# GLSIAM Conference Program

**Plenary Talks**

From PDEs to Information Science and Back

**Russel Caflisch, Director, Institute for Pure and Applied Mathematics**

*The arrival of massive amounts of data from imaging, sensors, computation and the internet brings with it challenges for information science. New methods for analysis and manipulation of big data have come from many fields of mathematical science. Among these, the application of ideas from PDEs, such as variational principles and numerical diffusion, to information science is the first focus of this presentation. The second focus is the emerging influence back to PDEs of very successful ideas from information science, such as sparsity and compressed sensing.*

Computational Analysis of Stochastic Reaction-Diffusion Processes

**Hans Othmer, University of Minnesota, Department of Mathematics**

**Reaction and diffusion processes are used to model chemical and biological processes over a wide range of spatial and temporal scales. Several routes to the diffusion process at various levels of description in time and space are discussed and the master equation for spatially-discretized systems involving reaction and diffusion is developed. We discuss an estimator for the appropriate compartment size for simulating reaction-diffusion systems and introduce a measure of fluctuations in a discretized system. We then describe a new computational algorithm for implementing a modified Gillespie method for compartmental systems in which reactions are aggregated into equivalence classes and computational cells are searched via an optimized tree structure. Finally, we discuss several examples that illustrate the issues that have to be addressed in general systems.**

**Variational Multiscale Models for Biomolecules**

**Guowei Wei, Michigan State University, Department of Mathematics**

A major feature of biological sciences in the 21st century will be their transition from phenomenological and descriptive disciplines to quantitative and predictive ones. Revolutionary opportunities have emerged for mathematically driven advances in biological research. However, the emergence of complexity in self-organizing biological systems poses fabulous challenges to their quantitative description because of the excessively high dimensionality. A crucial question is how to reduce the number of degrees of freedom, while preserving the fundamental physics in complex biological systems. This work focuses on a new variational multiscale paradigm for biomolecular systems. Under the physiological condition, most biological processes, such as protein folding, ion channel transport and signal transduction, occur in water, which consists of 65-90 percent of human cell mass. Therefore, it is desirable to describe macromolecules by discrete atomic and/or quantum mechanical variables; while treating the aqueous environment as a dielectric or hydrodynamic continuum. I will discuss the use of differential geometry for coupling microscopic and macroscopic scales on an equal footing. Based on the variational principle, we derive the coupled Poisson-Boltzmann, Nernst-Planck (or Kohn-Sham), Laplace-Beltrami and Navier-Stokes equations for the structure, dynamics and transport of ion-channel systems.

Abstracts of Contributed Talks

Approximating Off-Lattice Kinetic Monte Carlo

**Henry Boateng, University of Michigan**

We present an approximate off-lattice kinetic Monte Carlo method for simulating heteroepitaxial

growth. The model aims to retain the speed and simplicity of lattice based KMC methods while

capturing essential features that can arise in an off-lattice setting. Interactions between atoms are defined by an interatomic potential which determines the arrangement of the atoms. In this formulation we assign rates for configuration changes that are keyed to individual surface atoms in a fashion similar to bond counting schemes. The method is validated by simulations of heteroepitaxial growth, annealing of straine bilayer systems and a qualitative verification of Stoney's formula. The algorithm captures the effects of misfit on the growth modes, anti-correlation of quantum dots grown on both sides of a substrate, the effects of deposition flux on island size, the formation of edge dislocations (unlike lattice models) to relieve strain, and naturally incorporates intermixing.

Sensitivity analysis for spectral element based two-layered shallow water model

**Vani Cheruvu, University of Toledo**

A two layered shallow water model with various boundary conditions are considered for flow control. The system has much in common with limited area primitive equation model used in numerical weather prediction. An adjoint sensitivity method (ASM) is developed for this system to compute the gradient of a user-defined objective function that has the system control settings resulting in sensitivity expressions. Incorporating characteristic analysis in ASM results in new sensitivity expressions (CASM) which has additional information as compared to ASM. In this talk, I will present both the ASM and CASM analysis for two-layered shallow water model and compare the results with direct sensitivity results. This is a joint work with Dr. Tribbia from NCAR.

Compact optimal spline collocation methods for convection-diffusion problems

**Graeme Fairweather, Mathematical Reviews**

Methods involving smoothest spline collocation have been used frequently in the solution of various ordinary and partial differential equations. Often overlooked is the fact that, in their basic form, such methods yield suboptimal approximations. In recent years, several modified spline collocation (MSC) methods of optimal accuracy have been developed. In this talk, we discuss new MSC methods based on quadratic and cubic splines for the solution of convection-diffusion problems in one space variable. The approximate solutions are not only of optimal global accuracy but also exhibit superconvergence phenomena. Moreover, the methods are compact in that they require the solution of tridiagonal systems of linear equations whereas the MSC approach typically yields pentadiagonal linear systems. Results of numerical experiments are presented to demonstrate properties of the new methods, and extensions to multidimensional problems are outlined. This is joint work with Andreas Karageorghis of the University of Cyprus.

Combining Physical Resist Modeling and Self-Consistent Field Theory for Pattern Simulation in Directed Self-Assembly of Block Copolymers in Nanolithography

**Valeriy Ginzburg, The Dow Chemical Company**

In this presentation, we describe multi-scale modeling method combining PROLITH lithography

simulation with Self-Consistent Field Theory (SCFT) computation of the block copolymer Directed Self-Assembly (DSA). Within this method, we utilize PROLITH to predict the shape of a lithographic feature as function of process conditions. The results of that calculation are then used as input into SCFT simulation to predict the distribution of the matrix and etchable blocks of the DSA polymers (such as PS-b-PDMS or PS-b-PMMA) inside that feature. We applied this method to several simple cases (e.g., rectangular trench, cylindrical contact hole, and clover-leaf contact hole), and investigated the self-assembly of various polymers as function of their compositions. The new tool could therefore be applied to rapidly design and screen lithographic process conditions together with polymers used to shrink or rectify the features within the DSA technology.

Algorithms for numerical solution of the Goursat problem on a triangular domain with mixed boundary conditions

**Kun Gou, Michigan State University**

For the Goursat problem, we consider a triangular domain with mixed Dirichlet and impedance

boundary conditions imposed on it. We develop algorithms for its numerical solution mainly based on RungeKutta method and trapezoidal formula. Iterative techniques are constructed to compute some data for the nonlinear part of the differential equation and the impedance boundary condition. Error estimates are derived. Examples are presented to illustrate the viability of the method.

Predictive Simulation for Porous Media Flows on GPUs

**Arunasalam Rahunanthan, University of Toledo**

One of the most difficult tasks in subsurface characterization is the reliable characterization of

properties of the subsurface. A typical situation employs dynamic data integration such as partial production data to be matched with simulated responses associated with a set of permeability and porosity fields. The Markov Chain Monte Carlo (MCMC) methods are well suited for reconstructing the spatial distribution of permeability and porosity and quantifying associated uncertainties. However, because of the sequential nature of MCMC, it is often impractical to use MCMC to a problem that takes a long simulation time for a single realization. With the availability of inexpensive CPU-GPU high performance clusters, one could use the pre-fetching algorithm to parallelize an MCMC chain. In this talk, we discuss an MPI-CUDA implementation to parallelize an MCMC chain for predictive simulation in petroleum reservoirs.

Modeling within-host dynamics of influenza virus infection

**Libin Rong, Oakland University**

Influenza virus infection is a public health problem worldwide. The mechanisms underlying viral

control within an individual with an uncomplicated influenza virus infection are not fully understood. In this talk, I will address this by reviewing existing mathematical models and presenting a new model including immune responses. I will discuss the relative roles of target cell availability, and innate and adaptive immune responses in controlling the virus.

Numerical smoothness and error analysis of time-dependent PDEs

**Tong Sun, Bowling Green State University**

The talk will be a survey of my work on the topic in the last a few years. We will introduce the concept of numerical smoothness, show its necessity for achieving optimal order of approximation of PDE solutions, explain its implication as a new criterion on numerical schemes, and briefly summarize some error estimation results on nonlinear conservation laws. If time allows, we will discuss some general ways of representing the smoothness of numerical solutions for the purpose of error analysis. The last part will contain some unpublished contents.

Numerical Methods for High Dimensional Singular Perturbation Problems

**Weiqun Zhang, Wright State University**

Improved a priori bounds and boundary layer tests were extended to two dimensional singular

perturbation problems. Boundary layers along x-axis, y-axis, or both are considered. A robust

numerical scheme was developed. The numerical error is maintained at the same level for a family of singular perturbation problems with a constant number of mesh points. Numerical experiment supports the theory.

Interface-fitted adaptive mesh method for free interface problems with surface tension based on level-set formulation

**Xiaoming Zheng, Central Michigan University**

This work presents a novel two-dimensional interface-fittted adaptive mesh method to solve free interface problems using a level-set formulation. The interface-fitted mesh is achieved by two operations: (i) the projection of mesh nodes onto the interface and (ii) the insertion of mesh nodes right on the interface. This approach allows the use of the standard P1 finite element method to achieve almost second order accuracy for elliptic problems with jump conditions across the interface. Applications to the evolution of two free boundary problems, a sheared drop in Stokes flow and the growth of a solid tumor, are presented. In these applications, a level-set function is used to track the evolution of the interface and provide the interface location for the interface fitting. This is a joint work with Dr. John Lowengrub at UC Irvine.

Posters

Using Coiflets to Solve Integral Equations

**Yousef Al-Jarrah, Central Michigan University**

We solved a four different kind of integral equations, which are:

1- Fredholm integral equation of the first kind

2- Fredholm integral equation of the second kind

3- Volterra integral equation

4- Fredhol-Volterra integral equation.

Proved the convergence theorem for these methods. In addition, examples are also presented, and proposed similar method to solve 2-dimensional integral equation and application in image processing will be mentioned.

Alternative Energy through Fuel Cells: Computer Simulations & Dynamic Modeling

**Christopher Alkevicius, Central Michigan University**

In the recent years, society has become significantly dependent on renewable energy. While solar panels and wind turbines are some alternative ways to generate energy, fuel cells are often underutilized. It generates electric power, through electrochemical reactions between hydrogen and oxygen, thus serving as one of the most environment friendly technologies of the modern 21st century. With its ability to generate power in the range of 5-10,000 kilowatts, its market is expected to grow at a rate of 26% annually through 2023 [3, 4]. However, some of the current challenges with fuel cell design are lack of efficient models to study gas chamber flooding, a limiting factor in energy production. This poster focuses mainly on the operation, evolution and projected growth of polymer electrolyte membrane (PEM) fuel cell, and briefly outlines the gas chamber modeling and simulation using COMSOL multi-physics and MATLAB.

Data Analysis and Data Reduction of HPMC Gelation from Experimental Investigation

**Nalinda Almeida, Central Michigan University**

The addition of salts into HPMC solutions strongly influence the rheological properties of the resulting solutions, therefore broadly impact the manufacturing process of coating, fabrication of hard shell capsules, as well as producing oral-film strips. This study investigates the effects of mono and divalent salt on the viscoelastic properties of highly concentrated HPMC of low molecular weight solutions. Such study provides better understanding of the gelation mechanism of HPMC in the presence of various salt concentrations. The polymer solutions of 15 wt% concentration were made from METHOCEL Premium cellulose ethers (hypromellose, HPMC) of low-viscosity (low-molecularweight) grades of type 2906 and varying salt concentrations from 0.01M to 0.5M. Steady state and dynamic measurement data were obtained using AR2000 rheometer with concentric cylinder geometry. The reduction of gelation temperatures of hypromellose solutions was observed when salt was added into the solution. The amount of reduction on gelation temperature appear to linearly decrease with increasing concentration of salts and the linear dependence of salt concentration seem independent of

the type of salts investigated. Thermal analysis showed that the depression of melting temperature increased linearly with increasing salt concentrations.

Multiscale Quantum Dynamics in Continuum Model and Simulation for Proton Transport

**Duan Chen, Mathematical Biosciences Institute**

The present work presents a multiscale/multiphysical model for the molecular mechanism of proton transport in transmembrane proteins. We describe proton dynamics quantum mechanically via a density functional approach while implicitly model other solvent ions as a dielectric continuum to reduce the number of degrees of freedom. The densities of all other ions in the solvent are assumed to obey the Boltzmann distribution and the impact of protein molecular structure and its charge polarization on the proton transport is considered explicitly at the atomic level. We formulate a total free energy functional and the variational principle is employed to derive nonlinear governing equations for the proton transport system. A number of advanced numerical algorithms are utilized to implement the proposed model in a computationally efficient manner. A comparison with experimental data verifies the present

model predictions and validates the proposed model.

Differential Geometry Based Solvation Models

**Zhan Chen (University of Minnesota), Peter Bates, Nathan Baker and Guowei Wei**

Solvation is an elementary process in nature and is of paramount importance to more sophisticated chemical, biological and bio molecular processes. The understanding of solvatioin is an essential prerequisite for the quantitative description and analysis of biomolecular systems.

Computational Studies of Various Herbal Components Encapsulated in Glycodendrimers

**Pratik Chhetri, Central Michigan University**

With growing number of population and discovery of various diseases follows research in various areas to find cures for them. Scientists have been endlessly working to find treatment for various newly emerged and some age old uncured disease. Computational studies on various chemical and biological agents are one such approach to find answers to some yet unanswered problems in the medical world. Here, we study the energy components of various active ingredients of known herbs and spices (such as turmeric, capsicum, garlic, ginger, onion, bittermelon). For an effective and targeted drug delivery system, interactions of these ingredients have been studied with biodegradable glyco-dendrimers. Researchers are often looking to increase the loading capacity of a self-associating linear or block copolymer or dendrimers for a given hydro-phobic drug in order to introduce chemical moieties to the

backbone of the system. This will help more interaction between drugs and the polymeric system, which will lead to the negative heat of mixing, a measure of Florry-Huggins solubility parameter. The presence of benzyl carboxylate groups on the glycodendrimer induce the formation of additional inter or intra-molecular specific interactions (e.g. phi-phi interactions/hydrogen bonds, etc.), which in turn leads to the improvement of the drug loading capacity of the dendrimer.

An Exploration of Dynamical Systems with an Application in Cancer Growth

**Patrick Davis, Central Michigan University**

Complex systems are inherent in both engineered and natural situations. As we encounter these systems, there is a great need to design simple but accurate mathematical modeling and simulations that describe them in order to prevent serious errors and mistakes - which can serve to increase costs (both monetarily and otherwise). Currently, nowhere is such work needed more than in the medical field and in particular with cancer research. Toward that end, we propose a simple model using differential equations and concepts from dynamical systems to study cell development in an attempt to understand tumor growth (or any similar dynamical growth progression). The resulting model is solved numerically and then we perform a basic parameter analysis to determine the models relative validity.

A Treecode-Accelerated Boundary Integral Poisson-Boltzmann Solver for Electrostatics of Solvated Biomolecules

**Robert Krasny, University of Michigan**

We present a treecode-accelerated boundary integral (TABI) solver for electrostatics of solvated biomolecules described by the linear Poisson-Boltzmann equation. The method employs a wellconditioned boundary integral formulation for the electrostatic potential and its normal derivative on the molecular surface. The surface is triangulated and the integral equations are discretized by centroid collocation. The linear system is solved by GMRES iteration and the matrix-vector product is carried out by a Cartesian treecode which reduces the cost from $O(N^2)$ to $O(N\log N)$, where $N$ is the number of faces in the triangulation. The TABI solver is applied to compute the electrostatic solvation energy in two cases, the Kirkwood sphere and a solvated protein. We present the error, CPU time, and memory usage, and compare results for the Poisson-Boltzmann and Poisson equations. We show that the treecode approximation error can be made smaller than the discretization error, and we compare two versions of the treecode, one with uniform clusters and one with non-uniform clusters adapted to the molecular surface. For the protein test case, we compare TABI results with those obtained using the grid-based APBS code, and we also present parallel TABI simulations using up to eight processors. We find that the TABI solver exhibits good serial and parallel performance combined with relatively simple implementation, efficient memory usage, and geometric adaptability. This is joint work with Weihua Geng (University of Alabama, Tuscaloosa).

Reaction Diffusion Equation Applications

**Azza Abu Shams, Central Michigan University**

Solve, visualize and discuss importance of Reaction –Diffusion equations (linear/non-linear), using finite difference method (COMSOL) simulations and fractional derivatives withAdomian polynomials and its applications.

Variational Multiscale Modeling of Biomolecular Complexes

**Kelin Xia, Michigan State University**

Multiscale modeling is of paramount importance to the understanding of biomolecular structure,

function, dynamics and transport. Geometric modeling provides structural representations of molecular data from the Protein Data Bank (PDB) and the Electron Microscopy Data Bank (EMDB). Commonly used geometric models, such as molecular surface (MS), van der Waals surface, and solvent accessible surface are ad hoc devision of solvent and solute regions and lead to troublesome geometric singularities. We discuss our variational multiscale models and associated geometric modeling of biomolecular complexes, based on differential geometry of surfaces and geometric measure theory. Our models give rise to singularity-free surface representation, curvature characterization, electrostatic mapping, solvation energy and binding affinity analysis of biomolecules.

Mathematical Modeling of a Transverse Shear Deformation Thick Shell Theory

**Mohammad Zannon, Central Michigan University**

Three dimensional theory of elasticity in curvilinear coordinates are employed to understand the stressstrain deviations under various operating conditions in the mid-surface of thick shell. The relationship between forces, moments and stresses are given and the equations of motion are derived using Newtons 2nd law of motion and Hamilton's principle. The necessary theoretical assumptions are discussed to reduce the three dimensional (3-D) elasticity to two-dimensional (2-D) shell theory of third order to a first order approximations. Equilibrium equations are formulated using the equations of stress resultants to attain a system of differential equations and solved by Fourier series expansion. The results are compared with the exiting three- dimensional elasticity theory.

Differential Geometry Based Ion Transport Models

**Qiong Zheng, Michigan State University**

Ion channels are pore-forming proteins presented in cell membranes, usually allowing specific ions to pass across membranes and maintaining proper intracellular ion compositions. They are crucial to cell survival and function, and are key components in many biological processes such as nerve and muscle excitation, action potential generation and resting, sensory transduction, learning and memory, to name a few. Dysfunction ion channels can cause many diseases: deafness, blindness, migraine headaches, and cardiac arrhythmias. Ion channels are frequent targets for drug design.

A Viscoelastic Model of Blood Capillary Extension and Regression

**Xiaoming Zheng, Central Michigan University**

This work studies a fundamental problem in blood capillary growth: how the cell proliferation or death induces the stress response and the capillary extension or regression. The mathematical model treats the cell density as the growth pressure eliciting a viscoelastic response from the cells, which again induces extension or regression of the capillary. This model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.