歷史演講 PAST TALKS

probabilistic grammar for the grouping and labeling of parts and objects, when taken together with pose and part-dependent appearance models, constitutes a generative scene model and a Bayesian framework for image analysis. To the extent that the generative model generates features, as opposed to pixel intensities, the posterior distribution (i.e. the conditional distribution on part and object labels given the image) is based on incomplete information; feature vectors are generally insufficient to recover the original intensities. I will propose a way to learn pixel-level models for the appearances of parts. I will demonstrate the utility of the models with some experiments in Bayesian image classification.

We study a generalized two-component Camassa-Holm system which can be derived from the theory of shallow water waves moving over a linear shear flow. This new system also generalizes a class of dispersive waves in cylindrical compressible hyperelastic rods. In the first part of the talk, we show that this new system can still exhibit the ave-breaking phenomenon. We also establish a sufficient condition for global solutions. In the second part of the talk, we study the existence of solitary-wave solutions of a generalized two-component Camassa-Holm system. In addition to those smooth solitary-wave solutions, we show that there are solitary waves with singularities: peaked and cusped solitary waves. We also demonstrate that all smooth solitary waves are orbitally stable in the energy space.

In this talk, I will present a novel recovery theory of high frequency wave fields which combines both the phase space-based level set method and the Gaussian beam superposition. The method applies to all strictly hyperbolic and Schroedinger equations subject to highly oscillatory initial data. We are particularly concerned with the accuracy of the developed method in terms of the wavelength, the optimal error estimates in the appropriate norm are obtained. These results are valid for any number of spatial dimensions and are unaffected by the presence of caustics. This talk will conclude with several remaining challenges associated with applications. This talk is based on recent works joint with J. Ralston (UCLA), and with N. Tanushev (UT-Austin) and O. Runborg (KTH).

This is a study seminar to learn the uses of stochastic control in insurance. We plan to study the materials based on a book by Hanspeter Schmidli. The seminar gives a chance to learn the uses of stochastic control method in solving the optimization problems. The book in particular discusses some problems from insurance. They include: the optimal dividend in risk theory, minimizing the ruin probabilities, and the optimal investment problem. We plan to meet on Wednesday in the morning for every two weeks. The book by Schmidli consists of 4 chapters. The main issues discussed in this seminar are organized as follows. First, we will start with the concept of stochastic control in discrete time and introduce the discrete-time dynamic programming. The stochastic control in continuous time will be illustrated in the second part. Furthermore, several applications in insurance will be discussed after presenting Hamilton-Jacobi-Bellman approach. Then, the problems in life insurance with the continuous time case will be discussed, and the last is the asymptotic properties of the value function obtained from Hamilton-Jacobi-Bellman equation.

In this talk we start with a brief review on short time asymptotic behavior of implied volatility under various models. Particularly in the case of one dimensional diffusion model, using an expansion of the transition density function, we obtain the first and the second order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the first order and second order deviation of implied volatility from its leading value and obtain approximation which we numerically demonstrate to be highly accurate. The analysis is extended to degenerate diffusions using probabilistic methods, i.e., the so called principle of no feeling the boundary. This is based on a joint work with Jim Gatheral, Elton Hsu, Peter Laurence, and Cheng Ouyang.

The global (in time) regularity issue concerning the surface quasi-geostrophic (SQG) equation has recently attracted a lot of attention. Many studies have been devoted to this issue and important progress has been made. In particular, the global regularity for the SQG equation with critical dissipation has been successfully established. The global regularity issue for the inviscid SQG and the supercritical SQG equations has only been partially resolved. This talk presents recent theoretical and computational results on a family of active scalar models that generalize the SQG equations. It is based on collaborative work with Dongho Chae and Peter Constantin. The study of these model equations is partially motivated by an attempt to gain insight into the unresolved issues concerning the inviscid SQG and the supercritical SQG equations.

1. Design flow introduction: 1.1 Parasitic RC extraction for post-layout simulation. 1.2 Design and process information for Maxwell's equations (RC extraction). 2. Static analysis: (commerical RC extraction tool) 2.1 Poisson equation, 2.2 Numerical solution: method of moment, 2.3 Equivalent capacitance model. 3. Full-wave analysis: (commerical ElectroMagnetic field solver) 3.1 Maxwell's equations, 3.2 Numerical solution: spectral domain approach and Non-Uniform FFT, 3.3 2D and 3D applications. 4. Challenge in the 20nm technology 4.1 Double patterning technology (Graphic theory: 2-colorable). 4.2 Commercial electronic design automation (EDA) RC extraction tool: limitation and solution.

Extremal problems for quasiconformal mappings typically involve two domains Omega1 and Omega2 in n-space, (or two Riemann surfaces) for which there exists a quasiconformal mapping f:Omega1 --> Omega2, and ask for a quasiconformal map u:Omega1--> Omega2 which minimizes one or more of the dilation functions in a given class of competitors which are usually other quasiconformal mappings with same boundary data as f on a portion (or all) of the boundary of the domain Omega1 or in the same homotopy class as the given map f. In a joint work with Andrew Raich (UARK) we study this L^{\infty} variational problem from a PDE point of view which echoes some of G. Aronsson's work in the 60's, concerning absolute minimal extensions and the infinity Laplacian. We derive a non-linear, degenerate elliptic, PDE system as a formal limit of Euler Lagrange equations for L^p approximations for the dilation functional and we prove that solving such a system provides necessary and sufficient conditions for local forms of extremality of C^2 quasiconformal diffeomorphisms. We also study a family of gradient flows for L^p dilation and prove short time existence of solutions for smooth data.

The deficiency of Riemann Integral is that it cannot handle functions with singular points, which is why we need improper Riemann Integrals. The deficiency of Lebesgue Integral is that it cannot integrate highly oscillating functions with singular points because its primitive function needs to be absolutely continuous which means the total oscillations should be as small as we wish. In this talk, we will see how can easily extend the Riemann Integrals and Lebesgue Integrals by modifying Riemann Integration definition just a little,we will see some integrals that do not exist under Lebesgue sense but they exist under Henstock-Kurzweil (HK) sense, and we see how we compute such integrals. In addition, we discuss the relationship between the primitive functions of Lebesgue and HK integrable functions respectively. Under the HK integral, the Fundamental Theorem of Calculus holds in one dimension and Divergence Theorem holds in higher dimensions.

For the stationary equations, We will describe basic local bifurcation theory based on implicit function theorem and global bifurcation theorem based on topological degree. Examples include Turing bifurcation induced by diffusion or cross-diffusion, loops of stationary solutions in predator-prey systems, and also bounded branch connecting semi-trivial solutions in predator-prey systems. We will also present Hopf bifurcation theorem for reaction-diffusion systems, including the case of delayed reaction-diffusion equations. In particular, we show the existence of spatial non-homogeneous periodic orbits in predator-prey systems. Global bifurcation of periodic orbits will also be discussed.

In this talk, the 3D inviscid Primitive Equations related to weather prediction and oceanography are considered. Considering an expansion of the inviscid Primitive Equations in the vertical direction, we derive an in nite set of equations which have been studied in different contexts by Qingshan Chen, A. Rousseau, R. Temam and J. Tribbia. In this work, a new set of nonlocal boundary conditions have been implemented. Our goals are twofold; on the one hand, we want to numerically con rm that the proposed boundary conditions, proven suitable for the linearized equations, are also suitable for the nonlinear equations. On the other hand, we want to numerically verify the transparency property of the proposed boundary conditions. After introducing the necessary concepts, we will discuss the well-posedness of the equations for the zero mode (barotropic mode) which is not classic. Then, we will describe the applied numerical schemes and show results of numerical simulations performed in a nested environment.

In this talk, we introduce the zeta function of a finite complex arising from a p-adic reductive Lie group $G$(of some special types). This zeta function is an analogue of the local zeta function of varieties, such that it is a rational function and has a homological interpretation. We will study this zeta function from both combinatorial and representation-theoretic view points. In the end, we will discuss its relation to local L-factors of automorphic representations.

Reference: Hanspeter Schmidli, Stochastic Control in Insurance, This is a study seminar to learn the uses of stochastic control in insurance. We plan to study the materials based on a book by Hanspeter Schmidli. The seminar gives a chance to learn the uses of stochastic control method in solving the optimization problems. The book in particular discusses some problems from insurance. They include: the optimal dividend in risk theory, minimizing the ruin probabilities, and the optimal investment problem. We plan to meet on Wednesday in the morning for every two weeks. The book by Schmidli consists of 4 chapters. The main issues discussed in this seminar are organized as follows. First, we will start with the concept of stochastic control in discrete time and introduce the discrete-time dynamic programming. The stochastic control in continuous time will be illustrated in the second part. Furthermore, several applications in insurance will be discussed after presenting Hamilton-Jacobi-Bellman approach. Then, the problems in life insurance with the continuous time case will be discussed, and the last is the asymptotic properties of the value function obtained from Hamilton-Jacobi-Bellman equation.

Credit risk associated with credit derivatives has impacted our financial systems tremendously. The Gaussian copula plays a crucial role for the boost of credit derivative market because the correlation structure and its risk are easy to be built and calculated, respectively. Due to the simplicity of Gaussian copula, it has been blamed for the collapse of the trillion-dollar credit derivative market. In this talk, I will start from a Gaussian integration but in high dimension. An efficient importance sampling method to measure dependence tail risk is justified asymptotically and its numerical performance is compared with mvncdf.m, a Matlab code. Then we generalize these results to (1) estimation of probabilities associated with order statistics and multivariate Student-t distribution, and (2) the first passage time problem related to stochastic systems. In the end, I shall introduce a nonparametric Fourier transform method and its correction schemes for estimation of the dynamic volatility matrix. This allows a new era for accessing various problems related to volatility and correlation dynamics.

A fundamental fact on Fourier analysis is that the Fourier transform is a unitary transform from on the "time" space to on the "frequency(=phase)" space. A important class of singular integral operators is defined via the Fourier transform. For example the Hilbert transform is a typical example of sigular integral operators realized by the Fourier multiplier. A more advanced example will be a class of PSIDO. In this talk I will present a decomposition formula of the "frequency" space. This talk shall be oriented to appications of this new decomposition formula as well as the related problem of a.e. convergence of the Fourier transform. Finally, the material of this talk is the following paper by the speaker, which was published from Michigan Mathematical Journal. Maximal operator for pseudo-differential operators with homogeneous symbols.

A Banach space X is called a G.T.-space if every continuous linear operator from X to l_2 is 1-summing. In this talk, first we will give characterizations of G.T.-spaces in terms of tensor products. Then we will discuss some properties of Littlewood-Orlicz operators which are related to G.T.-spaces.

Let X be a finite space and π be an underlying probability on X. For any real-valued function f defined on X , we are interested in calculating the expectation of f under π. Let X(1), X(2), …, X(n),… be a Markov chain generated by some transition matrix P with invariant distribution π. The time average, 1/n(f (X(1))+ f (X(2))+…+ f (X(n))), is a reasonable approximation to the expectation, E_π[f (X)]. Which matrix P minimizes the asymptotic variance of 1/n(f (X(1))+ f (X(2))+…+ f (X(n)))? The answer depends on f. Rather than a worst-case analysis, we will identify the set of P that minimize the average asymptotic variance, averaged with respect to a uniform distribution on f. A simulation study of a new algorithm called locally optimal sampler is presented. This is a joint work with Ting-Li Chen, Wei-Kuo Chen, and Hui-Ming Pai.

Percolation or Self-Avoiding walk have remained to this day as one of the most outstanding interesting problems in statistical mechanics and probability. In this talk, I will introduce these models and describe some critical phenomenon for them.

There is increasing demand for models of time-varying and non-Gaussian dependencies for multivariate time-series. Available models suffer from the curse of dimensionality or restrictive assumptions on the parameters and the distribution. The promising class of models are the hierarchical Archimedean copulae (HAC) that allow for non-exchangeable and non-Gaussian dependency structures with a small number of parameters. In this paper we develop a novel adaptive estimation technique of the parameters and of the structure of HAC for time-series. The approach relies on a local change point detection procedure and a locally constant HAC approximation. Typical applications are in the financial area but also recently in the spatial analysis of weather parameters. We analyse the time varying dependency structure of stock indices and exchange rates. We find that for stock indices the copula parameter changes dynamically but the hierarchical structure is constant over time. Interestingly in our exchange rate example both structure and parameters vary dynamically. Keywords: copula; multivariate distribution; Archimedean copula; adaptive estimation.

In this talk, I try to give an overview of deterministic and stochastic optimal control theory as well as its relationship with partial differential equations. In the latter half, I also present some of my recent results related to this subject.

The price of a financial option equals the discounted expected payoff of the option under the risk-neutral measure. The density that reproduces the observed option price is called the risk-neutral or state price density and is used for a variety of important activities in finance, including providing an arbitrage-free tool for pricing complex and less liquid securities. The importance of understanding this density with respect to asset pricing and risk management has led to a competing number of approaches for making inference about the state price density. We start by proposing a finite- dimensional model for the state price density in a Bayesian framework. This modeling approach can be viewed as a Bayesian Quadrature model, where the locations and weights of support points in the finite-dimensional representation of the risk-neutral density are random variables. We assess the performance of the proposed model using simulation studies based on synthetic data and then by contrasting the method with a number of competing methods using S&P 500 index option data. In contrast to European options, American options can be exercised any time prior to maturity. We show how our Bayesian Quadrature approach can be extended to make inference for American options. To tackle this problem, we propose a Bayesian implied random tree model as an extension of the Bayesian Quadrature approach by building a unique binomial tree similar to Rubinstein (1994). The benefits of our approach are demonstrated via simulation study and empirical studies using S&P 100 index option data.

The discontinuous Galerkin (DG) method, which is a finite element method suitable for solving convection dominated partial differential equations (PDEs). This method has gained a lot of popularity in recent years because of its nice mathematical properties in stability and convergence, its flexibility to many different PDEs from applications, and its efficiency for adaptivity and parallel implementation. In the mini-course, we will give a general introduction and a detail implementation of the DG methods for solving time-dependent, convection-dominated PDEs, including the hyperbolic conservation laws, convection-diffusion equations, and Hamilton-Jacobi equations. We will discuss cell entropy inequalities, nonlinear stability of DG methods.

In the presentation we will first give an introduction to the discontinuous Galerkin (DG) method, which is a finite element method suitable for solving convection dominated partial differential equations (PDEs). This method has gained a lot of popularity in recent years because of its nice mathematical properties in stability and convergence, its flexibility to many different PDEs from applications, and its efficiency for adaptivity and parallel implementation. In the second part of the talk we will describe our recent work on a class of new limiters, called WENO (weighted essentially non-oscillatory) type limiters, for Runge-Kutta discontinuous Galerkin (RKDG) methods. The goal of designing such limiters is to obtain a robust and high order limiting procedure to simultaneously obtain uniform high order accuracy and sharp, non-oscillatory shock transition for the RKDG method. We adopt the following framework: first we identify the "troubled cells", namely those cells which might need the limiting procedure; then we replace the solution polynomials in those troubled cells by reconstructed polynomials using WENO methodology which maintain the original cell averages (conservation), have the same orders of accuracy as before, but are less oscillatory. These methods work quite well in our numerical tests for both one and two dimensional cases, which will be shown in the presentation.

In the context of risky assets driven by a general semimartingale, the hedging problem is formulated taking into account a penalization of the priors. A dual formulation of the problem is obtained, and the relations with stochatic optimal control are explored, using the theory of viscosity solutions.

The discontinuous Galerkin (DG) method, which is a finite element method suitable for solving convection dominated partial differential equations (PDEs). This method has gained a lot of popularity in recent years because of its nice mathematical properties in stability and convergence, its flexibility to many different PDEs from applications, and its efficiency for adaptivity and parallel implementation. In the mini-course, we will give a general introduction and a detail implementation of the DG methods for solving time-dependent, convection-dominated PDEs, including the hyperbolic conservation laws, convection-diffusion equations, and Hamilton-Jacobi equations. We will discuss cell entropy inequalities, nonlinear stability of DG methods.

The discontinuous Galerkin (DG) method, which is a finite element method suitable for solving convection dominated partial differential equations (PDEs). This method has gained a lot of popularity in recent years because of its nice mathematical properties in stability and convergence, its flexibility to many different PDEs from applications, and its efficiency for adaptivity and parallel implementation. In the mini-course, we will give a general introduction and a detail implementation of the DG methods for solving time-dependent, convection-dominated PDEs, including the hyperbolic conservation laws, convection-diffusion equations, and Hamilton-Jacobi equations. We will discuss cell entropy inequalities, nonlinear stability of DG methods.