Clarkson University

Guangming Yao

Division of Math and C.S.

Clarkson University

gyao@clarkson.edu

Office: SC 363

Office Phone: (315)268-6496

 

About Me      Teaching      Publications      Research Interests      Activities      Students      Useful Links      AWM at Clarkson      PME at Clarkson      MCCNNY and REU

 

 

 Current Research Interests:

 

1.      Radial Basis Function Neural Network and Machine Learning

 

2.      Stochastic Differential Equations and High-Dimension Partial Differential Equations

 

3.      High Performance Computing

 

4.      Radial Basis Function for Partial Differential Equations.

http://www.rbf-pde.org/rbf2.gif http://www.rbf-pde.org/pde2.gif 

 

INTRODUCTION:

The RBF-PDE is the Radial Basis Function for the solution of PDE. This is a meshless collocation method with global basis Functions. It is known to have exponentional convergence for interpolation problems. One can descretize nonlinear elliptic PDEs by RBF method. This results in modest size systems of nonlinear algebraic equations which can be efficiently solved by standard software such as LINPACK, LAPACK etc. Examples are published for 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.

OVERVIEW (by Ed KANSA)

The numerical solution of partial differential equations (PDEs) has been dominated by either finite difference methods (FDM), finite element methods (FEM), and finite volume methods (FVM). These methods can be derived from the assumptions of the local interpolation schemes. These methods require a mesh to support the localized approximations; the construction of a mesh in three or more dimensions is a non-trivial problem. Typically with these methods only the function is continuous across meshes, but not its partial derivatives.

In practice, only low order approximations are used because of the notorious polynomial snaking problem. While higher order schemes are necessary for more accurate approximations of the spatial derivatives, they are not sufficient without monotonicity constraints. Because of the low order schemes typically employed, the spatial truncation errors can only be controlled by using progressively smaller meshes. The mesh spacing, h, must be sufficiently fine to capture the functions partial derivative behavior and to avoid unnecessarily large amounts of numerical artifacts contaminating the solution. Spectral methods while offering very high order spatial schemes typically depend upon tensor product grids in higher dimensions.... (Complete overview is here:PDF)

RBF people

List of people with E-mail addresses, Web-links is being appended. If you want your name to be added to the list, please send E-mail to G. Yao at gyao@clarkson.edu.

Links - Web resources

Boundary element method Web page

5.      Applied Mathematics

1)      Excitation-Contraction Coupling Modeling

2)      Chemical Engineering Modeling

3)       Ground Water Modeling

 

1)     Excitation-Contraction Coupling Modeling

             

Fig. Left: Multiple t-tubule geometry and its surrounding box domain. Right: 3D views of the calcium concentrations at the calcium peak of 72 ms when the subcellular model in placed 0 micrometer away from the cell membrane.

Spatial-temporal calcium dynamics due to calcium release, buffering, and reuptaking plays a central role in studying excitation-contraction (E-C) coupling in both normal and diseased cardiac myocytes. We employ numerical algorithms to system of reaction-diffusion PDEs to model such calcium behaviors.

2)     Chemical Engineering Modeling

Wood burning in wintertime poses a significant risk to human health. Apportioning wood smoke during wintertime would be significantly useful. Levoglucosan and methyl nitrocatechol have been considered as primary and secondary markers respectively for apportioning wood smoke. We aim to use PDE model and numerical simulations to find the lifetime of wood smoke markers in wintertime conditions.

3)     Ground Water Modeling

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.

Let us try to test the performance of the cylinders produced by Carus Corporation. Please click here