Plenary Lectures, Posco B153

Plenary Lecture I, December 15, 10:00~11:00

Title: Global-Local-Integration-based Radial Basis Function Computational Methods

by Benny Y.C. HON

City University of Hong Kong

Abstract: In this talk, the recent development in global, local, and integration-based meshless computational methods via the use of radial basis function (RBF) will be presented. The local radial basis function computational method (LRBFCM) is an extension to solve large scale problems which has hindered the practical application of the global RBF method for years due to the ill-conditioning of the resultant full coefficient matrix. The LRBFCM has recently been applied to solve cavity flows problems with free surface and some non-local diffusion and phase field problems. Because of the intrinsic stable and accurate advantages of numerical integration and spectral convergence of kernels approximation, the finite integration-based radial basis function(FIM-RBF) method can solve multi-dimensional boundary value problems (BVPs) under irregular domain with various kinds of stiffness. The main idea of the FIM-RBF method is to transform the original partial differential equation into an equivalent integral equation whose approximation can be sought by standard numerical integration techniques. Unlike the use of finite quotient formula in the classical finite difference method (FDM), the FIM-RBF uses numerical quadrature formula to approximate the unknown solution and its derivatives which is unconditional stable. This completely avoids the well-known optimal roundoff-discretization tradeoff error in FDM. The FIM-RBF method has successfully been applied to solve inverse heat conduction problem and stiff problems. Numerical examples in 2D will be given to verify the efficiency and effectiveness of the proposed methods.

Plenary Lecture II, December 15, 11:10~12:10

Title: Cryptology, isoperimetric problems and shadows

by Gyula O. H. Katona

Alfréd Rényi Institute of Mathematics, Hungary

Abstract: The practical problem is the following. Objects should be labeled with some geometric pictures. To avoid easy falsification, the pictures are chosen randomly. That is, a space S with a distance d and a measure is given. The label of one object (picture) is a randomly chosen element of S. More precisely we will mark out some subsets Ａ_i of S and if the random point falls in Ａ_i then it can be used as a label of an object numbered i. The sets Ａ_i must satisfy certain properties as described below.

IfＡ⊂Ｓ, ０＜ε then define n(Ａ,ε)＝｛x∈Ｓ: d(Ａ,x) ≤ ε ｝. The family of subsets Ａ_1, ... , Ａ_m is called a geometric code with parameters ε and if (1) μ(Ａ_i) ≤ ρ holds for every i (1≤ i ≤ m), (2) the sets n(Ａ_i, ε) (1 ≤ i ≤ m) are pairwise disjoint, (3) α ≤ μ(∪Ａ_i )/μ(Ｓ).

A geometric code can be applied in the following way. Choose random elements of Ｓ according to μ. If x∈Ａ_i then the let its code ｃ(x) be the binary form of i. On the other hand, if x∈Ａ_i holds for no i then is a waste. (1) ensures that a random choice ofｃ(x) (not knowing x) reproduce it with a small probability. (2) implies that reading x with an error at most ε ｃ(x) still can be recovered. Finally assumption (3) is needed to lowerbound the probability of the waste. The problem is to find the maximum of ｍ, given Ｓ, ε, ρ, and α. We give an inequality what has to be satisfied among these parameters, supposing that (1) the measure of a ball is not changed by moving its center, (2) the space satisfies the Brunn-Minkowski inequality.

In our implementation a label is a rectangle containing many small circles (they have a three-dimensional nature, this is why it is hard to copy them), what can be represented by their centers. Because of the computational approximation, it can be supposed that these centers are elements of a grid in the rectangle. Therefore an element of the space Ｓ is a subset of the set of the points of the grid where the sizes of the subsets are between a lower and an upper bound. From practical experiences we know that some of the points can be "lost" during the control, therefore the distance ｄ should be defined accordingly. One

Ａ_i is therefore a family of subsets of the grid points. Roughly speaking the largest number of such families should be found in such a way that deleting a small number of points from one member of a family is different from a subset obtained by deleting the points from the member of another family. This leads to the usage of the theory of extremal problems of finite sets, especially the "shadow theory".

Let us illustrate the problem in a very-very special case. Choose two families, A and B of 3-element subsets of an n-element set in such a way, that deleting one element from an Ａ∈A and one element from B∈B, the so obtained two-element sets are different. Determine max min{|A|, |B|}. The complete asymptotic solution of this "easy-looking" problem will be presented.

Plenary Lecture III, December 16, 10:00~11:00

Title: Modeling multivariate insurance losses with risk and queueing theoretic applications

by Jae Kyung Woo

University of New South Wales, Business School, Australia

Abstract: Modeling multivariate insurance losses with risk and queueing theoretic applications In the first part of the talk, to model highly correlated losses such as catastrophe losses, a class of multivariate mixed Erlang distributions with different scale parameters is considered. Some distributional properties involving higher-order equilibrium distributions and residual lifetime distributions are derived and in turn, we apply these results to study stop-loss moments, premium calculation, and the risk allocation problem in insurance risk theory. This is a joint work with G.E. Willmot. In the second part, we consider an insurance portfolio containing several types of policies which may simultaneously face claims arising from the same catastrophe. A renewal counting process for the number of events causing claims and multivariate claim severities which are dependent on the occurrence time and/or the delay in reporting or payment is assumed. A unified model is proposed to study the time-dependent loss quantities. Furthermore, some numerical examples involving covariances and correlations of the different types of discounted aggregate (reported/unreported) claims until a fixed time are provided. If time permits, some recent developments regarding a particular renewal-reward process with multivariate discounted rewards (inputs) where the arrival epochs are adjusted by adding some random delays. Then this accumulated reward can be regarded as multivariate discounted Incurred But Not Reported (IBNR) claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in G=G=1 queues with correlated batch arrivals. This is a joint work with L. Rabehasaina.