arXiv subjects

Yat Tin Chow

Publications and source records attributed to Yat Tin Chow.

36 records · Page 2Linked to original sources

arXiv2020

Quantum ergodicity and localization of plasmon resonances

We are concerned with the geometric properties of the surface plasmon resonance (SPR). SPR is a non-radiative electromagnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface. It is known that the SPR oscillation is very sensitive to the material interface. However, we show that the SPR oscillation asymptotically localizes at places with high magnitude of curvature in a certain sense. Our work leverages the Heisenberg picture of quantization and quantum ergodicity first derived by Shnirelman, Zelditch, Colin de Verdière and Helffer-Martinez-Robert, as well as certain novel and more general ergodic properties of the Neumann-Poincaré operator to analyze the SPR field, which are of independent interest to the spectral theory and the potential theory.

math.AP↗

arXiv2020

A Direct Sampling Method for Simultaneously Recovering Inhomogeneous Inclusions of Different Nature

In this work, we investigate a class of elliptic inverse problems and aim to simultaneously recover multiple inhomogeneous inclusions arising from two different physical parameters, using very limited boundary Cauchy data collected only at one or two measurement events. We propose a new fast, stable and highly parallelable direct sampling method (DSM) for the simultaneous reconstruction process. Two groups of probing and index functions are constructed, and their desired properties are analyzed. In order to identify and decouple the multiple inhomogeneous inclusions of different physical nature, we introduce a new concept of mutually almost orthogonality property that generalizes the important concept of almost orthogonality property in classical DSMs for inhomogeneous inclusions of same physical nature. With the help of this new concept, we develop a reliable strategy to distinguish two different types of inhomogeneous inclusions with noisy data collected at one or two measurement events. We further improve the decoupling effect by choosing an appropriate boundary influx. Numerical experiments are presented to illustrate the robustness and efficiency of the proposed method.

math.NA↗

arXiv2020

On the zeroth law of turbulence for the stochastically forced Navier-Stokes equations

We consider three-dimensional stochastically forced Navier-Stokes equations subjected to white-in-time (colored-in-space) forcing in the absence of boundaries. Upper and lower bounds of the mean value of the time-averaged energy dissipation rate, $\mathbb{E} [\langle\varepsilon \rangle] $, are derived directly from the equations. First, we show that for a weak (martingale) solution to the stochastically forced Navier-Stokes equations, \[ \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L},\] where $G^2$ is the total energy rate supplied by the random force, $U$ is the root-mean-square velocity, $L$ is the longest length scale in the applied forcing function, and $Re$ is the Reynolds number. Under an additional assumption of energy equality, we also derive a lower bound if the energy rate given by the random force dominates the deterministic behavior of the flow in the sense that $G^2 > 2 F U$, where $F$ is the amplitude of the deterministic force. We obtain, \[\frac{1}{3} G^2 - \frac{1}{3} (2+ \frac{1}{Re})\frac{U^3}{L} \leq \mathbb{E} [\langle\varepsilon \rangle] \leq G^2 + (2+ \frac{1}{Re})\frac{U^3}{L}\,.\] In particular, under such assumptions, we obtain the zeroth law of turbulence in the absence of the deterministic force as, \[\mathbb{E} [\langle\varepsilon \rangle] = \frac{1}{2} G^2.\] Besides, we also obtain variance estimates of the dissipation rate for the model.

math.AP↗

arXiv2019

Multilevel Optimal Transport: a Fast Approximation of Wasserstein-1 distances

We propose a fast algorithm for the calculation of the Wasserstein-1 distance, which is a particular type of optimal transport distance with homogeneous of degree one transport cost. Our algorithm is built on multilevel primal-dual algorithms. Several numerical examples and a complexity analysis are provided to demonstrate its computational speed. On some commonly used image examples of size $512\times512$, the proposed algorithm gives solutions within $0.2\sim 1.5$ seconds on a single CPU, which is much faster than the state-of-the-art algorithms.

stat.CO↗

arXiv2018

Optimal Mesh Size for Inverse Medium Scattering Problems

An optimal mesh size of the sampling region can help to reduce computational burden in practical applications. In this work, we investigate optimal choices of mesh sizes for the identifications of medium obstacles from either the far-field or near-field data in two and three dimensions. The results would have applications in the reconstruction process of inverse scattering problems.

math.NA↗

arXiv2018

Optimal Human Navigation in Steep Terrain: a Hamilton-Jacobi-Bellman Approach

We present a method for determining optimal walking paths in steep terrain using the level set method and an optimal control formulation. By viewing the walking direction as a control variable, we can determine the optimal control by solving a Hamilton-Jacobi-Bellman equation. We then calculate the optimal walking path by solving an ordinary differential equation. We demonstrate the effectiveness of our method by computing optimal paths which travel throughout mountainous regions of Yosemite National Park. We include details regarding the numerical implementation of our model and address a specific application of a law enforcement agency patrolling a nationally protected area.

math.OC↗

arXiv2018

Algorithm for Hamilton-Jacobi equations in density space via a generalized Hopf formula

We design fast numerical methods for Hamilton-Jacobi equations in density space (HJD), which arises in optimal transport and mean field games. We overcome the curse-of-infinite-dimensionality nature of HJD by proposing a generalized Hopf formula in density space. The formula transfers optimal control problems in density space, which are constrained minimizations supported on both spatial and time variables, to optimization problems over only spatial variables. This transformation allows us to compute HJD efficiently via multi-level approaches and coordinate descent methods.

math.NA↗

arXiv2018

A Splitting Method For Overcoming the Curse of Dimensionality in Hamilton-Jacobi Equations Arising from Nonlinear Optimal Control and Differential Games with Applications to Trajectory Generation

Recent observations have been made that bridge splitting methods arising from optimization, to the Hopf and Lax formulas for Hamilton-Jacobi Equations with Hamiltonians $H(p)$. This has produced extremely fast algorithms in computing solutions of these PDEs. More recent observations were made in generalizing the Hopf and Lax formulas to state-and-time-dependent cases $H(x,p,t)$. In this article, we apply a new splitting method based on the Primal Dual Hybrid Gradient algorithm (a.k.a. Chambolle-Pock) to nonlinear optimal control and differential games problems, based on techniques from the derivation of the new Hopf and Lax formulas, which allow us to compute solutions at points $(x,t)$ directly, i.e. without the use of grids in space. This algorithm also allows us to create trajectories directly. Thus we are able to lift the curse of dimensionality a bit, and therefore compute solutions in much higher dimensions than before. And in our numerical experiments, we actually observe that our computations scale polynomially in time. Furthermore, this new algorithm is embarrassingly parallelizable.

math.OC↗

arXiv2018

Algorithm for Overcoming the Curse of Dimensionality for State-dependent Hamilton-Jacobi equations

In this paper, we develop algorithms to overcome the curse of dimensionality in possibly non-convex state-dependent Hamilton-Jacobi equations (HJ PDEs) arising from optimal control and differential game problems. The subproblems are independent and can be implemented in an embarrassingly parallel fashion. This is an ideal setup for perfect scaling in parallel computing. The algorithm is proposed to overcome the curse of dimensionality [1, 2] when solving HJ PDE. The major contribution of the paper is to change an optimization problem over a space of curves to an optimization problem of a single vector, which goes beyond [23]. We extend [5, 6, 8], and conjecture a (Lax-type) minimization principle when the Hamiltonian is convex, as well as a (Hopf-type) maximization principle when the Hamiltonian is non-convex. The conjectured Hopf-type maximization principle is a generalization of the well-known Hopf formula [11, 16, 30]. We validated formula under restricted assumptions, and bring our readers to [57] which validates that our conjectures in a more general setting after a previous version of our paper. We conjectured the weakest assumption is a psuedoconvexity assumption similar to [46]. The optimization problems are of the same dimension as that of the HJ PDE. We suggest a coordinate descent method for the minimization procedure in the generalized Lax/Hopf formula, and numerical differentiation is used to compute the derivatives. This method is preferable since the evaluation of the function value itself requires some computational effort, especially when we handle high dimensional optimization problem. The use of multiple initial guesses and a certificate of correctness are suggested to overcome possibly multiple local extrema since the optimization process is no longer convex. Our method is expected to have application in control theory and differential game problems, and elsewhere.

math.OC↗

arXiv2017

A partial Laplacian as an infinitesimal generator on the Wasserstein space

We study stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes is modeled after Brownian motion and plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, and we use it to define heat flow on the Wasserstein space. We verify a distinctive smoothing effect of this flow for a particular class of initial conditions. To this end, we will develop a theory of Fourier analysis and conic surfaces in metric spaces. We note that the use of the infinitesimal generators has been instrumental in proving various theorems for Mean Field Games, and we anticipate they will play a key role in future studies of viscosity solutions of PDEs in the Wasserstein space.

math.AP↗

arXiv2017

Cyclic Coordinate Update Algorithms for Fixed-Point Problems: Analysis and Applications

Many problems reduce to the fixed-point problem of solving $x=T(x)$. To this problem, we apply the coordinate-update algorithms, which update only one or a few components of $x$ at each step. When each update is cheap, these algorithms are faster than the full fixed-point iteration (which updates all the components). In this paper, we focus on the coordinate-update algorithms based on the cyclic selection rules, where the ordering of coordinates in each cycle is arbitrary. These algorithms are fast, but their convergence is unknown in the fixed-point setting. When $T$ is a nonexpansive operator and has a fixed point, we show that the sequence of coordinate-update iterates converges to a fixed point under proper step sizes. This result applies to the primal-dual coordinate-update algorithms, which have applications to optimization problems with nonseparable nonsmooth objectives, as well as global linear constraints. Numerically, we apply coordinate-update algorithms with the cyclic, shuffled cyclic, and random selection rules to $\ell_1$ robust least squares, a CT image reconstruction problem, as well as nonnegative matrix factorization. They converge much faster than the standard fixed-point iteration. Among the three rules, cyclic and shuffled cyclic rules are overall faster than the random rule.

math.OC↗

arXiv2016

Unique determination of the index of refraction from phase-less near-field data in two dimensions

Motivated by non-destructive testing of optical fiber, we consider the problem of determining the index of refraction of a two-dimensional medium from magnitude of the total field resulting from known incident plane waves at a fixed frequency. The finiteness of the fiber cross section allows us to assume that the index of refraction is variable over a compact set. The measurement set is a closed curve, e.g., a circle of fixed radius, containing the fiber cross section. We use complex geometric optics (CGO) solutions to show that the inverse problem for the unknown index of refraction is unique in $C^{k}$ for large enough $k$. The key tool in this work is the multi-term expansion of the CGO solutions which we rely on to address the uniqueness question.

math.AP↗

arXiv2015

Phased and phaseless domain reconstruction in inverse scattering problem via scattering coefficients

In this work we shall review the (phased) inverse scattering problem and then pursue the phaseless reconstruction from far-field data with the help of the concept of scattering coefficients. We perform sensitivity, resolution and stability analysis of both phased and phaseless problems and compare the degree of ill-posedness of the phased and phaseless reconstructions. The phaseless reconstruction is highly nonlinear and much more severely ill-posed. Algorithms are provided to solve both the phased and phaseless reconstructions in the linearized case. Stability is studied by estimating the condition number of the inversion process for both the phased and phaseless cases. An optimal strategy is suggested to attain the infimum of the condition numbers of the phaseless reconstruction, which may provide an important guidance for efficient phaseless measurements in practical applications. To the best of our knowledge, the stability analysis in terms of condition numbers are new for the phased and phaseless inverse scattering problems, and are very important to help us understand the degree of ill-posedness of these inverse problems. Numerical experiments are provided to illustrate the theoretical asymptotic behavior, as well as the effectiveness and robustness of the phaseless reconstruction algorithm.

math.NA↗

arXiv2015

Analysis on Non-negative Factorizations and Applications

In this work we perform some mathematical analysis on non-negative matrix factorizations (NMF) and apply NMF to some imaging and inverse problems. We will propose a sparse low-rank approximation of big positive data and images in terms of tensor products of positive vectors, and investigate its effectiveness in terms of the number of tensor products to be used in the approximation. A new concept of multi-level analysis (MLA) framework is also suggested to extract major components in the matrix representing structures of different resolutions, but still preserving the positivity of the basis and sparsity of the approximation. We will also propose a semi-smooth Newton method based on primal-dual active sets for the non-negative factorization. Numerical results are given to demonstrate the effectiveness of the proposed method to capture features in images and structures of inverse problems under no a-priori assumption on the data structure, as well as to provide a sparse low-rank representation of the data.

math.OC↗

arXiv2014

Super-resolution in Imaging High Contrast Targets from the Perspective of Scattering Coefficients

In this paper we consider the inverse scattering problem for high-contrast targets. We mathematically analyze the experimentally-observed phenomenon of super-resolution in imaging the target shape. This is the first time that a mathematical theory of super-resolution has been established in the context of imaging high contrast inclusions. We illustrate our main findings with a variety of numerical examples. Our analysis is based on the novel concept of scattering coefficients. These findings may help in developing resonant structures for resolution enhancement.

math-ph↗

arXiv2014

Direct Sampling Method for Diffusive Optical Tomography

In this work, we are concerned with the diffusive optical tomography (DOT) problem in the case when only one or two pairs of Cauchy data is available. We propose a simple and efficient direct sampling method (DSM) to locate inhomogeneities inside a homogeneous background and solve the DOT problem in both full and limited aperture cases. This new method is easy to implement and less expensive computationally. Numerical experiments demonstrate its effectiveness and robustness against noise in the data. This provides a new promising numerical strategy for the DOT problem.

math-ph↗

arXiv2013

The Concept of Heterogeneous Scattering Coefficients and Its Application in Inverse Medium Scattering

This work investigates the scattering coefficients for inverse medium scattering problems. It shows some fundamental properties of the coefficients such as symmetry and tensorial properties. The relationship between the scattering coefficients and the far-field pattern is also derived. Furthermore, the sensitivity of the scattering coefficients with respect to changes in the permittivity and permeability distributions is investigated. In the linearized case, explicit formulas for reconstructing permittivity and permeability distributions from the scattering coefficients is proposed. They relate the exponentially ill-posed character of the inverse medium scattering problem at a fixed frequency to the exponential decay of the scattering coefficients. Moreover, they show the stability of the reconstruction from multifrequency measurements. This provides a new direction for solving inverse medium scattering problems.

math.AP↗

arXiv2013

Optimal Shape Design by Partial Spectral Data

In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial eigenvalues of its Neumann-Poincaré operator, which are known as the Fredholm eigenvalues. We begin by proposing to recover the eigenvalues of the Neumann-Poincaré operator from the polarization tensor by means of the holomorphic functional calculus. Then we develop a regularized Gauss-Newton optimization method for the shape reconstruction process. We present numerical results to demonstrate the effectiveness of the proposed methods and to illustrate important properties of the Fredholm eigenvalues and their associated eigenfunctions. Our results are expected to have important applications in the design of plasmon resonances in nanoparticles as well as in the multifrequency or pulsed imaging of small anomalies.

math.OC↗