Nonlinear Simulation of Solid Tumor Growth with Chemotaxis and Complex Far-field Geometries

Min-Jhe Lu

Illinois Institute of Techonlogy

We present a model for solid tumor growth with chemotaxis and cell-to-cell adhesion. We consider the effect of the tumor microenvironment by the variability in spatial diffusion gradients, the uptake rate of nutrients inside/outside the tumor and the heterogeneous distribution of vasculature modeled using complex far-field geometries. Using a spectrally accurate boundary integral method with both direct and indirect formulations, our nonlinear simulation reveals that vascular heterogeneity plays an important role in the development of morphological instability in both nutrient-poor and nutrient-rich regions. This is a joint work with Prof. Chun Liu (Illinois Institute of Technology), Prof. John Lowengrub (University of California at Irvine) and Prof. Shuwang Li (Illinois Institute of Technology).

Golub-Kahan-Type-Tikhonov Reduction Methods Applied to Systems with Multiple Right-Hand Sides

Enyinda Onunwor

Kent State University

We are interested in large linear systems of the form AX = B,

where X=[x_1, …, x_k], B = [b_1, …, b_k], and A is a very large ill-conditioned matrix and its singular values decay to zero. We encounter problems like this in the modeling of geophysical applications like the direct current (DC) resistivity problem. Common approaches to solving these include block methods and seed conjugate gradient (CG) methods. This talk will introduce two algorithms based on an implicitly restarted Golub-Kahan bidiagonalization method. The first algorithm will compute a partial SVD, then apply the truncated SVD to each right-hand side. The second algorithm will compute the partial bidiagonalization with reorthogonalization, then apply Tikhonov regularization to the reduced system. The stopping criterion for both methods is based on the discrepancy principle. Results will be compared to seed CG methods.

Voronoi Neural Networks

Matthew Dixon

Illinois Institute of Technology

We introduce a multi-layer perceptron (MLP) which explicitly tessellates the domain with a Voronoi diagram rather than learns the partitions through least squares optimization. The primary advantage of such a network is that is equivalent to b-spine interpolation, where the degree of the b-spline matches that of the activation functions. Under Lipschitz continuity of the target function, we present various bounds on the global approximation error as a function of the number of hidden units. Numerical experiments are presented to demonstrate the convergence properties. Finally, some important future directions are discussed, such as the effect of multiple hidden layers on the approximation error.

The Distributed Kaczmarz Method

Fritz Keinert

Iowa State University

In 1937, Stefan Kaczmarz published an algorithm for iteratively solving a system of linear equations. The algorithm became popular in the early days of Computed Tomography, under the name Algebraic Reconstruction Technique (ART).

I will describe a new implementation where the equations are distributed among the nodes of a tree-shaped graph. Each iteration step consists of a dispersion and a pooling phase. During the outgoing dispersion phase, each node receives an approximate solution from its predecessor. It performs its calculation and passes the result on to its successors. In the pooling phase, the intermediate results gets aggregated and passed back to the root.

I will describe the implementation and convergence behavior of this algorithm, and talk about some potential applications.

Sensitivity of Singular and Ill-conditioned Linear Systems

Zhonggang Zeng

Northeastern Illinois University

Although linear systems in scientific computing can inevitably be singular, numerical solutions of such systems are rarely mentioned in the literature. Solving a singular linear system for an individual vector solution is known to be an ill-posed problem with a condition number infinity. On the other hand, the solution of a singular linear system can alternatively be considered a unique element in an affine Grassmannian and, from this perspective, its sensitivity is bounded so that the system can even be well-conditioned. If a singular linear system is given through empirical data that are sufficiently accurate with a tight error bound, a properly formulated general numerical solution uniquely exists in the same affine Grassmannian, enjoys Lipschitz continuity and approximates the underlying exact solution with an accuracy in the same order as the data. Furthermore, any backward accurate numerical solution vector is an accurate approximation to one of the infinitely many solutions of the underlying singular system. In this talk we present the error analysis, software tools, numerical examples and applications of singular linear systems.

High performance computing in fluid solid interaction with applications to biological flow problems

Jifu Tan

Northern Illinois University

Fluid flow is important for many biological problems, e.g., blood cell transport, drug delivery, and blood clotting, etc. One of challenges in modeling bio-transport phenomena is the multiphase nature of the flow, e.g., fluid: blood plasma; solid: cells and small particles. By taking advantage of high performance computers, we developed a large scale highly parallel open sourced fluid-solid interaction program to solve the coupled Multiphysics problem. The fluid field was solved by the Lattice Boltzmann method, while the large deformation of solids were simulated by a coarse grained molecular dynamics model. The coupling was achieved through the immersed boundary method (IBM). Through IBM, highly heterogeneous solvers can be coupled together with less software development time but with new functionalities. The code modeled both rigid and deformable solids exposed to flow. The model was validated extensively with the Jeffery orbits of an ellipsoid particle in shear flow, red blood cell stretching tests, and effective blood viscosity flowing in tubes. It demonstrated essentially linear scaling from 512 to 8192 processors for both strong and weak scaling cases on the supercomputers in the Argonne National Lab. Two bio-transport examples are given to demonstrate the capabilities of the code: flexible filament (drug carrier) transport in a flowing blood cell suspension and cancer cell transport and adhesion in a microfluidic device, highlighting the advantages and versatilities of the developed code.

Reduced Model of One-Dimensional Unsaturated Flow in Heterogeneous Soils

Ruowen Liu

University of Michigan

A reduced model for unsaturated flow in one-dimensional layered soils with local stochastic soil hydraulic conductivity is proposed. The reduced model relies on an improved closure assumption compared to straightforward truncation of moment equations for the statistics (expectation and covariance) of the volumetric water content, whereby certain higher-order moments can be expressed in terms of known quantities. The reduced model is compared numerically with

(1) direct numerical simulations using homogenized soil properties and

(2) agent-based stochastic direct numerical simulations, with soil hydraulic properties defined at the local (finite difference cell) level. Numerical results show that the reduced model successfully captures larger variations in water content of the unsaturated flow in heterogeneous soils compared to the solution resulting from homogenization.

A Modified Preconditioned Conjugate Gradient Method for a Nonsymmetric Elliptic Boundary Value Problem

Zhen Chao

University of Wisconsin-Milwaukee

We develop algorithms for solving nonsymmetric indefinite linear systems by considering the augmented linear systems resulting from a weighted linear least squares problem. Even though the augmented system is more ill-conditioned than the original linear system, one can construct preconditioned GMRES or PCG methods for solving these augmented systems capable of obtaining reasonable approximation of the solution in fewer iterations than the classical ILU preconditioned GMRES method for solving the original linear system. More specifically, we present some different preconditioners for these augmented systems, examine the spectral properties of these preconditioned augmented systems, and report numerical results to illustrate the effectiveness of these preconditioners.