Colloquium/Seminar
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Event(s) on December 2017
- Friday, 1st December, 2017
Title: Convergence rate analysis of an iterative scheme for fixed point problems Speaker: Mr. MA Yaonan, Department of Mathematics, Hong Kong Baptist University, HKSAR Time/Place: 10:00 - 11:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We focus on an iterative scheme for fixed point problems and prove its worst-case O(1/n) convegence rate measured by the iteration complexity. By applying it to solve a linearly constrained convex minimization problem, we obtain an iterative scheme which can be reduced to the alternating direction method of multipliers (ADMM) under some special parameters. Furthermore, its convergence rates for objective function value and residual are established. Numerical examples are presented to show its efficiency for lasso and image inpainting problems. - Friday, 1st December, 2017
Title: Pairwise-difference Least Absolute Deviation Regression and The Oracle Model Selection Theory Speaker: Mr. SHEN Wei, Department of Mathematics, Hong Kong Baptist University, HKSAR Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: A new method is proposed to do variable selection and coefficient estimation simultaneously in the linear regression. By the least absolute deviation regression based on pairwise-differenced data, we obtain the so-called PD-LAD estimator, which are expected to be as efficient as the asymptotic version of the composite quantile regression (CQR) with K approximates infinity [Zou and Yuan (2008)]. Using the adaptive lasso penalty, we establish the PD-LAD-oracular estimator enjoying the same advantages (like the robustness and safety) of the CQR-oracular estimator in Zou and Yuan (2008). For computation issues, we use the optimal linearized alternating direction method of multipliers proposed by He, Ma and Yuan (2017) recently to attain the PD-LAD-oracular estimator in practice. In the simulation our estimator performs at least as well as the CQR-oracle and sometimes even a little better. - Friday, 1st December, 2017
Title: Alternating direction method of multipliers for optimal control problem Speaker: Mr. SONG Yongcun, Department of Mathematics, Hong Kong Baptist University, HKSAR Time/Place: 14:00 - 15:00
FSC1111, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We are concerned with numerical solutions of the parabolic optimal control problem with state constraint. For state-constrained problem, some additional assumptions are required to guarantee the existence and regularity of Lagrange multipliers. The resulting optimality system leads to difficulties on the numerical implementation and theoretical analysis. The approach we discussed relies on the alternating direction method of multipliers(ADMM) and conjugate gradient(CG) method. This approach allows the decoupling of the state constraint and optimization problem. We prove the convergence of our algorithm without assuming the existence of Lagrange multipliers to tackle issues mentioned above. Furthermore, a worst-case O(1/k) convergence rate in an ergodic sense is established. After full discretization, the numerical results validate the efficiency of our proposed method. - Wednesday, 6th December, 2017
Title: MULTI-THRESHOLD ACCELERATED FAILURE TIME MODEL Speaker: Dr. Jialiang Li, Department of Statistics & Applied Probability, National University of Singapore, Singapore Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: A two-stage procedure for simultaneously detecting multiple thresholds and achieving model selection in the segmented accelerated failure time (AFT) model is developed in this paper. In the first stage, we formulate the threshold problem as a group model selection problem so that a concave 2-norm group selection method can be applied. In the second stage, the thresholds are finalized via a refining method. We establish the strong consistency of the threshold estimates and regression coefficient estimates under some mild technical conditions. The proposed procedure performs satisfactorily in our simulation studies. Its real world applicability is demonstrated via analyzing a follicular lymphoma data. - Wednesday, 6th December, 2017
Title: Preconditioning for Non-symmetric Toeplitz Matrices with Application to Time-dependent PDEs Speaker: Prof. Andrew Wathen, Mathematical Institute, University of Oxford, England, UK Time/Place: 16:30 - 17:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Gil Strang proposed the use of circulant matrices (and the FFT) for preconditioning symmetric Toeplitz (constant-diagonal) matrix systems in 1986 and there is now a well-developed theory which guarantees rapid convergence of the conjugate gradient method for such preconditioned positive definite symmetric systems developed by Raymond Chan, Michael Ng, Tony Chan and Eugene Tyrtyshnikov amongst others. In this talk we describe our recent approach which provides a preconditioned MINRES method with the same guarantees for real nonsymmetric Toeplitz systems regardless of the non-normality. We demonstrate the utility of these ideas in the context of time-dependent PDEs. This is joint work with Elle McDonald and Jen Pestana. - Friday, 8th December, 2017
Title: Matrix Problems and Techniques in Quantum Information Science Speaker: Prof. Chi-Kwong Li, Department of Mathematics, The College of William & Mary (USA) and Institute for Quantum Computing, University of Waterloo, U.S.A. Time/Place: 10:30 - 16:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Three lectures will be presented on some matrix problems and techniques in quantum information science. No quantum mechanics background is required. Audience should have the basic linear algebra knowledge. One may see the recent reprints and preprints of Li and his collaborators on arXiv to get some ideas about the topics he plans to present. Lecture 1 -- Optimization Problems in Quantum Information Science In the mathematical framework, quantum states are represented as density matrices, i.e., positive semidefinite matrices with trace one, and quantum operations are represented as trace preserving completely positive linear maps. In quantum information science, one often has to estimate various measures between different quantum states and quantum processes. One also needs to design quantum operations with special properties. These give rise to many optimization problems involving matrices and linear transformations. In this lecture, selected results, techniques, and open problems in this line of study will be described. Lecture 2 -- Numerical Ranges and Quantum Information Science In quantum mechanics, measurement operators or observable are represented as Hermitian matrices, and measurement are done by taking the inner product of the measurement operators and the states (represented as density matrices). The collection of such measurement values on states can be viewed as elements in the joint numerical range of the measurement operators. Also, in the study of quantum operations with special properties, and the quantum error correction codes of quantum channels, one can formulate the problems in terms of the higher rank numerical ranges of the Choi-Kraus operators of the quantum operations/channels. In this lecture, problems and results involving different kind of numerical ranges will be described. Lecture 3 -- Preserver Problems and Quantum Information Science Preserver problems concern the characterization of maps on matrices or operators with special properties. In connection to quantum information science, researcher are interested in maps that leave invariant some certain measures, relations, or subsets of quantum states or quantum systems. In this lecture, selected problems and results in such research will be described - Monday, 11th December, 2017
Title: An Elimination Game - Old and New Speaker: Prof. Esmond Ng, Applied Mathematics Department, Lawrence Berkeley National Laboratory, USA Time/Place: 10:30 - 11:30 (Preceded by Reception at 10:00am)
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We consider a game on an undirected graph, in which vertices are eliminated. When a vertex is eliminated, the incident edges are removed, but new edges may be added to the graph according to a prescribed rule. The edges that are added will eventually be removed at later stages of the game. The graph will be empty at end of the game. We are interested in the total number of new edges that the game sees throughout the elimination. The order in which the vertices are eliminated will affect the number of new edges added to the graph. Thus, we are interested in finding an elimination order that minimizes a function, which depends on the number of new edges added during the game. In this talk, we will provide an overview of the elimination game. We will also discuss some old and new results on the complexity of the game. This elimination game is relevant to the problem of computing a triangular factorization of a sparse matrix. - Monday, 11th December, 2017
Title: Numerical stability of deterministic and stochastic differential equations with piecewise continuous arguments Speaker: Prof. SONG Minghui, Department of Mathematics, Harbin Institute of Technology, China Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: This talk's aim is to study under what conditions the exact and numerical solution to deterministic/stochastic differential equations with piecewise continuous arguments (EPCAs / SEPCAs) share the property of asymptotic stability / mean-square exponential stability. In the first part, we concerned with the stability analysis of the Runge-Kutta methods for EPCAs. The stability regions for the Runge- Kutta methods are determined. The conditions that the analytic stability region is contained in the numerical stability region are obtained. In the second part, we show that under a global Lipschitz assumption an SEPCAs is exponentially stable in mean square if and only if for some sufficiently small stepsize the Euler–Maruyama (EM) method is exponentially stable in mean square. We then replace the global Lipschitz condition with a finite-time convergence condition and establish the same “if and only if” result. The important feature of this result is that it transfers the asymptotic problem into a finite-time convergence problem. - Monday, 18th December, 2017
Title: Solving Monotone Stochastic Variational Inequalities and Complementarity Problems by Progressive Hedging Speaker: Prof. SUN Jie, Department of Mathematics and Statistics, Faculty of Science and Engineering, School of Science, Curtin University, Australia Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The concept of a stochastic variational inequality has recently been extended to a format that covers, in particular, the optimality conditions for a multistage stochastic programming problem. One of the long-standing methods for solving such optimization problems under convexity is the progressive hedging algorithm. That approach is demonstrated here to be applicable also to solving multistage stochastic variational inequality problems under monotonicity, thus vastly increasing its range of applications. A game with uncertainty is presented as a special case and explored numerically in a quadratic two-stage formulation. - Wednesday, 20th December, 2017
Title: A model based joint sparsity approach for inverse elastic medium scattering Speaker: Dr. Abdul Wahab, Department of Bio and Brain Engineering, Korea Advanced Institute of Science and Technology, Korea Time/Place: 15:00 - 16:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: In this talk, I will focus on an inverse problem arising in Magnetic Resonance Elas-tography. A compressive sensing based algorithm will be presented for efficient and accurate re-construction of the spatial support and material parameters of multiple inhomogeneous elastic inclusions in a bounded elastic material. Only a few measurements of the elastic displacement field over a very coarse grid (in the sense of Nyquist sampling rate) will be taken into account, on contrary to classical algorithms assuming continuous or dense grid measurements. Our pro-posed algorithm is not only very accurate but also computationally efficient as it does not require any linearization or iterative proceedure. The breakthrough comes from a novel interpretation of Lippmann-Schwinger type integral representation of the displacement field in terms of unknown densities (connected to the displacement and strain fields inside the inclusions) having common and sparse spatial support on the location of inclusions. Accordingly, the support identification prob-lem is recast as a joint sparse recovery problem that renders such densities and the support of the inclusions simultaneously. Then, using the leverage of the learned densities and associated fields inside the inclusions, a linear inverse problem for quantitative evaluation of material parameters is formulated. The resulting problem is then converted to a noise robust constraint optimization problem. For numerical implementation, modified Multiple Sparse Bayesian Learning (M-SBL) algorithm and the Constrained Split Augmented Lagrangian Shrinkage Algorithm (C-SALSA) are used due to their robustness with respect to noise. The efficacy of the proposed framework will be manifested through a variety of numerical examples. The significance of this investigation is due to its pertinence for bio-medical imaging and non-destructive testing, wherein the real physical mea-surements are only available on a sub-sampled coarse grid [1]. The proposed algorithm is the first one tailored for parameter reconstruction problems in elastic media using highly under-sampled data.