歷史演講 PAST TALKS

Abstract: More and more large-scale imaging genetic studies are being widely conducted to collect a rich set of imaging, genetic, and clinical data in order to detect susceptibility genes for complexly inherited diseases including common mental disorders (e.g., schizophrenia) and neurodegenerative disorders, among many others. However, the development of statistical and computational methods for the joint analysis of complex imaging phenotypes, genetic data, and clinical data has fallen seriously behind the technological advances. The aim this work is to develop a functional mixed effects model (FMEM) for the joint analysis of high-dimensional imaging data with a set of genetic markers. Specifically, to accommodate the correlation structure of the marker set, we model the genetic effects as population-shared random effects with a common variance component (VC), whereas to accommodate spatial feature in imaging data, we spatially model varying associations between imaging measures in a three-dimensional (3D) volume (or 2D surface) with a set of covariates and the genetic random effects. We develop a two-stage estimation procedure to spatially and adaptively estimate the varying coefficient functions, while preserving its edges among different piecewise-smooth regions. To test hypothesis of interest, we provide two test statistics with well-controlled type I error and better power comparing to traditional voxel-based approach. Simulation studies and a real data analysis of the Alzheimer’s Disease Neuroimage Initiative (ADNI) show that FMEM significantly outperforms voxel-based approaches in terms of identification of activation regions.

古代巴比倫人在4000年前就用平方數來做乘法運算，希臘數學家發現平方數的一些巧妙性質，許多數學家：費馬、歐拉、拉瑪奴江都對平方數研究。聽眾可以知悉一些平方數奇妙美麗的性質，最後會了解兩年前印度數學家把平方數引進圖論標號創立「平方優美圖」理論。我會介紹我在這方面的工作，並提供一些研究課題。 聽眾：不需具備數論和圖論知識，只要對數學有一點興趣就行。

摘要詳如附件

演講摘要： 通過法國聖修伯里《小王子》(Le Petit Prince)裏面的主人公小王子，和講者一起遨遊古代、現代的時空之旅，來介紹不同文化民族及人們對數學的認識和發展。 本華曼德博(Benoit MMandelbrot)創造的分形幾何(Fractal Geometry)和法國數學家A杜阿迪(A.Douady)的兔子(rabbit)，體現講者對數學歷史及教育的看法，觸發聽眾的想象空間和探索數學課題的好奇心，分享講者從事數學科普工作的有趣經驗。如果您喜歡《愛麗絲夢遊記》、《平面國》、《哈利波特》這些魔幻故事，相信你一定會喜歡我的故事。歡迎6至97歲仍保有好奇及童稚之心的你，一同走進一個不一樣的數學世界。

時 間： Dec 16上午10:00~11:30，下午14:00~15:30 Dec 23上午10:00~11:30，下午14:00~15:30 摘要： Talk 1: Classical random number generators Linear Congruential Generator (LCG) Combination Generator and theoretical justifications Multiple Recursive Generator (MRG) Matrix Congruential Generator (MCG) Talk 2: Searching for large order MRGs Classes of efficient MRGs Efficient search algorithms Challenges to search for large order MRGs Talk 3: Parallel simulation using MRGs Classical fixed block or fixed leapfrog method Random leapfrog method for LCG Automatic generation method for large order MRG Talk 4: Secure random number generators Classical linear generators for computer simulation applications: LCG, MRG Random number/bit generators for security applications: BBS, RC4, HC-128 New SAFE (Secure And Fast Encryption), based on mutual shuffling method

Abstract: Total variation (TV) model for image restoration often suffers the staircasing effect. In this talk, we will introduce an adaptive TV model to overcome such drawback. Reference: Q. Chen, P. Montesinos, Q. S. Sun, P. A. Heng, D. S. Xia, Adaptive total variation denoising based on difference curvature, Image and Vision Computing, 28 (2010), pp. 298-306.

時 間： Dec 16上午10:00~11:30，下午14:00~15:30 Dec 23上午10:00~11:30，下午14:00~15:30 摘要： Talk 1: Classical random number generators Linear Congruential Generator (LCG) Combination Generator and theoretical justifications Multiple Recursive Generator (MRG) Matrix Congruential Generator (MCG) Talk 2: Searching for large order MRGs Classes of efficient MRGs Efficient search algorithms Challenges to search for large order MRGs Talk 3: Parallel simulation using MRGs Classical fixed block or fixed leapfrog method Random leapfrog method for LCG Automatic generation method for large order MRG Talk 4: Secure random number generators Classical linear generators for computer simulation applications: LCG, MRG Random number/bit generators for security applications: BBS, RC4, HC-128 New SAFE (Secure And Fast Encryption), based on mutual shuffling method

摘要詳如附件

Abstract: Image restoration is usually conducted by solving the minimizer of an energy functional with two terms: a data term and a regularization term. In this talk, the regularization term will be considered as a function of the magnitude of image gradient and we will quickly review some qualitative conditions on that function to ensure good restoration. Reference: G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, Second Edition, Springer 2006.

摘要： We consider the one-dimensional stochastic heat equation on [0,1] with periodic boundary condition driven by a space time white noise. We will show several moment estimates and the reasons why we need them.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: This talk introduces an iterative approach to global cluster synchronization of complex networks with a community structure. The units comprising the network are described by differential equations, and can be non-autonomous and involve time delays. In addition, units in the different communities can be governed by different equations. The coupling configuration of the network is rather general. Based on this approach, both delay-dependent and delay-independent criteria for global cluster synchronization are derived. We implement the present approach for a nonlinearly coupled neural network with heterogeneous coupling delays. Several numerical examples are given to show that neural networks can behave in a variety of new collective ways under the synchronization criteria. These examples also demonstrate that neural networks remain synchronized in spite of coupling delays between neurons across different communities; however, they may lose synchrony if the coupling delays between the neurons within the same community are too large, such that the synchronization criteria are violated.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： This paper studies first passage times of crossing a flat boundary under mixed exponential jump diffusion models. We establish a continuity correction for the joint distribution of a first passage time and its stopped underlying process when the event of crossing is monitored only discretely. The work is done by extending the Keener’s approach from partial differential equations to partial integro-differential equations. Unlike the exiting literature, the newly proposed correction amounts (regarding the boundary level) are different across different scenarios and related to both the diffusion and jump parts. Numerical results indicate that such scenario-dependent correction does improve the approximation performance especially when the monitoring frequency is low and the boundary is nearby. Similar conclusions are obtained to price a discrete barrier option, and to show how marketability affect security values. This is a joint work with S. Kou, S. F. Luo and H. C. Wong

TBA

[數學系畢業出路] 2016.11.15(二)上午10:10-11:40鴻經館M116 管理學院工業管理研究所 王啟泰老師帶您認識工業管理 *全程參與者提供生活知能學習時數2小時

Abstract: In this talk, we will first introduce the Il’in-Allen-Southwell (IAS) scheme, which is a robust finite difference scheme for treating scalar singularly perturbed convection-diffusion equations. The convergence of the IAS scheme is uniformly in the perturbation parameter in the discrete maximum norm. However, to our knowledge, it has never been achieved before in the literature to successfully construct an effective IAS scheme for strongly coupled systems of singularly perturbed convection-diffusion equations. Using the MHD duct flow as a model problem, we will then explain the difficulties that arise and give some ideas to capture the essence of the problem.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要：In a series of lectures, I will introduce some useful approaches to study investment problems (or portfolio optimization problems) in finance. In particular, I will consider Merton’s problem and some simple models to show the idea. An approach is the dynamic programming which is a classical powerful method used in control theory. This can be applied to Markovian models. The HJB (Hamilton-Jacobi-Bellman) equation can be derived, which is a nonlinear parabolic or elliptic partial differential equation. A solution of HJB equation can be used to construct an optimal strategy. The second approach is the martingale method. The method uses stochastic calculus and duality idea from convex analysis. In complete markets, this approach can be used to solve the problem without solving (complicated) nonlinear PDE. In incomplete market, the market completion is introduced. The use of BSDE (backward stochastic different equation) is another powerful approach. An adjoint equation is introduced. HJB equation is replaced by a BSDE, a stochastic differential equation, backward in time. The approach is very useful, in particularly, when the investment strategy has constraints. Here are some references for the materials that I will discuss. [1] W. H. Fleming and M. Soner (1993) Controlled Markov Processes and Viscosity Solution. [2] I. Karatzas and S. Shreve(1998), Methods of Mathematical Finance. [3] H. Pham (2002), Continuous Time Stochastic control and Optimization with Financial Applications.

摘要： This reading seminar is based on the book “Quantum Computation and Quantum Information” by Michael Nielsen and Issac Chuang.

Abstract: Phase retrieval, in the simplest terms, is solving the nonlinear system |Ax|=b where A is a given complex matrix, b a given nonnegative vector and x an unknown complex vector. In this course, we discuss recent progress in theory and algorithm for phase retrieval with coded diffraction patterns and application to coherent diffractive imaging.

Abstract: Magnetohydrodynamics (MHD) is the study of dynamics of electrically conducting fluids interacting with electromagnetic fields. Basically, the MHD equations can be obtained by coupling the incompressible Navier-Stokes equations for conducting fluid with Maxwell’s equations for electromagnetic field through Ohm’s law. In this talk, we will consider the steady incompressible MHD flowing through pipes under a transverse electromagnetic field. We will show that under some assumptions, the problem can be reduced to a strongly coupled system of singularly perturbed convection-diffusion equations.

摘要：In a series of lectures, I will introduce some useful approaches to study investment problems (or portfolio optimization problems) in finance. In particular, I will consider Merton’s problem and some simple models to show the idea. An approach is the dynamic programming which is a classical powerful method used in control theory. This can be applied to Markovian models. The HJB (Hamilton-Jacobi-Bellman) equation can be derived, which is a nonlinear parabolic or elliptic partial differential equation. A solution of HJB equation can be used to construct an optimal strategy. The second approach is the martingale method. The method uses stochastic calculus and duality idea from convex analysis. In complete markets, this approach can be used to solve the problem without solving (complicated) nonlinear PDE. In incomplete market, the market completion is introduced. The use of BSDE (backward stochastic different equation) is another powerful approach. An adjoint equation is introduced. HJB equation is replaced by a BSDE, a stochastic differential equation, backward in time. The approach is very useful, in particularly, when the investment strategy has constraints. Here are some references for the materials that I will discuss. [1] W. H. Fleming and M. Soner (1993) Controlled Markov Processes and Viscosity Solution. [2] I. Karatzas and S. Shreve(1998), Methods of Mathematical Finance. [3] H. Pham (2002), Continuous Time Stochastic control and Optimization with Financial Applications.

摘要： This reading seminar is based on the book “Quantum Computation and Quantum Information” by Michael Nielsen and Issac Chuang.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

Abstract: Finite-Element Method (FEM) is widely-used for solving various types of real-world scientific and engineering problems, such as structural analysis, fluid dynamics, electromagnetics, and etc. This lecture course provides brief introduction to procedures of FEM for steady-state heat conduction problems with iterative linear solvers and to parallel FEM. Lectures for parallel FEM will be focused on design of data structure for distributed local mesh files, which is the key issue for efficient parallel FEM. Solving large-scale linear equations with sparse coefficient matrices is the most expensive and important part of FEM and other methods for scientific computing, such as Finite-Difference Method (FDM) and Finite-Volume Method (FVM). Recently, families of Krylov iterative solvers are widely used for this process. In this short course, details of implementations of parallel Krylov iterative methods are provided along with parallel FEM. Moreover, lectures on programming for multicore architectures will be also given along with brief introduction to OpenMP and OpenMP/MPI Hybrid Parallel Programming Model. In the afternoon session, lectures for advanced topics, such as parallel preconditioning for ill-conditioned problems, and pipelined conjugate gradient algorithms towards post-peta/exascale supercomputers, will be provided 10:00-11:00 Introduction to FEM and Iterative Solvers 11:00-12:00 Parallel FEM, Hybrid Parallel Programming Model 13:30-14:30 Preconditioned Iterative Solvers for Ill-conditioned Problems 14:30-15:30 Pipelined Conjugate Gradient Algorithm

Abstract: The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{\alpha, -\alpha\}$, $\alpha\in[0,1)$, are called equiangular. The problem of determining the maximal size of $s$-distance sets in various spaces has a long history in mathematics. We determine a new method of bounding the size of an $s$-distance set in two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in $\mathbb{R}^n$ is $\frac{n(n+1)}2$ with possible exceptions for some $n=(2k+1)^2-3$, $k \in \mathbb{N}$. We also prove the universal upper bound $\sim \frac 2 3 n a^2$ for equiangular sets with $\alpha=\frac 1 a$ and, employing this bound, prove a new upper bound on the size of equiangular sets in an arbitrary dimension. Finally, we classify all equiangular sets reaching this new bound.

Abstract: In this talk we will demonstrate, in MATLAB, the implementation of some variational methods such as ALM [1], SB [2,3], and FTVd [4] for image denoising and deblurring. Experiments will also be included. References: [1] X. C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, in Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision, 2009, Springer. [2] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), pp. 323-343. [3] P. Getreuer, Rudin-Osher-Fatemi total variation denoising using split Bregman, Image Processing On Line, 2 (2012), pp. 74-95. [4] Y. Wang, W. Yin, and Y. Zhang, A fast algorithm for image deblurring with total variation regularization, CAAM Technical Report, 07-10 (2007), Rice University.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

The first part of this talk will be a brief introduction to complex symmetric operators. Recall that a bounded linear operator T on a Hilbert space H is said to be complex symmetric if T admits a symmetric matrix representation with respect to some orthonormal basis for H. The second pare of the talk will focus on the algebraic aspects of the theory of complex symmetric operators. More precisely, we shall discuss the complex symmetric generator problem for operator algebras, that is, the problem of determining which operator algebras can be generated by a single complex symmetric operator. For type I von Neumann algebras, properly infinite von Neumann algebras and a large class of finite von Neumann algebras, we give a complete answer.

Abstract: In this talk we will introduce the augmented Lagrangian method (ALM) [1, 2] for solving the Rudin-Osher-Fatemi total variation model [3]. The relation between ALM and split Bregman iteration will also be discussed. References: [1] X. C. Tai and C. Wu, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model, in Proceedings of the Second International Conference on Scale Space and Variational Methods in Computer Vision, 2009, Springer. [2] C. Wu and X. C. Tai, Augmented Lagrangian method, dual methods and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM Journal on Imaging Sciences, 1, (2008), pp. 248-272. [3] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp. 259-268.

摘要：In a series of lectures, I will introduce some useful approaches to study investment problems (or portfolio optimization problems) in finance. In particular, I will consider Merton’s problem and some simple models to show the idea. An approach is the dynamic programming which is a classical powerful method used in control theory. This can be applied to Markovian models. The HJB (Hamilton-Jacobi-Bellman) equation can be derived, which is a nonlinear parabolic or elliptic partial differential equation. A solution of HJB equation can be used to construct an optimal strategy. The second approach is the martingale method. The method uses stochastic calculus and duality idea from convex analysis. In complete markets, this approach can be used to solve the problem without solving (complicated) nonlinear PDE. In incomplete market, the market completion is introduced. The use of BSDE (backward stochastic different equation) is another powerful approach. An adjoint equation is introduced. HJB equation is replaced by a BSDE, a stochastic differential equation, backward in time. The approach is very useful, in particularly, when the investment strategy has constraints. Here are some references for the materials that I will discuss. [1] W. H. Fleming and M. Soner (1993) Controlled Markov Processes and Viscosity Solution. [2] I. Karatzas and S. Shreve(1998), Methods of Mathematical Finance. [3] H. Pham (2002), Continuous Time Stochastic control and Optimization with Financial Applications.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要：In a series of lectures, I will introduce some useful approaches to study investment problems (or portfolio optimization problems) in finance. In particular, I will consider Merton’s problem and some simple models to show the idea. An approach is the dynamic programming which is a classical powerful method used in control theory. This can be applied to Markovian models. The HJB (Hamilton-Jacobi-Bellman) equation can be derived, which is a nonlinear parabolic or elliptic partial differential equation. A solution of HJB equation can be used to construct an optimal strategy. The second approach is the martingale method. The method uses stochastic calculus and duality idea from convex analysis. In complete markets, this approach can be used to solve the problem without solving (complicated) nonlinear PDE. In incomplete market, the market completion is introduced. The use of BSDE (backward stochastic different equation) is another powerful approach. An adjoint equation is introduced. HJB equation is replaced by a BSDE, a stochastic differential equation, backward in time. The approach is very useful, in particularly, when the investment strategy has constraints. Here are some references for the materials that I will discuss. [1] W. H. Fleming and M. Soner (1993) Controlled Markov Processes and Viscosity Solution. [2] I. Karatzas and S. Shreve(1998), Methods of Mathematical Finance. [3] H. Pham (2002), Continuous Time Stochastic control and Optimization with Financial Applications.

Abstract: We use character sums over finite fields to give explicit formulae for the number of solutions of certain diagonal equations. We also show that if the value distribution of character sum $\sum_{x\in \mathbb{F}_q} \chi (ax^m+bx)$, $a,b\in \mathbb{F}_q$, is known, then one can obtain the number of solutions of the system of equations.

Abstract: In this talk we will introduce a fast total variation deconvolution algorithm, called FTVd [1][2], for image deblurring. The main computation at each iteration is three Fast Fourier Transforms (FFTs) applied to some quantities of the image size. The MATLAB implementation of this method will also be discussed. References: [1] Y. Wang, W. Yin, and Y. Zhang, A fast algorithm for image deblurring with total variation regularization, CAAM Technical Report, 07-10 (2007), Rice University. [2] Y. Wang, J. Yang, W. Yin, and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), pp. 248-272. [3] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp. 259-268.

參考: Brownian Motion and Stochastic Calculus 2/E, Ioannis Karatzas and Steven E. Shreve.

摘要： Efficient calculation of high-dimensional integrals is of critical importance in various disciplines, such as engineering, statistics, and finance. Monte Carlo simulation is indispensable for calculating high-dimensional integrals because it can avoid the curse of dimensionality. However, Monte Carlo simulation is notoriously known for its slow convergence, and hence additional variance reduction techniques are required to achieve the desire defficiency. In this talk, I would like to review Monte Carlo simulation and its applications in finance. Specifically, I will focus on recent developments and challenges of Monte Carlo simulation for Value-at-Risk evaluation, option pricing, and Greeks calculation.

Abstract: Using a special form of spanning surface for a knot, we give a formula for the coefficient of the z^2-term of the Alexander-Conway polynomial in terms of the sum of determinants of the blocks of 2x2 submatrices of the Seifert matrix, from which the topological meaning of the coefficient is revealed.

Abstract: Total variation (TV) regularization, introduced by Rudin, Osher, and Fatemi [1], is a technique often used in digital image processing for denoising, as well as a multitude of other imaging problems. The total variation of the image minimized subject to some fidelity constraints (i.e., some measures to ensure a close match to the noisy image) is remarkably effective at smoothing away noise in flat regions while preserving important details such as edges. Because of these advantages, many numerical methods were proposed to iteratively approximate the regularized minimization problem, which is today known as the ROF model. In this talk, we will briefly introduce the ROF model and focus on the split Bregman algorithm of Goldstein and Osher [2] for TV-regularized denoising. References: [1] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp. 259-268. [2] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), pp. 323-343. [3] P. Getreuer, Rudin-Osher-Fatemi total variation denoising using split Bregman, Image Processing On Line, 2 (2012), pp. 74-95.

Abstract: Total variation (TV) regularization, introduced by Rudin, Osher, and Fatemi [1], is a technique often used in digital image processing for denoising, as well as a multitude of other imaging problems. The total variation of the image minimized subject to some fidelity constraints (i.e., some measures to ensure a close match to the noisy image) is remarkably effective at smoothing away noise in flat regions while preserving important details such as edges. Because of these advantages, many numerical methods were proposed to iteratively approximate the regularized minimization problem, which is today known as the ROF model. In this talk, we will briefly introduce the ROF model and focus on the split Bregman algorithm of Goldstein and Osher [2] for TV-regularized denoising. References: [1] L. I. Rudin, S. Osher, and E. Fatemi, Nonlinear total variation based noise removal algorithms, Physica D, 60 (1992), pp. 259-268. [2] T. Goldstein and S. Osher, The split Bregman method for L1 regularized problems, SIAM Journal on Imaging Sciences, 2 (2009), pp. 323-343. [3] P. Getreuer, Rudin-Osher-Fatemi total variation denoising using split Bregman, Image Processing On Line, 2 (2012), pp. 74-95.