Developing fast and accurate parallel solver for multi-scales biochemical reacting systems

Bruno Bieth, Purdue University

Abstract

Application of mathematical models to real problems from the biological or medical arenas poses challenges in both the modeling and in the computation needed to get predictions from the model that has been formulated, especially if the problem is inherently multi-scaled. Those real-world applications involve situations where different physical phenomena acting on very different time scales occur simultaneously. The ordinary differential equations (ODEs) governing such situations are categorized as stiff ODEs. Stiffness is a challenging property of differential equations (DEs) that prevents conventional explicit numerical integrators from handling a problem efficiently. For such cases, stability (rather than accuracy) requirements dictate the choice of time step size to be very small. Considerable effort in coping with stiffness has gone into developing time-discretization methods to overcome many of the constraints of the conventional methods. Recently, there has been a renewed interest in exponential integrators that have emerged as a viable alternative for dealing effectively with stiffness of ODEs. Our attention has been focused on a drug based model governed by a system of stiff ODEs within the framework of an innovative drug-drug interaction method based on a three-level Bayesian meta-analysis model including Monte Carlo Markov Chain (MCMC) pharmacokinetic parameter estimation procedure. Underlying the parameter estimation procedure is a need for a fast integration method of the stiff pharmacokinetic equations. This thesis represents our effort to gain insight into the nature of the pharmacokinetic equations and their speedy and accurate integration. The first aspect of this study involves the development of an integration method that converts a system of differential equation into a system of Volterra integral equations using a quasilinearization step and incorporating initial conditions into the equations. Then we employ either a multi-stage linear interpolation or its combination with successive approximation method to find the solution to the Volterra integral equations. No polynomial interpolant is used in the development of the integration scheme and computation results show an improvement over existing methods for stiff-equations.

Degree

Ph.D.

Advisors

Chin, Purdue University.

Subject Area

Mathematics

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