Haomin Zhou

Wiener Chaos expansions and numerical solutions of randomly forced equations of fluid mechanics

In this paper, we propose a numerical method based on Wiener Chaos expansion and apply it to solve the stochastic Burgers and Navier-Stokes equations driven by Brownian motion. The main advantage of the Wiener Chaos approach is that it allows for the separation of random and deterministic effects in a rigorous and effective manner. The separation principle effectively reduces a stochastic equation to its associated propagator, a system of deterministic equations for the coefficients of the Wiener Chaos expansion. Simple formulas for statistical moments of the stochastic solution are presented. These formulas only involve the solutions of the propagator. We demonstrate that for short time solutions the numerical methods based on the Wiener Chaos expansion are more efficient and accurate than those based on the Monte Carlo simulations.

Total Variation Wavelet Inpainting

We consider the problem of filling in missing or damaged wavelet coefficients due to lossy image transmission or communication. The task is closely related to classical inpainting problems, but also remarkably differs in that the inpainting regions are in the wavelet domain. New challenges include that the resulting inpainting regions in the pixel domain are usually not geometrically well defined, as well as that degradation is often spatially inhomogeneous. We propose two related variational models to meet such challenges, which combine the total variation (TV) minimization technique with wavelet representations. The associated Euler-Lagrange equations lead to nonlinear partial differential equations (PDE’s) in the wavelet domain, and proper numerical algorithms and schemes are designed to handle their computation. The proposed models can have effective and automatic control over geometric features of the inpainted images including sharp edges, even in the presence of substantial loss of wavelet coefficients, including in the low frequencies. Existence and uniqueness of the optimal inpaintings are also carefully investigated.

Total variation improved wavelet thresholding in image compression

In this paper, we propose using partial differential equation (PDE) techniques in wavelet based image processing to reduce edge artifacts generated by wavelet thresholding. We employ minimization techniques, in particular the minimization of total variation, to modify the retained standard wavelet coefficients so that the reconstructed images have less oscillations near edges. Numerical experiments show that this approach improves the reconstructed image quality in wavelet compression and in denoising.

ENO-Wavelet Transforms for Piecewise Smooth Functions

We have designed an adaptive essentially nonoscillatory (ENO)-wavelet transform for approximating discontinuous functions without oscillations near the discontinuities. Our approach is to apply the main idea from ENO schemes for numerical shock capturing to standard wavelet transforms. The crucial point is that the wavelet coefficients are computed without differencing function values across jumps. However, we accomplish this in a different way than in the standard ENO schemes. Whereas in the standard ENO schemes the stencils are adaptively chosen, in the ENO-wavelet transforms we adaptively change the function and use the same uniform stencils. The ENO-wavelet transform retains the essential properties and advantages of standard wavelet transforms such as concentrating the energy to the low frequencies, obtaining maximum accuracy, maintained up to the discontinuities, and having a multiresolution framework and fast algorithms, all without any edge artifacts. We have obtained a rigorous approximation error bound which shows that the error in the ENO-wavelet approximation depends only on the size of the derivative of the function away from the discontinuities. We will show some numerical examples to illustrate this error estimate.

Total Variation Wavelet Thresholding

We propose using Partial Differential Equation (PDE) techniques in wavelet based image processing to remove noise and reduce edge artifacts generated by wavelet thresholding. We employ a variational framework, in particular the minimization of total variation (TV), to select and modify the retained wavelet coefficients so that the reconstructed images have fewer oscillations near edges while noise is smoothed. Numerical experiments show that this approach improves the reconstructed image quality in wavelet compression and in denoising.

On the Convergence Rate of a Quasi-Newton Method for Inverse Eigenvalue Problems

In this paper, we first note that the proof of the quadratic convergence of the quasi-Newton method as given in Friedland, Nocedal, and Overton [ SIAM J. Numer. Anal., 24 (1987), pp. 634--667] is incorrect. Then we give a correct proof of the convergence.

Total variation regularization for 3D reconstruction in fluorescence tomography: experimental phantom studies

Fluorescence tomography (FT) is depth-resolved three-dimensional (3D) localization and quantification of fluorescence distribution in biological tissue and entails a highly ill-conditioned problem as depth information must be extracted from boundary measurements. Conventionally, L2 regularization schemes that penalize the euclidean norm of the solution and possess smoothing effects are used for FT reconstruction. Oversmooth, continuous reconstructions lack high-frequency edge-type features of the original distribution and yield poor resolution. We propose an alternative regularization method for FT that penalizes the total variation (TV) norm of the solution to preserve sharp transitions in the reconstructed fluorescence map while overcoming ill-posedness. We have developed two iterative methods for fast 3D reconstruction in FT based on TV regularization inspired by Rudin-Osher-Fatemi and split Bregman algorithms. The performance of the proposed method is studied in a phantom-based experiment using a noncontact constant-wave trans-illumination FT system. It is observed that the proposed method performs better in resolving fluorescence inclusions at different depths.

Numerical solution of an inverse medium scattering problem with a stochastic source

This paper is concerned with the inverse medium scattering problem with a stochastic source, the reconstruction of the refractive index of an inhomogeneous medium from the boundary measurements of the scattered field. As an inverse problem, there are two major difficulties in addition to being highly nonlinear: the ill-posedness and the presence of many local minima. To overcome these difficulties, a stable and efficient recursive linearization method has been recently developed for solving the inverse medium scattering problem with a deterministic source. Compared to classical inverse problems, stochastic inverse problems, referred to as inverse problems involving uncertainties, have substantially more difficulties due to randomness and uncertainties. Based on the Wiener chaos expansion, the stochastic problem is converted into a set of decoupled deterministic problems. The strategy developed is a new hybrid method combining the WCE with the recursive linearization method for solving the inverse medium problem with a stochastic source. Numerical experiments are reported to demonstrate the effectiveness of the proposed approach.

GLOBAL OPTIMIZATIONS BY INTERMITTENT DIFFUSION

We propose an intermittent diffusion(ID) method to find global minimizers of a given scalar function g : R → R. The main idea is to add intermittent, instead of continuously diminishing, random perturbations to the gradient flow generated by g, so that the trajectories can quickly escape from the trap of one minimizer and then approach others. During this process, the associated Fokker-Planck equation alternates its type between hyperbolic and parabolic. We prove that by using the ID method one can find, with probability arbitrarily close to 1, a good approximation to the global minimizers in a finite time T provided T is sufficiently large. The convergence rate in probability follows a geometric series. We also prove that for any given finite set of minimizers, a trajectory of the ID method visits an arbitrary small neighborhood of each minimizer with positive probability. Numerical simulations show that the proposed method achieves clear improvements in terms of the time and the frequencies of visiting the global minimizers over some existing global optimization algorithms for many testing problems. School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, U.S. chow@math.gatech.edu Department of Mathematics, Tunghai University, Taichung 40704, Taiwan. tsyang@thu.edu.tw Georgia Institute of Technology, Atlanta, GA 30332, U.S. hmzhou@math.gatech.edu, this author is partially supported by NSF Faculty Early Career Development (CAREER) Award DMS0645266, and DMS-1042998.

A numerical study of void nucleation and growth in a flip chip assembly process

In this study, we develop mathematical models and numerical simulations of void nucleation and growth induced by the chemical reaction in the flip chip package assembly process using a no-flow underfill. During the thermal assembly process, the underfill chemically reacts to the oxidation of solders I/O on the chip, achieving interconnection between chip and substrate. The chemical reaction causes a large number of voids in the thermal reflow process. The voids have been considered as a critical defect, reducing the life of the thermal reliability. This study investigates the mechanism of void nucleation and growth based on classical bubble nucleation theory and bubble dynamics, respectively. This knowledge can provide a theoretical foundation to achieve a void-free assembly process and high reliability performance.