Abstract : Wave propagation is ubiquitous in our daily life, yet computing wave motion efficiently and accurately is still challenging in the high-frequency regime in many practical applications, such as nano-optics, material sciences, and geosciences. This mini-symposium gathers researchers in the field and provides a forum to exchange new ideas on recent theoretical and computational developments in high-frequency wave propagation and optics, as well as significant applications in medical and seismic tomography.
[03696] Learning based on data and numerical solutions for differential equations
Format : Talk at Waseda University
Author(s) :
Jin Cheng (Professor)
Yu Chen (Dr.)
Abstract : Numerical methods for partial differential equations are the important and powerful tols for the engineering problems. The study of numerical methods for PDE is one of the hottest research topics in the applied mathematics and computational mathematics. Based on ideas from data learning, we propose a new method for finding numerical solutions of differential equations effectively, especially for the Helmholtz equations with the high wave numbers. The main idea of this method is to construct the approximation solutions by utilizing the relevant information from the solution we already have, which include the explicit solutions, the measured data from the experiments, the expression of the basic solution and the numerical solution obtained from the numerical experiments, etc. This is also a fast and high-precision numerical algorithm. It is shown that, especially for high-frequency problems, this method provides feasible solutions.
[01633] Butterfly-compressed Hadamard-Babich Integrator for High-Frequency Helmholtz and Maxwell Equations in Inhomogeneous Media
Format : Talk at Waseda University
Author(s) :
Yang Liu (Lawrence Berkeley National Laboratory)
Jian Song (Michigan State University)
Robert Burridge (University of New Mexico)
Jianliang Qian (Michigan State University)
Abstract : We present a butterfly-compressed representation of the Hadamard-Babich (HB) ansatz for the Green's function of the high-frequency Helmholtz and Maxwell equations in smooth inhomogeneous media with arbitrary excitation sources. The proposed algorithm first solves the phase and HB coefficients via eikonal and transport equations using a coarse mesh, and then compresses the resulting HB interactions using several newly developed butterfly algorithms, leading to an optimal CPU complexity for any bounded 3D domains.
[02074] Development and Analysis a higher-order numerical method for Helmholtz equation with high wave number
Format : Talk at Waseda University
Author(s) :
Wenyuan Liao (University of Calgary)
Abstract : Numerical solution of the Helmholtz equation with high wave numbers is a challenging task due to several reasons, such as the large size and indefiniteness of the discrete linear system. In this talk, we introduce a new iterative method which is motivated by the idea of the normal equation method. We then solve the equivalent positive definite normal equation. Combined with acceleration techniques, the new method is efficient and accurate for solving the Helmholtz equation.
[02172] Fixed Angle Inverse Scattering For Velocity
Format : Talk at Waseda University
Author(s) :
Rakesh Rakesh (University of Delaware)
Abstract : An inhomogeneous acoustic medium is probed by a plane wave and the resultant time dependent wave is measured on the boundary of a ball enclosing the inhomogeneous part of the medium. We describe our partial results about the recovery of the velocity of the medium from the boundary measurement. This is a formally determined inverse problem for the wave equation, consisting of the recovery of a non-constant coefficient of the principal part of the operator from the boundary measurement.
[02057] Eulerian PDE methods for complex-valued eikonals in attenuating media
Format : Online Talk on Zoom
Author(s) :
Jiangtao Hu (Chengdu University of Technology)
Jianliang Qian (Michigan State University)
Shingyu Leung (The Hong Kong University of Science and Technology)
Abstract : Seismic waves in earth media usually undergo attenuation. In the regime of high-frequency asymptotics, a complex-valued eikonal is essential for describing wave propagation in attenuating media. Conventionally, it is computed by ray-tracing methods defined by ODEs, but those irregularly distributed results hinder their applications. This talk proposes a unified eulerian PDE for several popular real ray-tracing methods. We also develop a highly accurate numerical scheme using factorization and LxF-WENO scheme.
Davide Bianchi (Harbin Institute of Technology, Shenzhen)
Guanghao Lai (Harbin Institute of Technology, Shenzhen)
Wenbin Li (Harbin Institute of Technology, Shenzhen)
Abstract : We propose a non-stationary iterated network Tikhonov (iNETT) method for the solution of ill-posed inverse problems. The iNETT employs deep neural networks to build a data-driven regularizer, and it avoids the difficult task of estimating the optimal regularization parameter. To achieve the theoretical convergence of iNETT, we introduce uniformly convex neural networks to build the data-driven regularizer. Rigorous theories and detailed algorithms are proposed for the construction of convex and uniformly convex neural networks. In particular, given a general neural network architecture, we prescribe sufficient conditions to achieve a trained neural network which is component-wise convex or uniformly convex; moreover, we provide concrete examples of realizing convexity and uniform convexity in the modern U-net architecture. With the tools of convex and uniformly convex neural networks, the iNETT algorithm is developed and a rigorous convergence analysis is provided. Lastly, we show applications of the iNETT algorithm in 2D computerized tomography, where numerical examples illustrate the efficacy of the proposed algorithm.
[03580] Linearized Inverse Potential Problems at a High Frequency
Format : Talk at Waseda University
Author(s) :
Boxi XU (Shanghai University of Finance and Economics)
Abstract : We investigate the recovery of the potential function from many boundary measurements at a high frequency for linear or nonlinear equations. By considering such a linearized form, we obtain Hölder type stability which is a big improvement over logarithmic stability in low frequencies. Increasing stability bounds for these coefficients contain a Lipschitz term with a factor growing polynomially in terms of the frequency, a Hölder term, and a logarithmic term that decays with respect to the frequency as a power. Based on the linearized problem, a reconstruction algorithm is proposed aiming at the recovery of sufficiently many Fourier modes of the potential function. By choosing the high frequency appropriately, the numerical evidence sheds light on the influence of the growing frequency and confirms the improved resolution.
This is the joint work with Prof. Victor Isakov, Prof. Shuai Lu, Prof. Mikko Salo, and Mr. Sen Zou.
[03600] An Embedding Method for Hyperbolic Conservation Laws on Implicit Surfaces
Format : Talk at Waseda University
Author(s) :
Chun Kit Hung (The Hong Kong University of Science and Technology)
Shingyu Leung (The Hong Kong University of Science and Technology)
Abstract : In this talk, we will present a novel embedding method for solving scalar hyperbolic conservation laws on surfaces. The proposed method represents the interface implicitly through the zero-level set of its signed distance function and introduces a pushforward operator to extend the surface flux function to neighboring level surfaces. By solving an extended conservation law in a tubular neighborhood of the interface, it has been proven that the solution is the constant-normal extension of the surface conservation law. Numerical examples will be presented to demonstrate the accuracy and performance of the proposed method.
