Khamron Mekchay | Chulalongkorn University (original) (raw)

Papers by Khamron Mekchay

Symmetry

Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs dep... more Contingent claims, such as bonds, swaps, and options, are financial derivatives whose payoffs depend on uncertain future real values of underlying assets which emphasize various real-world applications. In general, valuations for contingent claims can be derived from the conditional expectations of underlying assets. For a simple process, the moments are usually directly obtained from its transition probability density function (PDF). However, if the transition PDF is unavailable in simple form, the derivations of the moments and the contingent claim prices will not be accessible in closed forms. This paper provides a closed-form formula for pricing contingent claims with nonlinear payoff under a no-arbitrage principle when underlying assets follow the extended Cox–Ingersoll–Ross (ECIR) process with the symmetry properties of the Brownian motion. The formula proposed here is a consequence of successfully solving an explicit solution for a system of recurrence partial differential eq...

We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for qua... more We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator: − ∇ · (α(x,∇u)∇u) = f (x) in Ω ⊂ R2, u = 0 on ∂Ω, where Ω is assumed to be a polygonal bounded domain in R2, f ∈ L2(Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H1-norm by an indicator η which is composed of L2-norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.

Research in the Mathematical Sciences

We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Belt... more We present an adaptive finite element method (AFEM) of any polynomial degree for the Laplace-Beltrami operator on C 1 graphs Γ in R d (d ≥ 2). We first derive residual-type a posteriori error estimates that account for the interaction of both the energy error in H 1 (Γ) and the surface error in W 1 ∞(Γ) due to approximation of Γ. We devise a marking strategy to reduce the total error estimator, namely a suitably scaled sum of the energy, geometric, and inconsistency error estimators. We prove a conditional contraction property for the sum of the energy error and the total estimator; the conditional statement encodes resolution of Γ in W 1 ∞. We conclude with one numerical experiment that illustrates the theory. Keywords. Laplace-Beltrami operator, graphs, adaptive finite element method, a posteriori error estimate, energy and geometric errors, bisection, contraction. 2000 MSC. Primary: 65 Numerical analysis. Secondary: 35 Partial differential equations. 1

A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-R... more A well-balanced scheme with total variation diminishing Runge-Kutta discontinuous Galerkin (TVD-RK DG) method for solving shallow water equations is presented. Generally, the flux function at cell interface in the TVD-RK DG scheme is approximated by using the Harten-Lax-van Leer (HLL) method. Here, we apply the weighted average flux (WAF) which is higher order approximation instead of using the HLL in the TVD-RK DG method. The consistency property is shown. The modified wellbalanced technique for flux gradient and source terms under the WAF approximations is developed. The accuracy of numerical solutions is demonstrated by simulating dam-break flows with the flat bottom. The steady solutions with shock can be captured correctly without spurious oscillations near the shock front. This presents the other flux approximations in the TVD-RK DG method for shallow water simulations.

The adaptive Discontinuous Galerkin (DG) method for solving the one-dimensional conservation equa... more The adaptive Discontinuous Galerkin (DG) method for solving the one-dimensional conservation equation is presented. In this paper we consider the advection equation, the Burgers’ equation, and the shallow water equations. To improve the accuracy of this method, two types of adaptive technique are employed. These are the adaptive polynomial(p-daptive) and the adaptive mesh(h-adaptive). Troubled cells needed to be refined are detected by two types of indicators, which are error and gradient indicators. The present schemes have been improved the accuracy of the solution during time integration process. For smooth solution the accuracy can be improve by the adaptive polynomial criteria, while the accuracy of moving shock, rarefaction and high gradient solution can be improved by the adaptive mesh scheme. High gradient area in the computational domain can be detected efficiently by both presented indicators Keywords—Adaptive Discontinuous Galerkin method, Advection equation, Burgers’ equ...

The purpose of this paper is a computational algorithm for simulation and visualization of flood ... more The purpose of this paper is a computational algorithm for simulation and visualization of flood inundation on natural topography. The algorithm is constructed based on the shallow water equations, which are solved numerically using an adaptive tree grid finite volume method that is also equipped with the dynamic domain defining technique. The algorithm is tested to simulate the flood inundation in Thailand. The results are compared with the non-adaptive finest grid simulation. The comparison shows that the algorithm can reduce the number of grid cells and the computational times, without much loss of accuracy in the results.

We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for qua... more We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator: −∇ · (α(x, ∇u)∇u) = f(x) in Ω ⊂ R 2 , u = 0 on ∂Ω, where Ω is assumed to be a polygonal bounded domain in R 2 , f ∈ L 2 (Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H 1 -norm by an indicator η which is composed of L 2 norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.

We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic... more We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic partial differential equations (PDEs) with vanish- ing boundary over a polyhedral domain in R d , d ≥ 2. Based on a posteriori error estimates using standard residual technique, we prove the contraction property for the weighted sum of the energy error and the error estimator between two consecu- tive iterations, which also leads to the convergence of AFEM. The obtained result is based on the assumptions that the initial mesh or triangulation is sufficiently refined and the nonlinear inhomogeneous term f(x,u(x)) is Lipschitz in the second variable.

Communications in Nonlinear Science and Numerical Simulation

International Journal for Computational Methods in Engineering Science and Mechanics

ABSTRACT Finite volume method with reconstruction and bottom modification techniques for simulati... more ABSTRACT Finite volume method with reconstruction and bottom modification techniques for simulating open channel flows in arbitrary cross-sectional areas is presented. These techniques are introduced to handle the difficulty in approximating water depth at wet/dry areas. Various numerical experiments with source terms are demonstrated to confirm the accuracy of numerical scheme. Further, we have applied the present method to simulate open channel flow in the Yom River, Phrae Province, Thailand. The simulation results are compared with measured data.

Journal of Physics: Conference Series

In this study, an explicit formula for conditional expectations of the product of polynomial and ... more In this study, an explicit formula for conditional expectations of the product of polynomial and exponential function of an affine transform is derived under the extended Cox-Ingersoll-Ross (ECIR) process. Moreover, we simplify the result to derive an explicit formula for the CIR process.

Foundations of Computational Mathematics

We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on param... more We present a new AFEM for the Laplace-Beltrami operator with arbitrary polynomial degree on parametric surfaces, which are globally W 1 ∞ and piecewise in a suitable Besov class embedded in C 1,α with α ∈ (0, 1]. The idea is to have the surface sufficiently well resolved in W 1 ∞ relative to the current resolution of the PDE in H 1. This gives rise to a conditional contraction property of the PDE module. We present a suitable approximation class and discuss its relation to Besov regularity of the surface, solution, and forcing. We prove optimal convergence rates for AFEM which are dictated by the worst decay rate of the surface error in W 1 ∞ and PDE error in H 1 .

Computational Geosciences

A well-balanced finite volume method for solving two-dimensional shallow water equations with wei... more A well-balanced finite volume method for solving two-dimensional shallow water equations with weighted average flux (WAF) is developed in this work to simulate flooding. Friction source terms are estimated with a semi-implicit scheme resulting in an efficient numerical method for simulating shallow water flows over irregular domains, for both wet and dry beds. A wet/dry cell tracking technique is also presented for reducing computational time. The accuracy of these methods are investigated by application to well-studied cases. For practical purposes, the developed scheme is applied to simulate the flooding of the Chao Phraya river from Chai Nat to Sing Buri provinces in Thailand during October 13–17, 2011. The numerical simulations yield results that agree with the existing data obtained from the satellite images.

Thai Journal of Mathematics, Feb 19, 2015

Itô processes are processes commonly used as a mathematical model in many fields. In order to est... more Itô processes are processes commonly used as a mathematical model in many fields. In order to estimate the unknown parameters of an Itô process based on the maximum likelihood method, the transitional probability density function (PDF) of the Itô process is needed. In fact, the transitional PDF is the solution of a Fokker-Planck equation subject to an initial condition in which the transitional PDF is set to be coincided with the Dirac delta function at the initial time. In this research, we applied the numerical method called the Laplace transform dual reciprocity method (LTDRM) to approximate the solution of the Fokker-Planck equations, corresponding to a one-dimensional Itô process. The key idea of the LTDRM for solving this type of problems is to transform the Dirac delta function into the Laplace space and then use the dual reciprocity method (DRM) to solve the transformed equation without approximating the Dirac delta function. The Stehfests algorithm is used to convert the solutions back into the transitional PDF. We tested and ran experiments on the OU and CIR models by comparing with exact transitional PDF. The tests show that our results using LTDRM give a very accurate approximation and can be used in the maximum likelihood estimation (MLE).

Siam Journal on Numerical Analysis, Jul 25, 2006

Thai Journal of Mathematics, Mar 25, 2015

We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic... more We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic partial differential equations (PDEs) with vanishing boundary over a polyhedral domain in R d , d ≥ 2. Based on a posteriori error estimates using standard residual technique, we prove the contraction property for the weighted sum of the energy error and the error estimator between two consecutive iterations, which also leads to the convergence of AFEM. The obtained result is based on the assumptions that the initial mesh or triangulation is sufficiently refined and the nonlinear inhomogeneous term f (x, u(x)) is Lipschitz in the second variable.

ScienceAsia, 2015

In this study a numerical algorithm is developed for simulation and visualization of rainfall ove... more In this study a numerical algorithm is developed for simulation and visualization of rainfall overland flows on natural topography. The algorithm is constructed based on the diffusion model of water flow which is solved numerically using an adaptive tree grid finite volume method which is equipped with dynamic domain defining. The developed algorithm is tested to simulate and visualize rainfall overland flows on natural topography showing the advantages of the algorithm in terms of computational costs and the simulation times without losing much accuracy in the results.

Symmetry

Research in the Mathematical Sciences

We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for qua... more We derived a posteriori error estimates for the Dirichlet problem with vanishing boundary for quasi-linear elliptic operator: −∇ · (α(x, ∇u)∇u) = f(x) in Ω ⊂ R 2 , u = 0 on ∂Ω, where Ω is assumed to be a polygonal bounded domain in R 2 , f ∈ L 2 (Ω), and α is a bounded function which satisfies the strictly monotone assumption. We estimated the actual error in the H 1 -norm by an indicator η which is composed of L 2 norms of the element residual and the jump residual. The main result is divided into two parts; the upper bound and the lower bound for the error. Both of them are accompanied with the data oscillation and the α-approximation term emerged from nonlinearity. The design of the adaptive finite element algorithm were included accordingly.

Communications in Nonlinear Science and Numerical Simulation

International Journal for Computational Methods in Engineering Science and Mechanics

Journal of Physics: Conference Series

Foundations of Computational Mathematics

Computational Geosciences

Thai Journal of Mathematics, Feb 19, 2015

Siam Journal on Numerical Analysis, Jul 25, 2006

Thai Journal of Mathematics, Mar 25, 2015

We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic... more We analyze a standard adaptive finite element method (AFEM) for second order semi-linear elliptic partial differential equations (PDEs) with vanishing boundary over a polyhedral domain in R d , d ≥ 2. Based on a posteriori error estimates using standard residual technique, we prove the contraction property for the weighted sum of the energy error and the error estimator between two consecutive iterations, which also leads to the convergence of AFEM. The obtained result is based on the assumptions that the initial mesh or triangulation is sufficiently refined and the nonlinear inhomogeneous term f (x, u(x)) is Lipschitz in the second variable.

ScienceAsia, 2015