# Applied Math Seminar

## Applied Math Seminar

**Title: **A New Block Preconditioner for Implicit Runge-Kutta Methods for Parabolic PDE Problems

**Abstract: **Explicit time integrators for parabolic PDE are subject to a restrictive timestep limit, so A-stable integrators are essential. It is well known that although there are no A-stable explicit linear multistep methods and implicit multistep methods cannot be A-stable beyond order two, there exist A-stable and L-stable implicit Runge-Kutta (IRK) methods at all orders. IRK methods offer an appealing combination of stability and high order; however, these methods are not widely used for PDE because they lead to large, strongly coupled linear systems. An s-stage IRK system has s-times as many degrees of freedom as the systems resulting from backward Euler or implicit trapezoidal rule discretization applied to the same equation set. In this talk, I will introduce a new block preconditioner for IRK methods, based on a block LDU factorization with algebraic multigrid subsolves for scalability. I will demonstrate the effectiveness of this preconditioner on two test problems, a 2-D heat equation and a model advection-difusion problem. I compare this preconditioner in condition number and eigenvalue distribution, and in numerical experiments with other preconditioners currently in the literature. Experiments are run using IRK methods with two to seven stages. We find that the new preconditioner outperforms the others, with the improvement becoming more pronounced as spatial discretization is refined and as temporal order is increased.

## Applied Math Seminar

**Title:** Data-driven hierarchical kernel matrix methods

**Abstract:** The explosion of datasets from diverse applications and the increasing computational power of computer hardware call for the need of scalable algorithms and software. In this talk, I will focus on the computational bottlenecks associated with fully populated kernel matrices that are ubiquitous in machine learning as well as scientific simulations. Those dense matrices usually induce large computation costs that scale quadratically or cubically with problem size. The complexity can be significantly reduced by exploiting the hierarchical rank structure inside the kernel matrices. Representing a kernel matrix in an appropriate hierarchical format enables (nearly) optimal storage and computations. I will demonstrate the newly developed data-driven techniques for hierarchical representations and compare their performance with state-of-the-art methods/software on several real-world applications.

## Applied Math Seminar

**Title:**Modeling and topological methods to better understand pattern formation in fish

**Abstract:**Many natural and social phenomena involve individual agents coming together to create group dynamics, whether the agents are drivers in a traffic jam, voters in an election, or locusts in a swarm. Self-organization also occurs at much smaller scales in biology, though, and here I will focus on elucidating how brightly colored cells interact to form skin patterns in zebrafish. Wild-type zebrafish are named for their dark and light stripes, but mutant zebrafish feature variable skin patterns, including spots and labyrinth curves. All these patterns form as the fish grow due to the interactions of tens of thousands of pigment cells. This leads to the question: how do cell interactions change to create mutant patterns? The longterm motivation for my work is to help shed light on this question and better link genes, cell behavior, and visible animal characteristics. Toward this goal, we combine different modeling approaches (including agent-based and continuum) to simulate pattern formation and make experimentally testable predictions. In this talk, I will overview our models and highlight how topological data analysis can be used to quantitatively describe self-organization

*in silico*and

*in vivo*.

## Applied Math Seminar

**Title:**A Self-consistent-field Iteration for Orthogonal Canonical Correlation Analysis

**Abstract:**We propose an efficient algorithm for solving orthogonal canonical correlation analysis (OCCA) in the form of trace-fractional structure and orthogonal linear projections. Even though orthogonality has been widely used and proved to be a useful criterion for visualization, pattern recognition and feature extraction, existing methods for solving OCCA problem are either numerically unstable by relying on a deflation scheme, or less efficient by directly using generic optimization methods. In this paper, we propose an alternating numerical scheme whose core is the sub-maximization problem in the trace-fractional form with an orthogonality constraint. A customized self-consistent-field (SCF) iteration for this sub-maximization problem is devised. It is proved that the SCF iteration is globally convergent to a KKT point and that the alternating numerical scheme always converges. We further formulate a new trace-fractional maximization problem for orthogonal multiset CCA and propose an efficient algorithm with an either Jacobi-style or Gauss-Seidel-style updating scheme based on the SCF iteration. Extensive experiments are conducted to evaluate the proposed algorithms against existing methods, including real-world applications of multi-label classification and multi-view feature extraction. Experimental results show that our methods not only perform competitively to or better than the existing methods but also are more efficient.

## Applied Math Seminar

**Title:****Boolean canalization in the micro and macro scales**

**Abstract:**Canalization is a property of Boolean automata that characterizes the extent to which subsets of inputs determine (canalize) the output. In this presentation, I describe the role of canalization as a determinant of the dynamical character of Boolean networks (BN). I consider two different measures of canalization introduced by Marques-Pita and Rocha, namely 'effective connectivity' and 'input symmetry,' in a three-pronged approach. First, we show that the mean 'effective connectivity,' a measure of the true mean in-degree of a BN, is a better predictor of the dynamical regime (order or chaos) of random BNs with homogeneous connectivity than the mean in-degree. Next, I combine effective connectivity and input symmetry in a single measure of 'unified canalization' by using a common yardstick of Boolean hypercube “dimension” — a form of fractal dimension. I show that the unified measure is a better predictor of dynamical regime than effective connectivity alone for BNs with large in-degrees. Finally, I introduce 'integrated effective connectivity' as a macro-scale extension of effective connectivity that characterizes the canalization present in BNs coarse-grained in time obtained by iteratively composing a BN with itself. I show that the integrated measure is a better predictor of long-term dynamical regime than just effective connectivity for a small class of BNs known as the elementary cellular automata. The results also help partly explain the chaotic properties of Rule 30 and why it makes sense to use it as a random number generator.

## Applied Math Seminar

**Title:** Statistics, Topology and Data Analysis

**Abstract:** In this talk, I will discuss how statistics and topological data analysis are beautifully complement each other to solve real data problems. As a paradigm, I will discuss supervised learning, and present a classification approach using a novel Bayesian framework for persistent homology. An application to materials science will be discussed.

**Bio:** Vasileios Maroulas is a Professor of Mathematics with joint appointments at the Business Analytics and Statistics, and the Bredesen Center’s Data Science Engineering at the University of Tennessee, Knoxville (UTK). He is a Senior Research Fellow at the US Army Research Lab, an Elected Member of the International Statistical Institute, and an Editor-in-Chief of Foundations of Data Science published by AIMS. Following his PhD graduation from the Statistics Department at the University of North Carolina at Chapel Hill in 2008, he continued as a Lockheed Martin Postdoctoral Fellow at the IMA at the University of Minnesota for two years until he joined UTK in 2010 as an Assistant Professor. Maroulas was also a Mathematical Sciences Leverhulme Trust Fellow at the University of Bath, UK during 2013-2014. His research interests span from computational statistics and machine learning to applied probability and computational topology and geometry with applications in data analysis and quantum computing. His methods have found applications in chemistry, neuroscience, materials science, and biology. His work has been funded by several federal agencies, including AFOSR, ARO, DOE, and NSF; by national labs and private foundations, including ARL, ORNL, the Simons Foundation, and the Leverhulme Trust in the UK; as well as by industry, including Eastman, and Thor Industries.

## Applied Math Seminar

**Title:** Mathematical modelling of blood coagulation system

**Abstract:** Blood is an important liquid organ performing transport functions. Any injury may lead to dangerous blood loss, but fortunately we have a reliable emergency blood coagulation system that quickly reacts to injuries and prevents massive blood loss. Disorders in blood coagulation may induce thrombosis, stroke, myocardial infarction and other complications, including lethal ones.

## Applied Math Seminar

**Title:**Modeling of Emergent Patterns Within Stem Cell Colonies

**Abstract:**The differentiation of stem cell colonies into specified tissue types is possible through local and long-distance intercellular communication; however, it is unclear which mechanisms take priority in context-specific situations. Here we consider human induced pluripotent stem cells (hiPSCs) whose therapeutic potential arises from their ability to differentiate into all germ lineages. Prior work in the literature suggests that both cell-autonomous and non-autonomous (e.g. positional) mechanisms determine cell fate during the differentiation of hiSPCs, producing patterns and other system-level features in the process. Informed by experimental data, we develop a collection of agent-based models (ABMs) whose agents (i.e. cells) are each equipped with local rules that govern how the agents interact with their environment and with each other. The purpose of each ABM is to simulate the early differentiation of hiPSCs according to a different set of biological assumptions, with some ABMs using a Boolean network to model potential mechanisms of intercellular communication. We also extend an existing mathematical framework by M. Yereniuk and S.D. Olson which formalizes ABMs to estimate long-term model behavior with respect to time. Our extensions introduce the birth and death of agents into the framework, and our estimates aim to establish connections between local interactions and certain system-level observations. Thus, we study both the emergent behaviors of our ABMs and the dynamics of the local rules governing each agent in order to ascertain which modes of intercellular communication determine cell fate.

## Applied Math Seminar

**Title:** Improved Training of Generative Adversarial Network

**Abstract:** The original Generative Adversarial Network was introduced by Ian Goodfellow et al. in 2014, together with a discriminator loss function, called binary cross-entropy. Later, other discriminator loss functions were developed: WGAN loss, hidge loss, Dragan loss, etc. We introduce a new family of discriminator loss functions. Experiments validated the effectiveness of our loss functions on unconditional image generation task.

## Applied Math Seminar

**Title:** Designing multistability with AND gates

**Abstract:** Systems of differential equations have been used to model biological systems such as gene and neural networks. A problem of particular interest is to understand and control the number of stable steady states. Here we propose conjunctive networks (systems of differential equations equations created using AND gates) to achieve any desired number of stable steady states. Our approach uses combinatorial tools to easily predict the number of stable steady states from the structure of the wiring diagram.