歷史演講 PAST TALKS

Abstract: We study a 1D model for the 3D incompressible Euler equations in axisymmetric eometries, which can be viewed as a local approximation to the Euler equations near the solid boundary of a periodic cylinder. We prove the local well-posedness of the model in standard energy spaces and establish its finite-time blowup for certain classes of smooth initial data. It is hoped that the results obtained on the 1D model will be useful to the analysis of the full 3D problem, whose loss of regularity in finite time has been reported in a recent numerical study.

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Abstract: We show that a compact operator A is a multiple of a positive operator if and only if the spectrum of AB is always a subset of the closure of the product of the numerical ranges W(A) and W(B) for any (rank one) operator B. The sufficiency may fail for a general operator A.

Abstract: A new theory is developed for Fortin interpolation of H(curl)-conforming elements, aiming at very weak solution with piecewise non H1 space regularity in the solution itself and its curl. The non H1 space regularity is typically due to irregular domain boundary and discontinuous anisotropic and inhomogeneous media. When compared with the earlier theory, the significant findings feature several aspects. The Fortin interpolation is suitable for piecewise and very low regular function with non H1 space regularity. The technique is new, with the multiple application of the regular-singular decomposition. Optimal error bounds are established for piecewise non H1 space solutions. Discrete compactness is also shown in the application of the new theory to Maxwell eigenvalue problem. When compared with the so-called co-chain bounded projection, the Fortin interpolation we have developed is suitable for piecewise non H1 space solutions which typically live in discontinuous, anisotropic and inhomogeneous media. Another important advantage is the Fortin interpolation preserves the discrete divergence. These points are not possessed by the co-chain bounded projection. As an application, we consider the recently developed delta-regularization edge or Nedelec element method with a small parameter delta for the double curl problem with divergence-free constraint. The problem is posed in discontinuous, anisotropic and inhomogeneous media. With the help of the Fortin interpolation, uniform in the parameter delta, optimal error convergence of the resultant finite element solution is shown. A series of numerical experiments are performed to illustrate the theoretical results. In addition, adaptive algorithms are developed. An analysis is carried out and the convergence and the optimality are obtained, uniformly with respect to the parameter delta. Likewise, the Fortin interpolation developed plays a key role in the analysis. Numerical results are presented to confirm the convergence and the optimality of the adaptive algorithm.

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

摘要： We had existence, uniqueness, moment estimates, regularity and did a little about chaotic characterization of the solution to stochastic fractional heat equations [SFHE]. In this talk, we will continue to work on chaotic characterization. This is an ongoing work with Conus, Joseph and Khoshnevisan.

Abstract: This is an ongoing joint work with Ming-Cheng Shiue. Under suitable smallness assumption on the initial data, we prove the long-time H^1 boundedness for the solutions of three dimensional primitive equations obtained by an implicit Euler scheme.

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Abstract: In this talk, we are concerned with studying some second order elliptic partial differential equations. We are devote to establishing some qualitative properties of solutions, including existence, uniqueness and structure of solutions to three specific types of nonlinear elliptic equations. First, we study a gradient constraint equation which is related to a stochastic optimal control problem. We offer the existence and uniqueness of positive radial solutions with certain behavior under weaker conditions on nonlinearity. Second, we consider a semilinear elliptic equation on the hyperbolic space. The asymptotic behavior, existence and uniqueness of positive singular solutions at the origin are proved. In addition, we discuss the structure of solutions of various types via the Pohozaev identity. Finally, we deal with the Hardy-Sobolev equations and investigate behaviors, existence and uniqueness of solutions for different exponents.

摘要: In this talk I would introduce three parts (if time is available): 1.How to be an actuary and actuary exam (SOA). 2.Traditional Actuary (Life Contingencies) 3.Nontraditional Actuary (Mortgage-Backed Securities) 參考: 1.中華民國精算學會網站 http://www.airc.org.tw/2011/index.php 2.SOA http://www.soa.org/ 3.Models for Quantifying Risk, Third Edition, 2009, Cunningham, R., Herzog, T. and London, R.L 4.Handbook of Fixed Income Securities, Fabozzi, F.J., Eighth Edition, 2012, McGraw-Hill 5.Financial Engineering & Computation: Principles, Mathematics, Algorithms, Y.-D. Lyuu, . Cambridge University Press, 2002.

摘要: Under the condition that the diffusion coefficients are uniformly elliptic, we study the Euler scheme of (non-Markovian) stochastic differential equations. While the strong approximation (i.e. approximation in Lp) and the weak approximation (i.e. approximation in terms of expectation) have been well studied, we study an approximation which has not been very much touched so far-- the approximation of probability densities. We are concerned with the following problems: do the corresponding densities converge and, if they do, in which Holder space the convergence takes place? Using differential analysis on the Wiener space (Malliavin's stochastic calculus of variations), especially Watanabe's theory of pullback of distributions and fractional Sobolev space over the Wiener space, we answer the above problems. To this end we localize fractional calculus of Wiener functionals. We obtain the relation between the Holder indexes of the spaces in which the convergence can take place and those of the spaces in which the coefficients lay in.

摘要： The calculation of multivariate normal probabilities is of great importance in many statistical and economic applications. This paper proposes a spherical Monte Carlo method with both theoretical analysis and numerical simulation. To begin with, the multivariate normal probability is rewritten via an inner radial integral and an outer spherical integral by the spherical transformation. For the outer spherical integral, we apply an integration rule by randomly rotating a predetermined set of well-located points. To find the desired set, we derive an upper bound for the variance of the Monte Carlo estimator and propose a set which is related to the kissing number problem in sphere packings. For the inner radial integral, we employ the idea of antithetic variates and identify certain conditions so that variance reduction is guaranteed. Extensive Monte Carlo experiments on some probabilities calculation confirm these claims. This is a joint work with Cheng-Der Fuh and Ming-Hsuan Kang.

摘要： In this talk I would talk about one dimensional Burgers' turbulence with white noise initial condition (interpreted in the sense of the weak derivative of two-sided Brownian motion). Basic solution properties will be investigated, and an open problem concerning the Hausdorff dimension of the complement of the constant intervals of the Lagrangian map of the solution will also be briefly introduced.

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摘要： For the past three seminars, we discussed the existence, uniqueness, moment estimates and regularity of the solution to stochastic fractional heat equations [SFHE]. In this forth talk, we will start to work on chaotic characterization of the solution to SFHE. This is an ongoing work with Conus, Joseph and Khoshnevisan.

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摘要： Camassa-Holm (CH) equation is one of the important integrable systems which describes the behaviors of shallow water waves. The long-time asymptotic behaviors of the solutions is studied by the Riemann-Hilbert methods. In this talk, we consider the CH equation with a specified initial data to obtain the scattering data which is non-reflectionless by solving the scattering problem for the Lax pair of CH equation. under this data, from numerical simulation we can obtain the long-time behaviors of solutions which guarantee the previous results.

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

摘要： Diaconis 和 Sallof-Coste 在2006年獲得了生滅鏈separation cutoff 發生的一個判別準則，此準則涉及所有特徵值。而我們將給出只依賴于生速和死速的顯式判別準則。進一步，我們用兩種方法來估計最快強平穩時的矩，從而得到另一個等價的separation cutoff顯式判別準則。關於單生過程separation cutoff 的相關結果我們基本完成，也將簡單介紹。此工作是與毛永華教授合作完成的。

摘要： 單生過程，作為最簡單過程－生滅過程的自然推廣，本身具有可靠的研究背景。由於單生過程至多有單個流出邊界極點，因此是對經典問題可預期有顯式判別準則的最大類過程。這些優點使得單生過程成為研究無窮維反應擴散過程的基本工具。單生過程通常是非對稱的，因此被視為非對稱過程的突出代表。而關於非對稱過程的研究，至今所得結果并不完善。但對於單生過程，研究成果相對完整。在這個系列演講中，我們將介紹單生過程經典問題的判別準則，包括唯一性，常返性，遍曆性和強遍曆性，還包括回返時的多項式階矩和指數階矩，平穩分布的表示等等。最后，介紹我們最近關於單生過程Poisson方程解的研究結果。這個成果顯示：以上所有問題可以有一個統一的處理，包括回返時，滅絕時的分布，滅絕概率等等。

摘要：In this presentation, we develop an efficient moving mesh weighted essentially non-oscillatory (WENO) method for one-dimensional hyperbolic conservation laws. The method is based on the quasi-Lagrange approach of moving mesh strategy with which the mesh is considered to move continuously in time. Several issues arising from the implementation of the scheme, including mesh smoothness, mesh movement restriction, and computation of transformation relations, and their effects on the accuracy of the underlying scheme have been addressed. Particularly, it is found that a least squares smoothing can be used to effectively smooth the mesh and the transformation relations can be computed using either high order finite differences or WENO applied to some geometric conservation laws. Moreover, mesh movement can cause WENO schemes to become unconditionally unstable. A simple strategy is used to restrict the mesh movement and recover the stability. Numerical results are presented to demonstrate the accuracy and shock-capturing ability of the new scheme.

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

摘要： For the past two seminars, we discussed the existence and uniqueness of the solution to stochastic fractional heat equations [SFHE] by appealing to stochastic Young’s inequality. We also had moment estimates and computed the Holder regularity in spatial variable. In this third seminar, we will continue on Holder regularity but in time variable.

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摘要： 單生過程，作為最簡單過程－生滅過程的自然推廣，本身具有可靠的研究背景。由於單生過程至多有單個流出邊界極點，因此是對經典問題可預期有顯式判別準則的最大類過程。這些優點使得單生過程成為研究無窮維反應擴散過程的基本工具。單生過程通常是非對稱的，因此被視為非對稱過程的突出代表。而關於非對稱過程的研究，至今所得結果并不完善。但對於單生過程，研究成果相對完整。在這個系列演講中，我們將介紹單生過程經典問題的判別準則，包括唯一性，常返性，遍曆性和強遍曆性，還包括回返時的多項式階矩和指數階矩，平穩分布的表示等等。最后，介紹我們最近關於單生過程Poisson方程解的研究結果。這個成果顯示：以上所有問題可以有一個統一的處理，包括回返時，滅絕時的分布，滅絕概率等等。

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

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摘要： The ready availability of advanced general mathematical software necessitates a revision of the approach to the teaching of mathematics to students of science and engineering, and to the learning of mathematics as a tool for application in various fields

摘要: Last semester, we discussed the existence and uniqueness of the solution to stochastic fractional heat equations [SFHE] by appealing to stochastic Young’s inequality. In this talk we will focus on local behaviors, such as regularity and localization.

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摘要: In this talk I want to present some recent results on random walks in random scenery which are observed through some U-statistics. The limit law is a Levy-sheet in the case when the random walk is transient. When the random walk is recurrent, the limit law is obtained through stochastic integration of the local time process with respect to the Levy-sheet. This is a joint work with M. Wendler and F. Pene.

2014 Lectures on Computational and Applied Mathematics

Speakers:
(a) Thomas Hou, Caltech, USA
(b) Quoqiang Shi, Tsinghua University, China

Topics:
(a) Data Driven Time-Frequency Analysis
(b) Data-Driven Stochastic Methods for Solving Stochastic PDEs
(c) Recent Progress on the Search of 3D Euler Singularities
(d) Finite Integral Method: A numerical approach for Poisson equation on point clouds

Time:

March 20(Thursday), 2014 PM 1:30-4:00

March 21, 2014 (Friday) AM 9:30-12:00 & PM1:30-4:00

Place: Hong-Jing Building M107

Abstract:
(1). Data Driven Time-Frequency Analysis
In this talk, we review some recent progress on our data-driven time-frequency analysis method for studying trend and instantaneous frequency of nonlinear and non-stationary data. This method is inspired by the Empirical Mode Decomposition method (EMD) and the recently developed compressed sensing theory. The main idea is to look for the sparsest representation of multiscale data within the largest possible dictionary consisting of intrinsic mode functions. This problem can be formulated as a nonlinear optimization problem. In order to solve this optimization problem, we propose a nonlinear matching pursuit method by generalizing the classical matching pursuit. One important advantage of this nonlinear matching pursuit method is it can be implemented very efficiently and is very stable to noise. Further, we provide a convergence analysis of our nonlinear matching pursuit method under certain scale separation assumptions. Extensive numerical examples will be given to demonstrate the robustness of our method and comparison will be made with the EMD/EEMD method. We also apply our method to study data without scale separation, data with intra-wave frequency modulation, and data with incomplete or under-sampled data.

(2). Data-Driven Stochastic Methods for Solving Stochastic PDEs.
Uncertainty arises in many complex real-world problems of scientific and engineering interests. Many of these problems involve multiple scales in both space and time, which may vary in several orders. We have recently developed an effective dynamically bi-orthogonal method (DyBO) for solving time-dependent stochastic PDEs. The main idea is to construct the most compact representation of the stochastic PDEs by tracking the Karhunen-Loeve expansion dynamically. We achieve this by deriving an equivalent set of evolution systems that govern the stochastic and deterministic basis without the need to form the covariance matrix or to compute its eigen-decomposition. Unlike other reduced model methods, our method constructs the reduced stochastic basis on-the-fly. We also propose an adaptive strategy to dynamically remove or add modes, and develop an efficient parallel algorithm for our method. Applications to solve 2D incompressible Navier-Stokes equations and the Boussinesq approximation with Brownian motion forcing demonstrate the effectiveness of the DyBO method. We will also introduce a data-driven method for solving stochastic elliptic problems with multiscale coefficient in a multi-query setting. A multiscale multilevel Monte-Carlo method has been developed to handle stochastic problems with high input dimensions. Our numerical experiments show that these methods could offer considerable computational saving over existing methods.

(3). Recent Progress on the Search of 3D Euler Singularities.
Whether the 3D incompressible Euler or Navier-Stokes equations can develop a finite time singularity from smooth initial data with finite energy is one of the most challenging questions in applied mathematics and fluid dynamics. We review some recent theoretical and computational studies of the 3D Euler equations which show that there is a subtle dynamic depletion of nonlinear vortex stretching due to local geometric regularity of vortex filaments. Our study suggests that the convection term could have a nonlinear stabilizing effect for certain flow geometry. This is demonstrated through two reduced models of the 3D incompressible Navier-Stokes equations. The first model is a new exact 1D model for the axisymmetric Navier-Stokes equation along the symmetry axis. We show that local flattening of the vortex structure and the effect of convection could lead to dynamic depletion of the vortex stretching term. In the second model, we derive a 3D model of the axisymmetric Navier-Stokes equation by removing the convection term from the reformulated Navier-Stokes equations. We show both numerically and analytically that the solution of this 3D model develops a finite time singularity from smooth initial data with finite energy. Finally we present a new class of solutions to the 3D Euler equations which could lead to a strong nonlinear alignment in the vortex stretching term and have the potential to develop a finite time singularity.

贊助單位:
1. 國立中央大學數學與理論物理中心
2. 中央研究院數學推動中心
3. 國立中央大學數學系
4. 國家理論科學研究中心

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