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Event(s) on January 2014
- Friday, 3rd January, 2014
Title: Robust Utility Maximisation Via Second Order BSDEs Speaker: Dr. ZHOU Chao, Department of Mathematics, National University of Singapore, Singapore Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The problem of robust utility maximisation in an incomplete market with volatility uncertainty is considered, in the sense that the volatility of the market is only assumed to lie between two given bounds. The set of all possible models (probability measures) considered here is non-dominated. We propose studying this problem in the framework of second order backward stochastic differential equations (2BSDEs for short) with quadratic growth generators. We show for exponential, power and logarithmic utilities that the value function of the problem can be written as the initial value of a particular 2BSDE and prove existence of an optimal strategy. Finally several examples which shed more light on the problem and its links with the classical utility maximization one are provided. In particular, we show that in some cases, the upper bound of the volatility interval plays a central role, exactly as in the option pricing problem with uncertain volatility models. - Tuesday, 7th January, 2014
Title: Eikonal Equation: An Elliptic Solver Approach Speaker: Prof. Roland Glowinski, University of Houston, USA Time/Place: 11:00 - 12:00 (Preceded by Reception at 10:30am)
RRS905, Sir Run Run Shaw Building, HSH Campus, Hong Kong Baptist UniversityAbstract: The main goal of this lecture is to discuss the numerical solution of the Dirichlet problem for the Eikonal equation (EKN) |nabla u| = 1 in Omega; u=g on partial Omega
where Omega is a bounded domain of bf{R}^2. Traditionally, the equation is considered as a nonlinear hyperbolic one leading to numerical methods taking advantage of this hyperbolic feature. Motivated by the numerical solution of systems of the formnabla bf{u}(x) in O(d),
completed by boundary conditions (with O(d) the group of the d x d orthogonal matrices), a problem considered by B. Dacorogna, we have investigated the solution of (EKN) through the use of finite elements and elliptic solvers. The main idea is to approximate (EKN) by a problem of Calculus of Variations involving a Ginzburg-Landau nonlinearity and a bi-harmonic regularization (linear or nonlinear). Next, a well-chosen initial value problem associated with the Euler-Lagrange equation of the above variational problem is solved by an operator-splitting method decoupling differential operators and nonlinearity. Assuming that (EKN) has multiple solutions, the above approach allows to compute at will the maximal or the minimal solution of (EKN). This approach is easy to implement via piecewise linear approximations, ideally suited to capture the solutions of this problem, and as such is easy to implement for domains Omega of arbitrary shape, including therefore curved boundaries. The results of numerical experiments will be presented. - Thursday, 9th January, 2014
Title: Geometric analysis on decouple domains in C^{n+1} Speaker: Prof. Der-Chen Chang, Department of Mathematics and Statistics, Georgetown University, USA Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Let $Sigma={im(z_{n+1})=sum_{j=1}^n {mathcal P}_j(z_j)}$ be a decoupled domain in ${bf C}^{n+1}$. Here ${mathcal P}_j$ are subharmonic but non-harmonic polynomials of degree $m_j$ with $m_1 < cdots < m_n$. In this talk, we shall discuss possible sharp estimates for the fundamental solution ${bf K}$ of $square_b$ defined on $Sigma$. We will see the size of the operator ${bf K}$ is controlled by {it at least $2$} non-equivalent Carnot-Carath'eodory metrics. It is not so difficult to show that $Z_jbar Z_ell{bf K}$ and $bar Z_j Z_ell{bf K}$ with $jnot= ell$ are bounded on $L^p_k(Sigma)$ for $1 < p < infty$, $kin {mathbb Z}_+$. However, for some $j$ and $ell$, {it e.g.,} $Z_1Z_1{bf K}$ and $bar Z_2bar Z_2{bf K}$ are {it not} bounded on $L^2(partialOmega)$. - Tuesday, 14th January, 2014
Title: Domain decomposition methods for time-dependent problems Speaker: Dr. Felix KWOK, Section de Mathmatiques, Universit de Genve, Switzerland Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: In many applications in science and engineering, one needs to solve large, sparse linear or nonlinear systems of algebraic equations that arise from the discretization of partial differential equations. Since the number of unknowns can be in the order of millions or even billions, it is essential to solve such problems in parallel. This is particularly true for time-dependent problems, where one must solve many such systems in order to track the evolution of the physical model. One natural idea for parallelization is to use domain decomposition methods, which involves dividing the computational domain into several subdomains, solving the subdomain problems in parallel, and iterating to obtain a consistent solution. In this talk, we show two different approaches for obtaining domain decomposition methods that converge quickly to the global solution of time-dependent problems. In the first approach, we modify the coupling conditions between subdomain boundaries to obtain methods that have the same computational cost per iteration, but which converge much more quickly than for classical conditions: such methods are known as optimized Schwarz methods. In the second approach, we decompose the computational domain of the time-dependent problem into space-time strips: this leads to the so-called Schwarz or Neumann-Neumann waveform relaxation methods, where the subdomain problems are also posed in space and time. The advantage of this approach is that one can choose the spatial and time discretization for each subdomain independently to suit the local features of the problem; moreover, such algorithms can be parallelized in time in a natural way. For each approach, we provide convergence estimates as well as numerical examples to illustrate our techniques. - Thursday, 16th January, 2014
Title: The relationship between boundary value problems and analysis on singular manifolds Speaker: Prof. B.-Wolfgang Schulze, Institut für Mathematik, Universität Potsdam, Germany Time/Place: 11:00 - 12:00
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The analysis on manifolds with singularities, such as conical points, boundaries, edges, higher corners, is an active field of research, motivated by applications in natural sciences and also in mathematics, e.g., geometry, topolgy, or index theory. We focus here on ellipticity of operators with a principal symbolic hierarchy, associated with the stratification of the underlying space. The basic new ideas can be derived from the pseudo-differential analysis of boundary value problems, say, with the transmission property at the boundary. Manifolds with smooth boundary are a special case of manifolds with edge where the inner normal can be regarded as a manifold with conical singularity, locally modelled close to the boundary (edge) by a half-space (wedge). In contrast to elliptic operators on an open smooth manifold it is by no means evident how parametrices of elliptic boundary value problems are organized in terms of their principal symbolic structure. However, the answer is given in a classical paper of Boutet de Monvel [2], see also the monograph of S. Rempel and B.-W. Schulze [3].The "right" answers, given by Boutet de Monvel in terms of pairs of interior and boundary symbol, the latter being operator-valued, stand in a strange contrast to what most of the present schools in singular analysis are able to do, namely, to give similar characterizations of parametrices in the case of manifolds with singularities. However, as we point out in our talk, it is a central issue to embed the "classical" analysis of elliptic boundary value problems into a well-organized analysis of elliptic operators on a singular manifold, in particular, to establish a calculus of operators with symbolic hierarchies and to perform the necessary applications and generalizations in singular cases. In that sense, the singular analysis is of a similar beauty as their more regular special cases, though full of new challenges and possibilities of future activities, see, in particular, the monograph [4], or the joint paper [2] of D.-C. Chang, N. Habal, and B.-W. Schulze, and the references there. [1] L. Boutet de Monvel, Boundary problems for pseudo-differential operators, Acta Math. 126 (1971), 11-51. [2] D.-C. Chang, N. Habal, and B.-W. Schulze, The edge algebra structure of the Zaremba problem, NCTS Preprints in Mathematics 2013-6-002, Taiwan, 2013. appeared in J. Pseudo-Differ Oper Appl., DOI:10.1007/s11868-013-0088-7 [3] S. Rempel and B.-W. Schulze, Index theory of elliptic boundary problems, Akademie-Verlag, Berlin, 1982. [4] B.-W. Schulze, Boundary value problems and singular pseudo-differential oper-ators, J. Wiley, Chichester, 1998. - Monday, 20th January, 2014
Title: Gauss-Jacobi spectral method for fractional option pricing Speaker: Ms. GUO Xu, Department of Mathematics, Hong Kong Baptist University , Hong Kong Time/Place: 10:30 - 11:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The classical Black-Scholes pricing model, which is based on the standard geometric Brownian motion, has independent and Gaussian log-returns. However, most of the recent statistical studies on log-returns suggested that the Brownian motion hypothesis may not be always consistent. In view of the unpleasant history of the traditional Black-Scholes model in real life applications, one of the ongoing topics of mathematical finance today is to find better models for the pricing model, which might be changed from the standard Black-Scholes diffusion equation and would have favorable empirical results. On those financial models that have already been proposed, this paper focuses on the finite moment log-stable process. An efficient high-order spectral method will be employed for solving the resulting fractional partial differential equation. By simulating the pricing of European and American options, we illustrate the exibility and accuracy of the method by comparing to some previously proposed finite difference schemes. Our results indicate that the global character of the proposed method is well-suited to fractional partial integral-differential equations and can naturally engage the global behavior of the solution into account. When moving from integer-order to fractional-order pricing models, the proposed method is an attractive alternative without extra computational cost. - Monday, 20th January, 2014
Title: Circular coloring of graphs Speaker: Prof. Xuding Zhu, Department of Mathematics, Zhejiang Normal University, China Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Graph coloring are used as models for many scheduling problems. For some periodic scheduling problems, circular coloring of graphs provides an ideal model. The concept of circular chromatic number has been studied extensively by graph theorists in the past two decades. The same concept, under a different name, has also been studied by computer scientists working in the area of resource sharing systems. In this talk, I shall survey problems, results and methods used in the study of circular coloring of graphs. - Tuesday, 21st January, 2014
Title: CMIV Colloquium - Multilinear Subspace Learning: Adaptation of Tensor Decompositions for Learning Compact Features from Big Data Speaker: Dr. Haiping Lu, Department of Computer Science, Hong Kong Baptist University , Hong Kong Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Tensor decomposition is a traditional and active research topic in the mathematical community. Recently, it has attracted increasing interests from other fields, such as machine learning, especially with the growing importance of big data. To succeed in this big data era, it is important to learn compact features for efficient processing. Most big data are multidimensional and can be represented as tensors. Based on a recent book, this talk focuses on the adaptation of classical tensor decompositions for learning compact features from big data based on tensors. In particular, we study multilinear subspace learning, a dimension reduction technique developed for tensors. Multilinear subspace learning (MSL) directly maps input tensors to a low-dimensional subspace, without reshaping into high-dimensional vectors. It preserves data structure, obtains more compact features, and processes big data more efficiently. The mapping can be done through tensor-to-tensor projection and tensor-to-vector projection, which are adaptations of Tucker decomposition and the canonical polyadic decomposition (PARAFAC/CANDECOMP), respectively. We will examine MSL algorithms and MSL feature characteristics, explore various MSL applications, and outline future research directions in learning compact features for big data analytics via tensors. - Tuesday, 21st January, 2014
Title: Total Variation Based Tensor Decomposition for Multi-Dimensional Data with Time Dimension Speaker: Mr. CHEN Chuan, Department of Mathematics, Hong Kong Baptist University , Hong Kong Time/Place: 14:30 - 15:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: The main aim of our research is to study tensors for multi-dimensional data with time dimension which arises in many scientiﬁc and engineering applications. We are interested in determining a tensor decomposition CANDECOMP where the relevant attributes in the consecutive time points are obtained in the decomposition components. Our idea is to make use the total variation regularization term for time dimension to derive the required tensor decomposition. We employ alternating direction method of multipliers to solve the required optimization problem, and study its convergence. Numerical examples on synthetic and real data sets are used to demonstrate that the proposed total variation based tensor decomposition model can provide better and interesting results than the standard CANDECOMP. - Tuesday, 21st January, 2014
Title: Sparse-MIML: A Sparsity-Based Multi-Instance Multi-Learning Algorithm Speaker: Mr. SHEN Chenyang, Department of Mathematics, Hong Kong Baptist University , Hong Kong Time/Place: 15:30 - 16:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Multi-Instance Multi-Label (MIML) learning is one of challenging research problems in machine learning. The main aim of this paper is to propose and develop a novel sparsity-based MIML learning algorithm. Our idea is to formulate and construct a transductive objective function for labels indicator to be learned by using the method of random walk with restart that exploits the relationships among instances and labels of objects, and computes the affinities among the objects. Then sparsity can be introduced in the labels indicator of the objective function such that relevant and irrelevant objects with respect to a given class can be distinguished. The resulting sparsity-based MIML model can be given as a constrained convex optimization problem, and it can be solved very efficiently by using the augmented Lagrangian method. Experimental results on benchmark data have shown that the proposed sparse-MIML algorithm is computationally efficient, and effective in label prediction for MIML data. We demonstrate that the performance of the proposed method is better than the other testing MIML learning algorithms. - Wednesday, 22nd January, 2014
Title: Lecture on Mathematics: Game Coloring of Graphs Speaker: Prof. Xuding Zhu, Zhejiang Normal University, China Time/Place: 16:30 - 17:30 (Preceded by Reception at 16:00pm)
RRS905, Sir Run Run Shaw Building, HSH Campus, Hong Kong Baptist UniversityAbstract: Graph coloring is a fascinating research area in graph theory, with many challenging open problems, deep results and sophiscated tools. The most well-known result is the Four Color Theorem: In every map, the countries (connected regions in the plane) can be colored with four colors so that any two neighboring countries have distinct colors. In this talk, we consider a variation of the coloring problem: a coloring produced through a game. Take an arbitrary map and a set of colors. Two players, Alice and Bob, alternate their turns in coloring the countries, subject to the constraint that neighboring countries must be assigned distinct colors. Alice wins the game if eventually every country is coloured, and Bob win if there is a country uncoloured but all the colors have been used by its neighboring countries. The problem is how many colors are needed for Alice to have a winning strategy. I shall explain some tools used in the study of this problem, and survey development in this area. - Thursday, 23rd January, 2014
Title: Wind Generation Development and Incentive for Dispatchable Thermal Generation Investment Speaker: Prof. Chi Keung WOO, Department of Economics, Hong Kong Baptist University , Hong Kong Time/Place: 10:30 - 11:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: We use a large Texas database to quantify the effect of rising wind generation on the payoffs of a tolling agreement for natural-gas-fired generation of electricity. We find that while a 20% increase in wind generation may not have a statistically-significant effect, a 40% increase can reduce the agreement’s average payoff by 8% to 13%. Since natural-gas-fired generation is necessary for integrating large amounts of intermittent wind energy into an electric grid, our finding contributes to the policy debate of capacity adequacy and system reliability in a restructured electricity market that will see large-scale wind-generation development. - Tuesday, 28th January, 2014
Title: Tensor computations and applications in information sciences Speaker: Prof. Lars Eldén, Department of Mathematics, Linköping University, Sweden Time/Place: 11:30 - 12:30
FSC1217, Fong Shu Chuen Library, HSH Campus, Hong Kong Baptist UniversityAbstract: Analyses of data organized as matrices are ubiquitous and well supported by theory and algorithms. In information sciences data are often organized in more than two categories, and it is often unnatural to reorganize the data as a matrix. Therefore there is a need for theory and algorithms for tensor computations. We discuss generalizations of the matrix singular value decomposition to tensors, in particular the best rank-(p,q,r) approximation of a tensor. The analysis and computation of this decomposition need be done in terms of differential-algebraic concepts, and thus we present a Newton method and perturbation theory on the Grassmann manifold. We also outline the generalization of matrix Krylov methods for computations with large and sparse tensors. We briefly discuss applications in pattern recognition, information retrieval, and the simultaneous clustering of graphs. This is joint work with Berkant Savas.