Titles & Abstracts
Locating Multiple Multiscale Electromagnetic Scatterers by a Single Far-field Measurement
Hongyu Liu, University of North Carolina at Charlotte
Abstract:
In this talk, we shall describe a inverse scattering scheme of locating multiple scatterers. The method could work in an extremely general setting. The underlying scatterer could consist of multiple components, and the number of the components and the physical property of each component are not required to be known in advance. Moreover, the scatterer may include, at the same time, components of small size and regular size compared to the detecting EM wavelength. For regular-sized components, we need impose a certain generic condition. The method is based on a series of indicator functions.
Second Moment Analysis for Stochastic Interface Problems Based on Low-rank Approximation
Jingzhi Li, South University of Science and Technology of China
Abstract:
In this talk, we present a new numerical method to solve stochastic elliptic interface problems with random interfaces. Shape calculus is first employed to derive the shape-Taylor expansion in the framework of the asymptotic perturbation approach. Given the mean field and the two-point correlation function of the random interface, we can thus quantify the mean field and the variance of the random solution in terms of certain orders of the perturbation amplitude by solving a deterministic elliptic interface problem and its tensorized counterpart with respect to the reference interface. Error estimates are derived for the interface-resolved finite element approximation in both, the physical and the stochastic dimension. In particular, a fast finite difference scheme is proposed to compute the variance of random solutions by using a low-rank approximation based on the pivoted Cholesky decomposition. Numerical experiments are presented to validate and quantify the method.
Multiplicative Noise Removal via a Learned Dictionary
Yumei Huang, Lanzhou University
Abstract:
Multiplicative noise removal is a challenging image processing problem, and most existing methods are based on the Maximum A Posteriori formulation and the logarithmic transformation of multiplicative denoising problems into additive denoising problems. On the other hand, sparse representations of images have shown to be efficient approaches for image recovery. Following this idea, we propose to learn a dictionary from the logarithmic transformed image, and then to use it in a variational model built for noise removal. Extensive experimental results suggest that in terms of visual quality, PSNR and mean absolute deviation error, the proposed algorithm outperforms state-of-the-art methods.
A Convex Variational Model for Restoring Blurred Images with Multiplicative Noise
Tieyong Zeng, Hong Kong Baptist University
Abstract:
In this talk, a new variational model for restoring blurred images with multiplicative noise is proposed. Based on the statistical property of the noise, a quadratic penalty function technique is utilized in order to obtain a strictly convex model under a mild condition, which guarantees the uniqueness of the solution and the stabilization of the algorithm. For solving the new convex variational model, a primal-dual method is proposed and its convergence is studied. The talk ends with a report on numerical tests for the simultaneous deblurring and denoising of images subject to multiplicative noise. A comparison with other methods is provided as well.
Computational Methods for Quasiconformal Surface Maps and Their Optimization
Alvin Tsz Wai Wong, University of California, Irvine
Abstract:
The manipulation of surface homeomorphisms is an important aspect in 3D modeling and surface processing. Every homeomorphic surface map can be considered as a quasiconformal map, with its local non-conformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasiconformal maps are of great interest in mathematical study and real applications. Efficient and accurate computational construction of desirable quasiconformal maps between general surfaces is crucial. However, existing computational works on construction of quasiconformal maps to or from a compact domain require global parametrization onto the plane, and have difficulty to be directly applied to maps between arbitrary surfaces. This talk presents numerical algorithms for computing quasiconformal homeomorphisms between arbitrary Riemann surfaces using discrete Beltrami flow, which is a vector field corresponding to the adjustment to the intrinsic Beltrami differential of the map. The vector field is defined by a partial differential equation (PDE) in local conformal coordinates. Based on this formulation and a composition formula, we can compute the Beltrami flow of any homeomorphism adjustment as a vector field on the target domain defined from the source domain, with appropriate boundary conditions and correspondences. Numerical results will be presented to show the robustness and efficiency of our algorithms for adjusting surface homeomorphisms, and their use in surface map optimization.
Quantification of Exciton Diffusion Length in Organic Solar Cells
Jingun Chen, University of California, Santa Barbara
Abstract:
Exciton diffusion length (EDL) is an important parameter to characterize the external quantum efficiency of organic solar cells. Diffusion model is used to compute EDLs of different materials under photoluminescence and photocurrent measurements. Results obtained are consistent with other models. Due to the particularity of some materials, it is observed that energy transfer effect must be included to produce good results.
Partial Expansion of a Lipchitz Domain and Some Applications
Frederick Weifeng Qiu, City University of Hong Kong
Abstract:
We show that a Lipchitz domain can be expanded solely near a part of its boundary, assuming that the part is enclosed by a piecewise C1 curve. The expanded domain as well as the extended part are both Lipchitz. We apply this result to prove a regular decomposition of standard vector Sobolev spaces with vanishing traces only on part of the boundary. Another application in the construction of low-regularity projectors into finite element spaces with partial boundary conditions is also indicated.
Multiscale Method Coupling Network and Continuum Models in Porous Media
Chia Chieh Chu, National Tsing Hua University
Abstract:
In this talk, we present a numerical multiscale method for coupling a conservation law for mass at the continuum scale with a discrete network model that describes the pore scale flow in a porous medium. We developed single-phase flow algorithm and extended it to two-phase flow, for the situations in which the saturation profile go through a sharp transition from fully saturated to almost unsaturated states. Our coupling method for the pressure equation uses local simulations on small sampled network domains at the pore scale to evaluate the continuum equation and thus solve for the pressure in the domain. We present numerical results for single-phase flows with nonlinear flux-pressure dependence, as well as two-phase flow.
Rigidity of Infinite Hexagonal Triangulation of Plane
Jian Sun, Tsinghua University
Abstract:
In this talk, I will describe the concept of PL conformal transformation of the metric on triangulated surfaces, and present our recent result on the rigidity of the infinite hexagonal triangulation of the plane under PL conformal transformation.
This is the joint work with Tianqi Wu, Feng Luo and David Xianfeng Gu.
Optimizing Radiotherapy
Bin Dong, The University of Arizona
Abstract:
Mathematical modeling and scientific computing are very important in improving the quality of radiotherapy. I will specifically talk about the following two topics.
The first topic is on optimal marker selection for tumor motion estimation in lung cancer radiotherapy. We propose a novel mathematical model and an efficient algorithm to automatically determine the optimal number and locations of fiducial markers on patient’s surface (typically on the chest) for predicting lung tumor motion. Experiments on the 4DCT data of 4 lung cancer patients have shown that usually 6-7 markers are selected on patient’s external surface. Using these markers, the lung tumor positions can be predicted with an average 3D error of approximately 1mm. Both the number of markers and the prediction accuracy are clinically acceptable, indicating that our method can be used in clinical practice.
The second topic is on accurate radiation dose delivery in volumetric modulated arc therapy (VMAT) in cancer radiotherapy. It can be described as an optimization problem, where beam parameters, such as directions, shapes, and intensities, can be adjusted in simulations to yield desired dose distributions. This can be practically implemented by the VMAT setup which involves the use of a full-rotation trajectory of the beam about the patient along with a multi-leaf collimator for beam shape sculpting, with notable advantages in shortened treatment time. Treatment plan optimization in this setting, however, can be quite complicated due to constraints arising from the equipment involved. We introduce a variational model in the VMAT setup for the optimization of beam shapes and intensities under these constraints. We apply a binary level-set strategy to represent beam shapes and a fast sweeping technique to satisfy beam intensity variation limits. Our numerical tests on real data reveal that our algorithm shows great promise in the generation of desired dose distributions for treatment plans in cancer radiotherapy.
An A Posteriori Error Estimate for Local C0 Discontinuous Galerkin Approximations of Kirchhoff Plates
Yifeng Xu, Shanghai Normal University
Abstract:
A residual-type a posteriori error estimate is given for a local C0 discontinuous Galerkin (LCDG) method for the Kirchhoff bending plate clamped on the boundary. The estimator is both reliable and efficient with respect to the moment-field approximation error in an energy norm. Some numerical experiments are provided to verify theoretical results.
Variational Image Fusion with High Order Gradient Information
Fang Li, East China Normal University
Abstract:
Image fusion has become an active issue in image processing and computer vision owing to the availability of multisensor data in many fields. The main goal of image fusion is to integrate multiple source images of the same scene into a highly informative single image which is more suitable for human or computer vision. We propose a variational image fusion method based on the first and second-order gradient information. Firstly, we select the target first-order and second-order gradient information from the source images by some salience criterion. Then we build our model by requiring that the first-order and second-order gradients of the fused image match the target gradients correspondingly in the sense of L1 norm, and meanwhile the fused image should be close to each of the source image in the sense of L2 norm. Theoretically we can prove that the variational model has a unique minimizer once the gradient information is fixed. In the numerical implementation, we take use of split Bregman method to get an efficient algorithm. Moreover, four-direction difference scheme is proposed to discrete gradient operator, which can dramatically enhance the fusion quality. A number of experiments and comparisons with some popular existing methods demonstrate the performance of the proposed model in various image fusion applications.
Joint work with T. Zeng (Hong Kong Baptist University).
Homotopy Methods for Solving Mixed Trigonometric Polynomial Systems
Bo Dong, Dalian University of Technology
Abstract:
A mixed trigonometric polynomial system, which rather frequently occurs in applications, is a polynomial system where every monomial is a mixture of some variables and sine and cosine functions applied to the other variables. This class of systems can be transformed to polynomial systems through variable substitution and adding quadratic equations, and then solved by the existent polynomial solving methods. However, the existent methods increase the dimension of the problems due to the introduced additional variables, and more importantly, they do not adequately take advantage of the special structure of the transformed polynomial systems. In this talk, two different classes of methods which do not increase the dimension of the problem are presented. One class of methods, named direct homotopy methods, do not introduce additional variables, and every monomial in the constructed homotopies and start systems is a mixture of the variables and trigonometric functions applied to variables. The other class of methods, which are indirect methods, exploit the special structure of the transformed polynomial systems from mixed trigonometric polyno-mial systems to construct e±cient symbolic-numerical methods for solving polynomial systems. These two methods complement each other, and their combination leads to an e±cient solving method for a challenging practical problem. Numerical results show that our methods are more e±cient than the existent state-of-art methods for mixed trigonometric polynomial systems.
Numerical Methods for American Option Pricing
Kai Zhang, Jilin University
Abstract:
In this talk, we present a fast solver for computing Black-Scholes equation of American options. Perfectly matched layer (PML) method is proposed to truncate the unbounded domain into a bounded computational domain, and under some weak assumptions on the PML medium parameters, it is shown that the solution of truncated PML problem converges to the solution of unbounded Black-Scholes equation in concerned domain and exponentially decays in perfectly matched layer. Using front-fixing transformation, the treatment of the free boundary is solved by Newton's method based on some properties of optimal exercise boundary. Finite element method and discontinuous Galerkin method are used to solve the resulting problems. Numerical simulations are presented to test the performance of proposed algorithm. Comparisons to results obtained by previous approaches indicate its high accuracy and efficiency.
On Preconditioned Iterative Methods for Sinc Systems of Linear Third-Order ODEs
Zhiru Ren, Chinese Academy of Sciences
Abstract:
We study the preconditioned iterative methods for the linear system arising from the sinc discretizations of the linear third-order ordinary differential equations.
First, we discretize the boundary value problems of linear third-order ordinary differential equations by sinc methods and prove that the discrete solutions converge exponentially to the true solutions. The discrete solution is determined by a linear system with the coefficient matrix being a combination of Toeplitz and diagonal matrices. The system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by banded matrices. We demonstrate that the eigenvalues of the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system.
In order to overcome the difficulty causing by the third-order terms in solving the linear third-order ordinary differential equations, we introduce variable replacements to transform the third-order ordinary differential equations into systems of two second-order ordinary differential equations. The system of order-reduced ordinary differential equations is solved by sinc discretization methods and the discrete solution is proved to be convergent to the true solution of the ordinary differential equation exponentially. The discrete solution is actually determined by a linear system with the coefficient matrix being block two-by-two structure and each block being a combination of Toeplitz and diagonal matrices. This class of linear systems can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by the block-diagonal matrix. We demonstrate that the eigenvalues of an approximate to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the linear system. Numerical examples show that the order-reduced method outperforms the direct sinc method in solving these problems.
This is a joint work with Zhong-Zhi Bai and Raymond H. Chan.
Two-Level Iteration Penalty Methods for the Incompressible Flows
Feng Shi, Shenzhen Institutes of Advanced Technology
Abstract:
In this talk, we present some iteration penalty methods for solving stationary Navier-Stokes equations. By using the techniques of two level and linearization of nonlinear convection term based on Stokes/Oseen/Newton iteration, we derive the two-level Stokes/Oseen/Newton iteration penalty methods. Also, we prove some error estimates, and present some numerical experiments to demonstrate the convergence and efficiency of these three methods.
This is a joint work with Rong An from Wenzhou University.
Confounder Correction in Biological Data Classification
Limin Li, Xi'an Jiao-Tong University
Abstract:
Classifying biological data into different groups is a central task of bioinformatics, for instance, to predict the function of a gene or protein, the disease state of a patient or the phenotype of an individual based on its genotype. However, it is unclear how to correct for confounding factors such as population structure, age or gender or experimental conditions. In this talk, I will present our recent confounder correction approaches and some results in the application of bioinformatics.
Central Discontinuous Galerkin Methods and their Applications to Solve Magnetohydrodynamic and Shallow Water Equations
Liwei Xu, Chongqing University
Abstract:
In this talk, we first review the central discontinuous Galerkin (CDG) methods on overlapping meshes (Liu-Shu-Tadmor-Zhang, 2007). Then, we discuss our development of CDG methods, including divergence-free, positivity-preserving and well-balanced schemes. These schemes have been applied to solve the Magnetohydrodynamic (MHD) and shallow water equations, and corresponding numerical results will be presented to validate the schemes. This is a joint work with Prof. Fengyan Li at RPI, Prof. Philippe Guyenne at UD, Prof. Jianxian Qiu at XMU and Dr. Maojun Li at Beijing CSRC.
Noise Drives Sharpening of Expression Boundaries in the Zebrafish Hindbrain
Lei Zhang, City University of Hong Kong
Abstract:
Morphogens provide positional information for spatial patterns of gene expression during development. However, stochastic effects such as local fluctuations in morphogen concentration and noise in signal transduction make it difficult for cells to respond to their positions accurately enough to generate sharp boundaries between gene expression domains. In this talk, I will present a multiscale stochastic model to investigate a novel noise attenuation mechanism during the development in the zebrafish hindbrain. Computational analyses of spatial stochastic models show, surprisingly, that a combination of noise in RA concentration and noise in hoxb1a/krox20 expression promotes sharpening of boundaries between adjacent segments. In particular, fluctuations in RA initially induce a rough boundary that requires noise in hoxb1a/krox20 expression to sharpen. This finding suggests a novel noise attenuation mechanism that relies on intracellular noise to induce switching and coordinate cellular decisions during developmental patterning.
Teichmüller Extremal Map and its Application
Ronald Lui, The Chinese University of Hong Kong
Abstract:
Registration, which aims to find an optimal 1-1 correspondence between shapes (or images), is important in different areas such as in computer vision and medical imaging. Conformal mappings have been widely used to obtain a diffeomorphism between shapes that minimizes angular distortion. Conformal registrations are beneficial since it preserves the local geometry well. However, when extra constraints (such as landmark constraints) are enforced, conformal mappings generally do not exist. This motivates us to look for an optimal quasi-conformal registration, which satisfies the required constraints while minimizing the conformality distortion. Under suitable condition on the constraints, a unique diffeomporphism, called the Teichmüller extremal mapping between two surfaces can be obtained, which minimizes the maximal conformality distortion. In this talk, an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmüller mapping will be presented. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmüller mapping. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the Linear Beltrami Solver (LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration. Using the proposed method, the Teichmüller mapping can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. This proposed algorithm can also be practically applied to real applications. In the second part of my talk, I will present how Teichmüller extremal mapping can be used for brain landmark matching registration, constrained texture mapping and face recognition.
