Krylov implicit integration factor discontinuous Galerkin methods on sparse grids for high dimensional reaction-diffusion equations

Computational costs of numerically solving multidimensional partial differential equations (PDEs) increase significantly when the spatial dimensions of the PDEs are high, due to large number of spatial grid points. For multidimensional reaction-diffusion equations, stiffness of the system provides additional challenges for achieving efficient numerical simulations. In this paper, we propose a class of Krylov implicit integration factor (IIF) discontinuous Galerkin (DG) methods on sparse grids to solve reaction-diffusion equations on high spatial dimensions. The key ingredient of spatial DG discretization is the multiwavelet bases on nested sparse grids, which can significantly reduce the numbers of degrees of freedom. To deal with the stiffness of the DG spatial operator in discretizing reaction-diffusion equations, we apply the efficient IIF time discretization methods, which are a class of exponential integrators. Krylov subspace approximations are used to evaluate the large size matrix exponentials resulting from IIF schemes for solving PDEs on high spatial dimensions. Stability and error analysis for the semi-discrete scheme are performed. Numerical examples of both scalar equations and systems in two and three spatial dimensions are provided to demonstrate the accuracy and efficiency of the methods. The stiffness of the reaction-diffusion equations is resolved well and large time step size computations are obtained

A HYBRID METHOD FOR STIFF REACTION-DIFFUSION EQUATIONS.

The second-order implicit integration factor method (IIF2) is effective at solving stiff reaction-diffusion equations owing to its nice stability condition. IIF has previously been applied primarily to systems in which the reaction contained no explicitly time-dependent terms and the boundary conditions were homogeneous. If applied to a system with explicitly time-dependent reaction terms, we find that IIF2 requires prohibitively small time-steps, that are relative to the square of spatial grid sizes, to attain its theoretical second-order temporal accuracy. Although the second-order implicit exponential time differencing (iETD2) method can accurately handle explicitly time-dependent reactions, it is more computationally expensive than IIF2. In this paper, we develop a hybrid approach that combines the advantages of both methods, applying IIF2 to reaction terms that are not explicitly time-dependent and applying iETD2 to those which are. The second-order hybrid IIF-ETD method (hIFE2) inherits the lower complexity of IIF2 and the ability to remain second-order accurate in time for large time-steps from iETD2. Also, it inherits the unconditional stability from IIF2 and iETD2 methods for dealing with the stiffness in reaction-diffusion systems. Through a transformation, hIFE2 can handle nonhomogeneous boundary conditions accurately and efficiently. In addition, this approach can be naturally combined with the compact and array representations of IIF and ETD for systems in higher spatial dimensions. Various numerical simulations containing linear and nonlinear reactions are presented to demonstrate the superior stability, accuracy, and efficiency of the new hIFE method

Theory, Computation, and Modeling of Cancerous Systems

This dissertation focuses on three projects. In Chapter 1, we derive and implement the compact implicit integration factor method for numerically solving partial differential equations. In Chapters 2 and 3, we generalize and analyze a mathematical model for the nonlinear growth kinetics of breast cancer stem cells. And in Chapter 4, we develop a novel mathematical model for the HER2 signaling pathway to understand and predict breast cancer treatment. Due to the high order spatial derivatives and stiff reactions, severe temporal stability constraints on the time step are generally required when developing numerical methods for solving high order partial differential equations. Implicit integration method (IIF) method along with its compact form (cIIF), which treats spatial derivatives exactly and reaction terms implicitly, provides excellent stability properties with good efficiency by decoupling the treatment of reaction and spatial derivatives. One major challenge for IIF is storage and calculation of the potential dense exponential matrices of the sparse discretization matrices resulted from the linear differential operators. The compact representation for IIF (cIIF) was introduced to save the computational cost and storage for this purpose. Another challenge is finding the matrix of high order space discretization, especially near the boundaries. In Chapter 1, we extend IIF method to high order discretization for spatial derivatives through an example of reaction diffusion equation with fourth order accuracy, while the computational cost and storage are similar to the general second order cIIF method. The method can also be efficiently applied to deal with other types of partial differential equations with both homogeneous and inhomogeneous boundary conditions. Direct numerical simulations demonstrate the efficiency and accuracy of the approach. Cancer stem cells are responsible for tumor survival and resurgence and are thus essential in developing novel therapeutic strategies against cancer. Mathematical models can help understand cancer stem and differentiated cell interaction in tumor growth, thus having the potential to aid in designing experiments to develop novel therapeutic strategies against cancer. In Chapter 2, by using theory of functional and ordinary differential equations, we study the existence and stability of non-linear growth kinetics of breast cancer stem cells. First we provide a sufficient condition for the existence and uniqueness of the solution for non-linear growth kinetics of breast cancer stem cells. Then we study the uniform asymptotic stability of the zero solution. By using linearization techniques, we also provide a criteria for uniform asymptotic stability of a non-trivial steady state solution with and without time delays. We present a theorem from complex analysis that gives certain conditions which allow for this criteria to be satisfied. Next we apply these theorems to a special case of the system of functional differential equations that has been used to model non-linear growth kinetics of breast cancer stem cells. The theoretical results are further justified by numerical testing examples. Consistent with the theories, our numerical examples show that the time delays can disrupt the stability. All the results can be easily extended to study more general cell lineage models. Solid tumors are heterogeneous in composition. Cancer stem cells (CSCs) are a highly tumorigenic cell type found in developmentally diverse tumors that are believed to be resistant to standard chemotherapeutic drugs and responsible for tumor recurrence. Thus understanding the tumor growth kinetics is critical for developing novel strategies for cancer treatment. In Chapter 3, the moment stability of nonlinear stochastic systems of breast cancer stem cells with time-delays is investigated. First, based on the technique of the variation- of-constants formula, we obtain the second order moment equations for the nonlinear stochastic systems of breast cancer stem cells with time-delays. By the comparison principle along with the established moment equations, we can get the comparative systems of the nonlinear stochastic systems of breast cancer stem cells with time-delays. Then moment stability theorems are established for the systems with the stability properties for the comparative systems. Based on the linear matrix inequality (LMI) technique, we next obtain a criteria for the exponential stability in mean square of the nonlinear stochastic systems for the dynamics of breast cancer stem cells with time-delays. Finally, some numerical examples are presented to illustrate the efficiency of the results. Over-expression of human epidermal growth factor receptor 2 (HER2) plays a role in regulation of cancer stem cell (CSC) population in breast cancer. Current cancer therapy includes drugs that block HER2, however, patients can develop anti-HER2 drug resistance. Downstream of HER2 is nuclear factor κB (NFκB). The aberrant regulation of NFκB leads to cancer growth, which makes it a promising target for cancer therapy, especially for those who have developed resistance to anti-HER2 treatment. In Chapter 4 we develop a novel mathematical model that represents the dynamics of the HER2 signaling pathway. By integrating experimental data with model simulations, we discover that interleukin-1 (IL1), which is downstream of HER2, is responsible for NFκB activation. We perform global sensitivity analysis on the model to identify key reactions. Our modeling effort shows that IL1 is critical in NFκB regulation, especially in the absence of HER2, making it a potential target in treating breast cancer for patients who have developed resistance to anti-HER2 drugs