Guangming Yao

A localized approach for the method of approximate particular solutions

The method of approximate particular solutions (MAPS) has been recently developed to solve various types of partial differential equations. In the MAPS, radial basis functions play an important role in approximating the forcing term. Coupled with the concept of particular solutions and radial basis functions, a simple and effective numerical method for solving a large class of partial differential equations can be achieved. One of the difficulties of globally applying MAPS is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points, thereby affecting our ability to solve large-scale science and engineering problems. In this paper we develop a localized scheme for the method of approximate particular solutions (LMAPS). The new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation. In this paper, this local numerical scheme is used for solving large-scale problems, up to one million interpolation points. Three numerical examples in two-dimensions are used to validate the proposed numerical scheme.

Implicit local radial basis function interpolations based on function values

In this paper we propose two fast localized radial basis function fitting algorithms for solving large-scale scattered data interpolation problems. For each given point in the given data set, a local influence domain containing a small number of nearest neighboring points is established and a global interpolation is performed within this restricted domain. A sparse matrix is formulated based on the global interpolation in these local influence domains. The proposed methods have achieved both low computational cost and minimal memory storage. In comparison with the compactly supported radial basis functions, the proposed fitting algorithms are highly accurate. The numerical examples have provided strong evidence that the two proposed algorithms are indeed highly efficient and accurate. In the two proposed algorithms, we have successfully solved a large-scale interpolation problem with 225,000 interpolation points in two dimensional space.

Existence and stability of stationary waves of a population model with strong Allee effect

We investigate the existence and stability of stationary waves of a nonlocal reaction-diffusion population model with delay, nonlocality and strong Allee effect. By reducing the model, the conditions for existence of stationary wavefront, wave pulse and inverted wave pulse are established. Then we show that the stationary waves of the reduced model are also the stationary waves of the general model. The global stability of the stationary waves is illustrated by numerically solving the general model for different sets of parameter values.

An improved localized method of approximate particular solutions for solving elliptic PDEs

In this paper we improve the localized method of approximate particular solutions (LMAPS) in Yao et?al. (2011) by utilizing the polyharmonic splines (PS) radial basis function (RBF) for solving elliptic partial differential equations (PDEs). LMAPS has been widely circulated since it is published in 2010. The multiquadric (MQ) has been considered as the most popular choice among all RBFs. However, adjusting the shape parameter is a critical issue when utilizing the original LMAPS. In this paper, we modified LMAPS by combining conditionally positive definite RBF-PS and an additional low degree of polynomial basis in the localization process. The accuracy of the proposed LMAPS is significantly improved. We can simply increase the order of PS to achieve even higher accuracy. Other than the unexpected high accuracy, there is no need to deal with the difficult issues of choosing optimal shape parameter. This is a huge advantage in the RBF simulations of PDEs. In the numerical experiments, we will present the pros and cons of improved LMAPS (ILMAPS) using PS and some commonly used RBFs (MQ, Matern, and Gaussian) versus the original LMAPS (OLMAPS).

The localized method of approximated particular solutions for near-singular two- and three-dimensional problems

In this paper, the localized method of approximate particular solutions (LMAPS) using radial basis functions (RBFs) has been simplified and applied to near-singular elliptic problems in two- and three-dimensional spaces. The leave-one-out cross validation (LOOCV) is used in LMAPS to search for a good shape parameter of multiquadric RBF. The main advantage of the method is that a small number of neighboring nodes can be chosen for each influence domain in the discretization to achieve high accuracy. This is especially efficient for three-dimension problems. There is no need to apply adaptivity on node distribution near the region containing spikes of the forcing terms. To examine the performance and limitations of the method, we deliberately push the spike of the forcing term to be extremely large and still obtain excellent results. LMAPS is far superior than the compactly supported RBF (Chen et al. 2003) for such elliptic boundary value problems.

Radial basis function simulation of slow-release permanganate for groundwater remediation via oxidation

An emerging strategy for remediation of contaminated groundwater is the use of permanganate cylinders for contaminant oxidation. The cylinders, which are only a few inches in diameter, can be placed in wells or pushed directly into the subsurface. This work focuses on the modeling and simulation of the reactive process to better understand the design of a group of cylinders for large scale contaminated sites. The underlying model is a coupled system of nonlinear partial differential equations accounting for advection, dispersion, and reactive transport for a contaminant and the permanganate in two spatial dimensions. Radial Basis Functions collocation method is used to simulate different spatial arrangements of the cylinders to understand the behavior of the system and gain insight into designing a remediation strategy for a large-scale contaminated region. Since the radial basis function collocation method is a meshless method, the locations of the cylinders are not tied to a numerical grid, making it an attractive choice for determining optimal placement. Our focus is to (1) identify a domain of influence measuring the effectiveness of the injected cylinders, (2) understand the placement for multiple cylinders required to effectively clean-up a given domain, and (3) determine a protocol for injecting multiple cylinders over time. We provide numerical results showing that domain of influence is a way to measure the effectiveness of installed cylinders. Domain of influence of one through three sources are simulated. Placement of two cylinders for an area of 13 f t by 3 f t and three sources for an area of 26 f t by 6 f t are sufficient to clean the contaminant within a reasonable time period. The average concentrations of oxidant and contaminant are simulated for the cases of a third cylinder is installed at different time and locations.

Parallel acceleration for modeling of calcium dynamics in cardiac myocytes.

Spatial-temporal calcium dynamics due to calcium release, buffering, and re-uptaking plays a central role in studying excitation-contraction (E-C) coupling in both healthy and defected cardiac myocytes. In our previous work, partial differential equations (PDEs) had been used to simulate calcium dynamics with realistic geometries extracted from electron microscopic imaging data. However, the computational costs of such simulations are very high on a single processor. To alleviate this problem, we have accelerated the numerical simulations of calcium dynamics by using graphics processing units (GPUs). Computational performance and simulation accuracy are compared with those based on a single CPU and another popular parallel computing technique, OpenMP.