11 citations · 56 across the 26 of their papers we have counts for
27 papers
Fourier Multi-Component and Multi-Layer Neural Networks: Unlocking High-Frequency Potential
Shijun Zhang, Hongkai Zhao, Yimin Zhong +1
The architecture of a neural network and the choice of its activation function are both fundamental to its performance. Equally important is ensuring that these two elements are we…
Structured and Balanced Multi-Component and Multi-Layer Neural Networks
Shijun Zhang, Hongkai Zhao, Yimin Zhong +1
In this work, we propose a balanced multi-component and multi-layer neural network (MMNN) structure to accurately and efficiently approximate functions with complex features, in te…
Why Shallow Networks Struggle to Approximate and Learn High Frequencies
Shijun Zhang, Hongkai Zhao, Yimin Zhong +1
In this work, we present a comprehensive study combining mathematical and computational analysis to explain why a two-layer neural network struggles to handle high frequencies in b…
How much can one learn from a single solution of a PDE?
Hongkai Zhao, Yimin Zhong
Linear evolution PDE , where is a strongly elliptic operator independent of time, is studied as an example to show if one can supe…
How much can one learn a partial differential equation from its solution?
Yuchen He, Hongkai Zhao, Yimin Zhong
In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the…
A data-driven and model-based accelerated Hamiltonian Monte Carlo method for Bayesian elliptic inverse problems
Sijing Li, Cheng Zhang, Zhiwen Zhang +1
In this paper, we consider a Bayesian inverse problem modeled by elliptic partial differential equations (PDEs). Specifically, we propose a data-driven and model-based approach to…
Quantitative PAT with simplified approximation
Hongkai Zhao, Yimin Zhong
The photoacoustic tomography (PAT) is a hybrid modality that combines the optics and acoustics to obtain high resolution and high contrast imaging of heterogeneous media. In this w…
A Dual Iterative Refinement Method for Non-rigid Shape Matching
Rui Xiang, Rongjie Lai, Hongkai Zhao
In this work, a simple and efficient dual iterative refinement (DIR) method is proposed for dense correspondence between two nearly isometric shapes. The key idea is to use dual in…
Efficient and Robust Shape Correspondence via Sparsity-Enforced Quadratic Assignment
Rui Xiang, Rongjie Lai, Hongkai Zhao
In this work, we introduce a novel local pairwise descriptor and then develop a simple, effective iterative method to solve the resulting quadratic assignment through sparsity cont…
A fast algorithm for time-dependent radiative transport equation based on integral formulation
Hongkai Zhao, Yimin Zhong
In this work, we introduce a fast numerical algorithm to solve the time-dependent radiative transport equation (RTE). Our method uses the integral formulation of RTE and applies th…
Solving Phase Retrieval via Graph Projection Splitting
Ji Li, Hongkai Zhao
Phase retrieval with prior information can be cast as a nonsmooth and nonconvex optimization problem. We solve the problem by graph projection splitting (GPS), where the two proxim…
Separability of the kernel function in an integral formulation for anisotropic radiative transfer equation
Kui Ren, Hongkai Zhao, Yimin Zhong
We study in this work an integral formulation for the radiative transfer equation (RTE) in anisotropic media with truncated approximation to the scattering phase function. The inte…
A data-driven approach for multiscale elliptic PDEs with random coefficients based on intrinsic dimension reduction
Sijing Li, Zhiwen Zhang, Hongkai Zhao
We propose a data-driven approach to solve multiscale elliptic PDEs with random coefficients based on the intrinsic low dimension structure of the underlying elliptic differential…
Instability of an inverse problem for the stationary radiative transport near the diffusion limit
Hongkai Zhao, Yimin Zhong
In this work, we study the instability of an inverse problem of radiative transport equation with angularly averaged measurement near the diffusion limit, i.e. the normalized mean…
Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval
Ji Li, Jian-Feng Cai, Hongkai Zhao
We aim to find a solution to a system of quadratic equations of the form , , e.g., the well-known NP-har…
Intrinsic Complexity And Scaling Laws: From Random Fields to Random Vectors
Jennifer Bryson, Hongkai Zhao, Yimin Zhong
Random fields are commonly used for modeling of spatially (or timely) dependent stochastic processes. In this study, we provide a characterization of the intrinsic complexity of a…
A hybrid adaptive phase space method for reflection traveltime tomography
Hongkai Zhao, Yimin Zhong
We present a hybrid imaging method for a challenging travel time tomography problem which includes both unknown medium and unknown scatterers in a bounded domain. The goal is to re…
A hybrid approach to solve the high-frequency Helmholtz equation with source singularity in smooth heterogeneous media
Jun Fang, Jianliang Qian, Leonardo Zepeda-Núñez +1
We propose a hybrid approach to solve the high-frequency Helmholtz equation with point source terms in smooth heterogeneous media. The method is based on the ray-based finite eleme…
Learning Dominant Wave Directions For Plane Wave Methods For High-Frequency Helmholtz Equations
Jun Fang, Jianliang Qian, Leonardo Zepeda-Núñez +1
We present a ray-based finite element method (ray-FEM) by learning basis adaptive to the underlying high-frequency Helmholtz equation in smooth media. Based on the geometric optics…
Modified Virtual Grid Difference for Discretizing the Laplace-Beltrami Operator on Point Clouds
Meng Wang, Shingyu Leung, Hongkai Zhao
We propose a new and simple discretization, named the Modified Virtual Grid Difference (MVGD), for numerical approximation of the Laplace-Beltrami (LB) operator on manifolds sample…
Fast alternating bi-directional preconditioner for the 2D high-frequency Lippmann-Schwinger equation
Leonardo Zepeda-Núñez, Hongkai Zhao
This paper presents a fast iterative solver for Lippmann-Schwinger equation for high-frequency waves scattered by a smooth medium with a compactly supported inhomogeneity. The solv…
Variational Hamiltonian Monte Carlo via Score Matching
Cheng Zhang, Babak Shahbaba, Hongkai Zhao
Traditionally, the field of computational Bayesian statistics has been divided into two main subfields: variational methods and Markov chain Monte Carlo (MCMC). In recent years, ho…
Hamiltonian Monte Carlo Acceleration Using Surrogate Functions with Random Bases
Cheng Zhang, Babak Shahbaba, Hongkai Zhao
For big data analysis, high computational cost for Bayesian methods often limits their applications in practice. In recent years, there have been many attempts to improve computati…
Precomputing Strategy for Hamiltonian Monte Carlo Method Based on Regularity in Parameter Space
Cheng Zhang, Babak Shahbaba, Hongkai Zhao
Markov Chain Monte Carlo (MCMC) algorithms play an important role in statistical inference problems dealing with intractable probability distributions. Recently, many MCMC algorith…