歷史演講 PAST TALKS

Abstract: Multiscale problems arise in many scientific and engineering disciplines. A typical example is the modelling of flow in a porous medium containing a number of low and high permeability embedded in a matrix. Due to the high degrees of variability and the multiscale nature of formation properties, not only is a complete analysis of these problems extremely difficult, but also numerical solvers require an excessive amount of CPU time and storage. In this talk, I will introduce multiscale numerical methods for the elliptic equations arising in interface and two-phase flow problems. The model problems we consider are motivated by the multiscale computations of flow and transport of two-phase flow in strongly heterogeneous porous media. Although the analysis is carried out for simplified model problems, it does provide valuable insight in designing accurate multiscale methods for more realistic applications. In the first part, I will introduce a multiscale finite element method which is able to accurately capture solutions of elliptic interface problems with high contrast coefficients by using only coarse quasiuniform meshes, and without resolving the interfaces. The method has an optimal convergence rate of in the energy norm and in the L_2 norm. The hidden constants in these estimates are independent of the ``contrast'' (i.e. ratio of largest to smallest value) of the PDE's coefficients. I conduct some numerical experiments to confirm the optimal rate of convergence of the proposed method and its independence from the aspect ratio of the coefficients. In the second part, we discuss a recent work on 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 assume that over the same physical domain there exists an effective mass conservation equation at the continuum scale which could have been solved can be solved efficiently. Our coupling method uses local simulations on sampled domains at network scale to evaluate the continuum equation and thus solve for the pressure in the domain. In the case where classical homogenization applies, we prove convergence of the proposed multiscale solutions to the homogenized equations. We also extended the method 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.

Abstract: An operator algebra A acting on a Hilbert space is said to have the closability property if every densely de‾ned linear transformation commuting with A is closable. In this talk, I will review the results obtained in the ‾eld and other related properties. Then I will give necessary and su±cient conditions for a normal operator N such that the von Neumann algebra generated by N has the closability property. I will also characterize the operator T of class C0 for which the algebra H1(T) = fu(T) : u 2 H1g has the closability property which is shown to be equivalent to two signi‾cant properties in the class C0.

摘要： In credit risk modeling, Gaussian and Student’s t variates arise primarily from the copula method to retain certain correlation structures among defaultable assets. We propose efficient importance sampling algorithms to estimate lower tail probabilities of these two variates in any finite dimension. Variances of importance sampling estimators are shown asymptotically optimal by means of the large deviation theory and a truncation argument. Numerical comparisons with commercial codes, such as mvncdf.m and mvtcdf.m in Matlab, demonstrate robustness and efficiency of our proposed algorithms. Moreover, the flexibility of these algorithms can be seen from an application of probability estimation for the nth-to-default, i.e., the nth order statistic, given a credit portfolio.

摘要： In this lecture, I am going to go through very basic things in SPDEs by Walsh's approach. We will define the integral with respect to a martingale measure and define the solutions to stochastic heat equations. All materials needed will be covered, such as white noise, martingale measures.

Abstract: In this talk, I will mention the following results in my thesis. 1. Degree unique graphs 2. Embedding results for medians 3. Minimum status 4. Variance spectrum

Reference： Hanspeter Schmidli, Stochastic Control in Insurance

Abstract: This talk presents a very recent result asserting the global (in time) regularity of classical solutions to the 2D anisotropic Boussinesq equations with only vertical dissipation. The Boussinseq equations concerned here model many geophysical flows such as atmospheric fronts and ocean circulations. Mathematically the 2D Boussinesq equations serve as a lower- dimensional model of the 3D hydrodynamics equations. In fact, the 2D Boussinesq equations retain some key features of the 3D Euler and Navier-Stokes equations such as the vortex stretching mechanism. In the last few years the global regularity problem on the 2D Boussinesq equations with partial dissipation has attracted considerable attention. The global regularity problem for the 2D anisotropic Boussinesq equations with only vertical dissipation is very challenging due to the lack of control on the horizontal derivatives. To solve this problem, we bound the derivatives in terms of the L^\infty -norm of the vertical velocity v and prove that the L^r-norm of v for any r<\infty does not grow faster than the square root of r log r. A delicate interpolation inequality connecting L^\infty-norm of v and the L^r-norm of v then yields the desired global regularity. This ia a joint work with C. Cao.

Abstract: In this talk, I will first introduce notions (rational, log terminal, log canonical singularities, and multiplier ideals) arising from birational geometry in characteristic 0. I will then turn to F-singularities (F-rationality, strong F- regularity, and F-purity) and test ideals, which are defined using the Frobenius morphism in positive characteristics and are important concepts in tight closure theory. The fascinating connections between these two areas have been discovered and explored for decades. Many theorems in one area were motivated by the ideas (or even proved by the techniques) from the other area. The last part of this talk is to illustrate the powerfulness of these connections by mentioning the following three works of mine. 1. Determinantal varieties are log terminal in the sense of de Fernex and Hacon. 2. New proof of normality and rationality of singularities of Schubert varieties. 3. (jointed with K. Schwede and W. Zhang) The correspondence between Cartier modules and certain generalization of ideals defining unions of log canonical centers on toric varieties.

Abstract: In the talk, we will discuss an important class of continuous linear program known as Infinite transportation problems. These problems seek the least cost policy for transferring mass having a given distribution in one space to obtain another given distribution of mass, possibly in another space. A duality theorem will be discussed. An algorithm for the semi-infinite transportation problem is given and proved its convergence to its optimal solution. As for a special class of Infinite transportation problem, a continuous assignment problem and its algorithm will also be discussed.

Reference： Hanspeter Schmidli, Stochastic Control in Insurance

Abstract: Queueing management for the processor-sharing (PS) queue has been applied to the telecommunication and internet systems, so will be to the software-as-a-service (SaaS) business. As the pioneer queueing management paper by Naor in 1969 is for first-come-first-serve (FCFS) queues, we introduce its extension in this talk not only from FCFS to PS, but also from single-class to multiple-class queues. Consider an observable PS queue with an utility function of the reward of receiving service and the cost of waiting in system. We study the decision of to queue or not to queue from both the customer’s (self) and the system administrator’s (social) points of view, and obtain conditions of existence and uniqueness of Nash equilibrium strategy. These strategies are shown to be different with an economical explanation. We also discuss methods to induce the self-optimal decision approaching the social-optimal decision.

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

Abstract: In this talk, we first derive the CR version of Lichnerowicz formula. Secondly, by applying the CR Lichnerowicz formula, we are able to obtain a parallel result of the eigenvalue estimate for the Dirac operator in Riemannian geometry. That is, in a three-dimensional closed pseudohermitian manifold with vanishing pseudohermitian torsion, any eigenvalue of the Dirac operator has a uniform lower bound.

Abstract: (I) Motivated from credit risk modeling, this paper extends the two-dimensional first passage time problem studied by Zhou (2001) to any finite dimension by means of Monte Carlo simulation. We provide an importance sampling method to estimate the joint default probability, and apply the large deviation principle to prove that the proposed importance sampling is asymptotically optimal. Our result is an alternative to the interacting particle systems proposed by Carmona, Fouque, and Vestal (2009). (II) In credit risk modeling, Gaussian and Student’s t variates arise primarily from the copula method to retain certain correlation structures among defaultable assets. We propose efficient importance sampling algorithms to estimate lower tail probabilities of these two variates in any finite dimension. Variances of importance sampling estimators are shown asymptotically optimal by means of the large deviation theory and a truncation argument. Numerical comparisons with commercial codes, such as mvncdf.m and mvtcdf.m in Matlab, demonstrate robustness and efficiency of our proposed algorithms. Moreover, the flexibility of these algorithms can be seen from an application of probability estimation for the nth-to-default, i.e., the nth order statistic, given a credit portfolio.

Reference： Hanspeter Schmidli, Stochastic Control in Insurance

Abstract: Chern classes play a fundamental role in modern geometry. For example, they appear in important theorems like Gauss-Bonnet and Grothendieck-Riemann-Roch. For algebraic vector bundles over projective varieties, we are able to define their Chern classes in morphic cohomology by a geometric construction due to Friedlander and Lawson. I will explain how these morphic Chern classes coincide with original Chern classes by the natural transformation from morphic cohomology to singular cohomology.

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

Abstract: In this lecture, we briefly introduce the Monge-Kantorovich Problem and the Knothe-Rosenblatt rearrangement as an application. If time permitted, we also discuss the stochastic control analogue. The lectures will cover the following. Chapter 1. Monge-Kantorovich Problem 1 Monge's problem 2 Kantorovich's problem 3 Duality Theorem 4 Solution to Kantorovich's problem 5 A simple proof of Duality Theorem 6 One-dimensional case Chapter 2. Knothe-Rosenblatt rearrangement 7 Knothe-Rosenblatt rearrangement 8 Brunn-Minkowskii's inequality

In this talk, we consider a stochastic control problem for diffusion process. We discuss dynamic programming approach. This includes the validity of dynamic programming principle, the derivation of HJB equation, concept of viscosity solution for HJB equation and formulation of the verification theorem. We only discuss them briefly in this short note. Reference：1. Hanspeter Schmidli, Stochastic Control in Insurance 2. W. H. Fleming and H.M. Soner, Control Markov Processes and Viscosity Solution

Abstract: In this talk, we are concerned with the elliptic equation involving the Sobolev-Hardy type nonlinearities with the singularity on the boundary. As a result, we can show the existence of a positive solution provided that the mean curvature at the singularity is strictly negative, where we put the Sobolev type and the Sobolev-Hardy nonlinearities at the same time to the equation.

Abstract: In this talk, I would review the notion of contact open books, give some examples, and use them to study certain symplectomorphisms. This is based on a joint work with F. Ding and O.van Koert.

摘要： Metric entropy, a concept invented by Kolmogorov, plays a fundamental role on optimal statistical estimation. It determines how well an unknown function in a given target class can be learned. Efforts have been directed at obtaining flexible learning procedures that optimally adapt to various possible characteristics of the data generating mechanism. A question that addresses the issue of how far one can go in this direction is: Given a regression procedure, however sophisticated it is, how many regression functions are estimated accurately? In this work, for a given sequence of prescribed estimation accuracy (in sample size), we give an upper bound (in terms of metric entropy) on the number of regression functions for which the accuracy is achieved. Interesting consequences on adaptive and sparse estimations are also given.

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

Abstract: In the previous talks, we have learned that the HJB equation can be derived from a deterministic control problem. This is a nonlinear first order equation that a solution in general is not smooth. Moreover, the equation in general has multiple solutions. We introduced the concept of viscosity solution for HJB equation. We will show that the value function of a deterministic control problem is the unique viscosity solution of its HJB equation. Reference： 1. Hanspeter Schmidli, Stochastic Control in Insurance 2. W. H. Fleming and H.M. Soner, Control Markov Processes and Viscosity Solution

Abstract: We consider the perpetual American strangles in the geometric jump-diffusion models. We assume further that the jump distribution is a mixture of exponential distributions. To solve the corresponding optimal stopping problem for this option, by using the ODE approach, we derive a system of equations that is equivalent to the associated free boundary problem with smooth pasting condition. We verify the existence of the solutions to these equations. Then, in terms of the solutions together with a verification theorem, we solve the optimal stopping problem and hence find the optimal exercise boundaries and the rational prices for the perpetual American strangles. In addition we work out an algorithm for computing the optimal exercise boundaries and the rational prices of these American contracts.

Reference： Brownian Motion and Stochastic Calculus 2/E, Ioanni Karatzas and Steven E. Shreve.

Abstract: In this talk, we will study the Bergman kernel for a degeneration of Riemann surfaces with constant curvature -1.

In the previous talks, we have learned that the HJB equation can be derived from a deterministic control problem. This is a nonlinear first order equation that a solution in general is not smooth. Moreover, the equation in general has multiple solutions. In this talk we will introduce the concept of viscosity solution.. Under general condition, we can prove that the value function of the control problem is the unique viscosity solution of HJB equation. Reference：1. Hanspeter Schmidli, Stochastic Control in Insurance 2. W. H. Fleming and H.M. Soner, Control Markov Processes and Viscosity Solution

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

In the first talk we consider deterministic control problem. We formulate the dynamicprogramming priniciple. For discrete time model, in principle one can use this tosolve the problem. That is, a complicated problem due to longer time period becomeseasier to treat by consider the problem in several steps, in each step we have a simplerproblem to solve. In continuous time model, the same idea can be applied, but it ismore useful to use the dynamic programming principle in an infinitesimal form. Theresult is the HJB (Hamilton-Jacobi-Bellman) equation, which is a nonlinear partialdifferential equation. In the second talk, we discuss HJB equation ,the generalizedsolution and verification theorem . The latter is a theorem stating that a solution ofcontrol problem can be derived from a solution of HJB equation.. We will alsointroduce viscosity solution. We discuss the uniqueness of the solution. Reference： 1. Hanspeter Schmidli, Stochastic Control in Insurance 2. W. H. Fleming and H.M. Soner, Control Markov Processes and Viscosity Solutions

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

Dyanimic programming is a very useful idea used in the study of control problem. We first consider a deterministic control problem to show the idea. We derive the dynamic programming principle and then the dynamic programming equation (or Hamilton-Jacobi-Bellman equation, HJB equation). We discuss the verification theorem. The theorem state that a solution of HJB equation is the value function of the control problem. It also shows that a control derived from the solution is optimal. The HJB equation usually is a PDE. We also briefly discuss the concept of viscosity solution. This is very useful for HJB equation. The similar ideacan be also used to study the stochastic control problem. We can formally derivedynamic programming principle and HJB equation. There is a major difference between deterministic and stochastic control problem. A rigorous proof of the dynamic programming principle usually may require some continuity property of the value function. Reference：1. Hanspeter Schmidli, Stochastic Control in Insurance 2. W. H. Fleming and H.M. Soner, Control Markov Processes and Viscosity Solutions

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

This mini-course aims at a quick but self-contained introduction of the mathematical theory and computer algorithms for mixed finite element method. The content includes the origin of mixed problem or saddle-point problem for which there are several simultaneous independent variables as the solutions, and the functional analysis of the well-posedness for the general mixed variational problem in Hilbert spaces, and the mixed finite element approximation theory (well-posedness, convergence and error estimates), with application to second-order elliptic problem and Stokes problem, including the introduction of H(div)-conforming elements, and the implementation issue of the resultant algebraic system from the mixed finite element problem by using the Uzawa algorithm (one of the conjugate gradient algorithm applied to the Lagrange multiplier in the mixed problem). This mini-course is for graduates in applied and computational mathematics, numerical analyst, engineers in computing, and interested researchers from interdisciplinary scientific and engineering computing.

The numerical range W(A) of a matrix A and the associated ternary form FA(t, x, y) are useful to treat a pair of dual algebraic curves appeared in some problems in the area of Mathematical Physics.