierarchical factorization of the dense system matrix, permitting application of the inverse in $\mathcal O(n)$ time, where $n$ is the number of unknowns on the surface. The factorization itself also scales linearly with the system size, albeit with a somewhat larger constant. The scheme is built on a level-restricted adapt…
The Ohio State University（俄亥俄州立大学）；Binghamton University（宾汉姆顿大学）；University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）；Virginia Tech（弗吉尼亚理工大学）
Given their ability to effectively learn non-linear mappings and perform fast inference, deep neural networks (NNs) have been proposed as a viable alternative to traditional simulation-driven approaches for solving high-dimensional eigenvalue equations (HDEs), which are the foundation for many scientific applications. Unfortunately, for the learned models in these scientific applications to achieve generalization, a large, diverse, and preferably annotated dataset is typically needed and is computationally expensive to obtain. Furthermore, the learned models tend to be memory- and compute-inte…
The paper introduces a very simple and fast computation method for high-dimensional integrals to solve high-dimensional Kolmogorov partial differential equations (PDEs). The new machine learning-based method is obtained by solving a stochastic weighted minimization with stochastic gradient descent which is inspired by a high-order weak approximation scheme for stochastic differential equations (SDEs) with Malliavin weights. Then solutions to high-dimensional Kolmogorov PDEs or expectations of functionals of solutions to high-dimensional SDEs are accurately approximated without suffering from t…
A fast direct solver for scattering from periodic structures with multiple material interfaces in two dimensions
Courant Institute, New York University（纽约大学库朗数学科学研究所）；Department of Mathematics, Stanford University（斯坦福大学数学系）；Department of Mathematics, Ewha Womans University（梨花女子大学数学系）
From multi-layered problems to multiple two-layered problems: a novel frequency-time hybrid multiple-scattering integral equation solver
University of Electronic Science and Technology of China（电子科技大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；Nanyang Technological University（南洋理工大学）
The integral equation approach to partial differential equations (PDEs) provides significant advantages in the numerical solution of the incompressible Navier-Stokes equations. In particular, the divergence-free condition and boundary conditions are handled naturally, and the ill-conditioning caused by high order terms in the PDE is preconditioned analytically. Despite these advantages, the adoption of integral equation methods has been slow due to a number of difficulties in their implementation. This work describes a complete integral equation-based flow solver that builds on recently develo…
$O(N)$ Iterative and $O(NlogN)$ Fast Direct Volume Integral Equation Solvers with a Minimal-Rank ${\cal H}^2$-Representation for Large-Scale $3$-D Electrodynamic Analysis
School of Electrical and Computer Engineering, Purdue University（普渡大学电气与计算机工程学院）
Well-posed boundary integral equation formulations and Nyström discretizations for the solution of Helmholtz transmission problems in two-dimensional Lipschitz domains
University of New Hampshire（新罕布什尔大学）；New Jersey Institute of Technology（新泽西理工学院）
