Jingwei Zhang

Algorithm 905: SHEPPACK: Modified Shepard Algorithm for Interpolation of Scattered Multivariate Data

Scattered data interpolation problems arise in many applications. Shepard’s method for constructing a global interpolant by blending local interpolants using local-support weight functions usually creates reasonable approximations. SHEPPACK is a Fortran 95 package containing five versions of the modified Shepard algorithm: quadratic (Fortran 95 translations of Algorithms 660, 661, and 798), cubic (Fortran 95 translation of Algorithm 791), and linear variations of the original Shepard algorithm. An option to the linear Shepard code is a statistically robust fit, intended to be used when the data is known to contain outliers. SHEPPACK also includes a hybrid robust piecewise linear estimation algorithm RIPPLE (residual initiated polynomial-time piecewise linear estimation) intended for data from piecewise linear functions in arbitrary dimension m. The main goal of SHEPPACK is to provide users with a single consistent package containing most existing polynomial variations of Shepard’s algorithm. The algorithms target data of different dimensions. The linear Shepard algorithm, robust linear Shepard algorithm, and RIPPLE are the only algorithms in the package that are applicable to arbitrary dimensional data.

A modified uniformization method for the solution of the chemical master equation

The chemical master equation is considered an accurate description of general chemical systems, and especially so for gene regulatory networks and protein-protein interaction networks. However, solving chemical master equations directly is considered computationally intensive. This paper discusses an efficient way of solving the chemical master equation for some prototypical problems in systems biology. Comparisons between this new approach and some traditional approaches, especially Monte-Carlo algorithms, are also presented, and show that under certain conditions the new approach performs better than Monte-Carlo algorithms.

Adaptive aggregation method for the Chemical Master Equation

The chemical master equation, which is often considered as an accurate stochastic description of general chemical systems, usually imposes intensive computational requirements when used to characterize molecular biological systems. The major challenge comes from the curse of dimensionally, which has been tackled by a few research papers. The essential goal is to aggregate the system efficiently with limited approximation error. This paper presents an adaptive way to implement the aggregation process using information collected from Monte Carlo methods. Numerical results show the effectiveness of the proposed algorithm despite the lack of explicit estimation of approximation error.

RADIAL BASIS FUNCTION COLLOCATION FOR THE CHEMICAL MASTER EQUATION

The chemical master equation (CME), formulated from the Markov assumption of stochastic processes, offers an accurate description of general chemical reaction systems. This paper proposes a collocation method using radial basis functions to numerically approximate the solution to the CME. Numerical results for some systems biology problems show that the collocation approximation method has good potential for solving large-scale CMEs.

A Modified Uniformization Method for the Chemical Master Equation

The chemical master equation is considered an accurate description of general chemical systems, and especially so for modeling cell cycle and gene regulatory networks. This paper proposes an efficient way of solving the chemical master equation for some prototypical problems in systems biology. A comparison between this new approach and some traditional approaches is also given.

Adaptive aggregation method for the chemical master equation

The chemical master equation, which is often considered as an accurate stochastic description of general chemical systems, usually imposes intensive computational requirements when used to characterize molecular biological systems. The major challenge comes from the curse of dimensionally, which has been tackled by a few research papers. The essential goal is to aggregate the system efficiently with limited approximation error. This paper presents an adaptive way to implement the aggregation process using information collected from Monte Carlo methods. Numerical results show the effectiveness of the proposed algorithm despite the lack of explicit estimation of approximation error.