More Like this

1. Robust fast direct integral equation solver for three-dimensional doubly periodic scattering problems with a large number of layers

   https://doi.org/10.1016/j.jcp.2023.112573  

   Wu, Bowei; Cho, Min Hyung (December 2023, Journal of Computational Physics)

   Frequency-domain wave scattering problems that arise in acoustics and electromagnetism can be often described by the Helmholtz equation. This work presents a boundary integral equation method for the Helmholtz equation in 3-D multilayered media with many doubly periodic smooth layers. 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 specialized high-order quadrature is developed to discretize the underlying surface integral operators to keep the number of unknowns per layer small. We achieve overall linear computational complexity in the number of layers by reducing the linear system into block tridiagonal form and then solving the system directly via block LU decomposition. The new solver is capable of handling a 100-interface structure with 961.3k unknowns to 10−5 accuracy in less than 2 hours on a desktop workstation. 

   more » « less
2. Learning domain-independent Green’s function for elliptic partial differential equations

   https://doi.org/10.1016/j.cma.2024.116779  

   Negi, Pawan; Cheng, Maggie; Krishnamurthy, Mahesh; Ying, Wenjun; Li, Shuwang (March 2024, Computer Methods in Applied Mechanics and Engineering)

   Lorenzis, Laura (Ed.)

   Green’s function characterizes a partial differential equation (PDE) and maps its solution in the entire domain as integrals. Finding the analytical form of Green’s function is a non-trivial exercise, especially for a PDE defined on a complex domain or a PDE with variable coefficients. In this paper, we propose a novel boundary integral network to learn the domain independent Green’s function, referred to as BIN-G. We evaluate the Green’s function in the BIN-G using a radial basis function (RBF) kernel-based neural network. We train the BIN-G by minimizing the residual of the PDE and the mean squared errors of the solutions to the boundary integral equations for prescribed test functions. By leveraging the symmetry of the Green’s function and controlling refinements of the RBF kernel near the singularity of the Green function, we demonstrate that our numerical scheme enables fast training and accurate evaluation of the Green’s function for PDEs with variable coefficients. The learned Green’s function is independent of the domain geometries, forcing terms, and boundary conditions in the boundary integral formulation. Numerical experiments verify the desired properties of the method and the expected accuracy for the two-dimensional Poisson and Helmholtz equations with variable coefficients. 

   more » « less
3. Data-driven low-rank approximation for the electron-hole kernel and acceleration of time-dependent GW calculations

   https://doi.org/10.1038/s41524-025-01680-9  

   Hou, Bowen; Wu, Jinyuan; Chang_Lee, Victor; Guo, Jiaxuan; Liu, Luna Y; Qiu, Diana Y (December 2025, npj Computational Materials)

   Many-body interactions are essential for understanding non-linear optics and ultrafast spectroscopy of materials. Recent first principles approaches based on nonequilibrium Green’s function formalisms, such as the time-dependent adiabatic GW (TD-aGW) approach, can predict nonequilibrium dynamics of excited states including electron-hole interactions. However, the high-dimensionality of the electron-hole kernel poses significant computational challenges. Here, we develop a data-driven low-rank approximation for the electron-hole kernel, leveraging localized excitonic effects in the Hilbert space of crystalline systems to achieve significant data compression through singular value decomposition (SVD). We show that the subspace of non-zero singular values remains small even as the k-grid grows, ensuring computational tractability with extremely dense k-grids. This low-rank property enables at least 95% data compression and an order-of-magnitude speedup of TD-aGW calculations. Our approach avoids intensive training processes and eliminates time-accumulated errors, seen in previous approaches, providing a general framework for high-throughput, nonequilibrium simulation of light-driven dynamics in materials. 

   more » « less
4. Operator SVD with Neural Networks via Nested Low-Rank Approximation

   Ryu, Jongha; Xu, Xiangxiang; Erol, H. S.; Bu, Yuheng; Zheng, Lizhong; Wornell, Gregory W (December 2023, Proc. NeurIPS 2023 Workshop on Machine Learning and the Physical Sciences (ML4PS))

   This paper proposes an optimization-based method to learn the singular value decomposition (SVD) of a compact operator with ordered singular functions. The proposed objective function is based on Schmidt’s low-rank approximation theorem (1907) that characterizes a truncated SVD as a solution minimizing the mean squared error, accompanied with a technique called nesting to learn the ordered structure. When the optimization space is parameterized by neural networks, we refer to the proposed method as NeuralSVD. The implementation does not require sophisticated optimization tricks unlike existing approaches. 

   more » « less
5. Higher-order Hamilton–Jacobi perturbation theory for anisotropic heterogeneous media: dynamic ray tracing in ray-centred coordinates

   https://doi.org/10.1093/gji/ggab152  

   Iversen, Einar; Ursin, Bjørn; Saksala, Teemu; Ilmavirta, Joonas; de Hoop, Maarten V (May 2021, Geophysical Journal International)

   null (Ed.)

   SUMMARY Dynamic ray tracing is a robust and efficient method for computation of amplitude and phase attributes of the high-frequency Green’s function. A formulation of dynamic ray tracing in Cartesian coordinates was recently extended to higher orders. Extrapolation of traveltime and geometrical spreading was demonstrated to yield significantly higher accuracy—for isotropic as well as anisotropic heterogeneous 3-D models of an elastic medium. This is of value in mapping, modelling and imaging, where kernel operations are based on extrapolation or interpolation of Green’s function attributes to densely sampled 3-D grids. We introduce higher-order dynamic ray tracing in ray-centred coordinates, which has certain advantages: (1) such coordinates fit naturally with wave propagation; (2) they lead to a reduction of the number of ordinary differential equations; (3) the initial conditions are simple and intuitive and (4) numerical errors due to redundancies are less likely to influence the computation of the Green’s function attributes. In a 3-D numerical example, we demonstrate that paraxial extrapolation based on higher-order dynamic ray tracing in ray-centred coordinates yields results highly consistent with those obtained using Cartesian coordinates. Furthermore, in a 2-D example we show that interpolation of dynamic ray tracing quantities along a wavefront can be done with much better consistency in ray-centred coordinates than in Cartesian coordinates. In both examples we measure consistency by means of constraints on the dynamic ray tracing quantities in the 3-D position space and in the 6-D phase space. 

   more » « less