Talk Title and Abstract

Plenary Talks (in an alphabetical order of the speaker’s last name)

Talk title: Complex problems call for simplicity

Speaker: Malgorzata Peszynska, Oregon State University

Talk abstract: Computational simulations of complex problems motivated directly by physical or biological phenomena or by the associated engineering or medical solutions involve so many pieces (sometimes ill-defined), that it is hard to define one robust unifying strategy that would lead to useful solutions to realistic scenarios. In general, the choice of a particular algorithm or discretization is based usually on what is known about the accuracy and efficiency of algorithms for simple(r) subsystems, under simplifying assumptions; therefore, the properties of such schemes do not necessarily carry over to the coupled systems, or their particular significance fades in view of the overall high complexity of the problem.

In this talk we present arguments for "simplicity" as the key strategy for a particular class of complex problems for which we aim to set up an "in silico" laboratory to replace empirical constitutive laws at Darcy scale by upscaled porescale simulations in complicated geometries such as between the grains of porous materials or in human tissue at a spatial scale of nano- to millimeters. We present some rigorous analyses of the underlying schemes, demonstrate the limits of the assumptions, while we illustrate the talk with simulations of real life phenomena. This is joint work with many students and collaborators to be named in the talk.

Talk title: On data-driven predictor-corrector optimization algorithm for stable parameter estimation in epidemiology

Speaker: Alexandra Smirnova, Georgia State University

Talk abstract: This project was initiated during Collaborative Workshop for Women in Mathematical Biology (sponsored by the United Health Group) in Minnetonka, MN, in June 2022. A group of 6 women (2 senior faculty and 4 junior researchers) considered a nonlinear constrained optimization problem motivated by various challenges of stable parameter estimation in epidemiology. A hybrid regularized predictor-corrector scheme was introduced that builds upon all-at-once formulation, recently developed by Kaltenbacher and her co-authors for general constrained least squares problems (2016), the generalized profiling methods by Ramsay, Hooker, Campbell, and Cao (2007) for estimating parameters in nonlinear ordinary differential equations, and the so-called traditional route for solving nonlinear ill-posed problems, pioneered by Bakushinsky (1991). Similar to all-at-once approach, our proposed algorithm does not require an explicit deterministic or stochastic trajectory of system evolution. At the same time, the predictor-corrector framework of the new method avoids the difficulty of dealing with large solution spaces resulting from all-at-once make-up, which inevitably leads to oversized Jacobian and Hessian approximations. Therefore, our predictor-corrector algorithm (PCA) has the potential to save time and storage, which is critical when multiple runs of the iterative scheme are carried out for uncertainty quantification. The new algorithm takes full advantage of the iterative regularization framework, and it is not limited to the constraints in the form of differential equations (or systems of differential equations). Theoretical findings are illustrated with numerical experiments carried out with real data on COVID-19 pandemic.

Talk title: Mechanisms of SARS-CoV-2 Evolution and Transmission

Speaker: Guowei Wei, Michigan State University

Talk abstract: Discovering the mechanisms of SARS-CoV-2 evolution and transmission is one of the greatest challenges of our time. By integrating artificial intelligence (AI), viral genomes isolated from patients, tens of thousands of mutational data, biophysics, bioinformatics, and algebraic topology, the SARS-CoV-2 evolution was revealed to be governed by infectivity-based natural selection in early 2020 (J. of Mole. Biol. 2020, 432, 5212-5226). Two key mutation sites, L452 and N501 on the viral spike protein receptor-binding domain (RBD), were predicted in summer 2020, long before they occur in prevailing variants Alpha, Beta, Gamma, Delta, Kappa, Theta, Lambda, Mu, and Omicron. Recent studies identified a new mechanism of natural selection: antibody resistance (J. Phys. Chem. Lett. 2021, 12, 49, 11850–11857). AI-based forecasting of Omicron’s infectivity, vaccine breakthrough, and antibody resistance was later nearly perfectly confirmed by experiments (J. Chem. Inf. Model. 2022, 62, 2, 412–422). The replacement of dominant BA.1 by BA.2 in later March was predicted in early February (J. Phys. Chem. Lett. 2022, 13, 17, 3840–3849). On May 1, 2022, persistent Laplacian-based AI projected Omicron BA.4 and BA.5 to become the new dominating COVID-19 variants (arXiv:2205.00532). This prediction became reality in late June. Our model offers accurate prediction of mutational impacts on the efficacy of monoclonal antibodies (mAbs).

Invited Talks (in an alphabetical order of the speaker’s last name)

Talk title: A Switch Point Algorithm Applied to a Harvesting Problem

Speaker: Summer Atkins, Louisiana State University

Talk abstract: We study an optimal control problem that determines a harvesting strategy of a fish population over a spatial region. The problem is linear in the control. When parameters to the problem are set to where a singular subarc exists, chattering may occur. Chattering is the phenomenon in which the optimal control oscillates infinitely many times over a finite region. Such a solution cannot be realistically implemented. In this presentation, we consider a switch point algorithm, a procedure in which we solve the problem with respect to the switches rather than the control. We use the switch point algorithm to find suboptimal harvesting strategies that can be realistically applied. Additionally, we use Aitken extrapolation to approximate the boundary of the chattering region of the optimal control and its corresponding optimal value.

Talk title: A Stable Mimetic Finite-Difference Discretization for Convection-Dominated Diffusion Equations

Speaker: Casey Cavanaugh, Louisiana State University

Talk abstract: Convection-diffusion equations arise in a variety of applications such as particle transport, electromagnetics, and magnetohydrodynamics. Simulation of the convection-dominated case, even with high-fidelity techniques, is particularly challenging due to sharp boundary layers and shocks causing jumps and discontinuities in the solution, and numerical issues such as loss of the maximum principle in the discretization. These complications cause instabilities, admitting large oscillations in the numerical solutions when using traditional methods. Drawing connections to the simplex-averaged finite element method (S. Wu and J. Xu, 2020), we develop a mimetic finite-difference (MFD) discretization using exponentially averaged coefficients to guarantee monotonicity of the scheme and stability of the solution as the diffusion coefficient approaches zero. The finite-element framework allows for transparent analysis of the MFD, such as proving well-posedness and deriving error estimates from the finite-element setting. Numerical tests are presented confirming the stability of the method and verifying the error estimates.

Talk title: Tensor structured sketching for constrained least squares problems

Speaker: Ke Chen, University of Maryland

Talk abstract: Constrained least squares problems arise in many applications. The memory and computation costs are usually expensive with high-dimensional input data. We employ the so-called “sketching” strategy to project the least squares problem into a space of a lower “sketching dimension" via a random sketching matrix. The key idea of sketching is to reduce the dimension of the problem as much as possible while maintaining the approximation accuracy. In this talk, we will focus on least square problems with tensor data matrix. Such structure is often present in linearized inverse PDE problems and tensor decomposition optimizations. To match with the tensor structures of the problem, we utilize a general class of row-wise tensorized sub-Gaussian matrices as sketching matrices. We provide theoretical guarantees on the sketching dimension in terms of error criterion and probability failure rate. For unconstrained linear regressions, we obtain an optimal estimate for the sketching dimension. For optimization problems with general constraint sets, we show that the sketching dimension depends on a statistical complexity that characterizes the geometry of the underlying problems. Our theories are demonstrated and validated in a few concrete examples, including unconstrained linear regression and sparse recovery problems.

Talk title: Finite Element Methods for Phase Field Crystal Models

Speaker: Amanda E. Diegel, Mississippi State University

Talk abstract: A relatively new class of mathematical models known as phase field crystal models has emerged as a way to simulate physical processes where automic- and microscales are tightly coupled. In this talk, we present numerical schemes for two such models which rely on a C0 interior penalty finite element method spacial discretization. We show that the numerical methods are unconditionally energy stable and unconditionally convergent and support our conclusions with a few numerical experiments.

Talk title: A robust discretization technique for three dimensional Helmholtz problems

Speaker: Adrianna Gillman, University of Colorado, Boulder

Talk abstract: The ability to robustly and efficiently solve Helmholtz problems has been plagued by the so-called pollution effect and the introduction of artificial resonances by the discretization. The recently developed the Hierarchical Poincare-Steklov (HPS) method has demonstrated that it does not observe either of these shortcomings for two dimensional problems. Additionally, a robust coupling technique for scattering problem involving local deviations from constant coefficient which utilizes a Dirichlet-to-Neumann operator built by the HPS method has been developed. In this presentation, we will demonstrate that the extension of the HPS method to three dimensional problems is just as robust as the two-dimensional solution technique. We will also present ongoing work towards making the solution technique efficient for three dimensional problems. Currently utilizing an iterative solver, the method is able to solve a mid-frequency scattering problem with variable medium to 9 digits of accuracy with a billion unknowns in 40 minutes.

Talk title: Modeling lamprey locomotion, sensory feedback, and spinal injuries in an immersed boundary framework.

Speaker: Christina Hamlet, Bucknell University

Talk abstract: Lampreys are fish that serve as model systems for locomotion and neurophysiology studies. To study the relations between neural signaling and swimming, we employ a multi-scale, integrative, computational neuromechanical model of an anguilliform (eel-like) swimmer fully coupled to a viscous, incompressible fluid in an immersed boundary framework. This model is driven by a set of coupled oscillators capable of receiving sensory feedback from changes in the body. We use this model to perform computational experiments which examine the effects of feedback on swimming stability and the role feedback from the body may play in recovering from spinal injuries. Our results show, in some cases, feedback amplification below a spinal lesion is sufficient to partially or entirely restore normal swimming behavior.

Talk title: Domain decomposition methods for flow and transport problems in fractured porous media

Speaker: Thi Thao Phuong Hoang, Auburn University

Talk abstract: This talk is concerned with efficient numerical methods for compressible fluid flow and linear transport problems in a heterogeneous porous medium that contains fractures. The fracture represents a fast pathway (i.e., with high permeability) and is modeled as a hypersurface embedded in the porous medium. We develop fast-convergent and accurate global-in-time domain decomposition methods for such a reduced fracture model, in which smaller time step sizes in the fracture can be coupled with larger time step sizes in the rock matrix. Each method leads to a space-time interface problem which is solved iteratively and globally in time. Numerical results for the non-immersed and partially immersed fracture problems are presented to illustrate and compare the convergence and accuracy of the proposed methods with nonconforming time grids.

Talk title: Computational Models for Bioseparations Processes

Speaker: Lea Jenkins, Clemson University

Talk abstract: The demand for biopharmaceuticals has increased over the past few decades. There are currently over 200 biotherapeutics on the market, with many more in trials or in the pipeline. Many of the recent biotherapeutics are monoclonal antibody therapies, with the treatment for COVID-19 being one of the more famous examples. Monoclonal antibody therapies are quickly becoming the standard of care for various cancers, anti-immune disorders, and chronic inflammatory diseases. This increase in demand requires the development of novel adsorptive chromatography media to ensure high-volume throughput of purified product. These media use multiple modes of interaction with the product to recover it selectively from impurities in the feed solution, leading to mathematically complex models to describe the adsorption process. In this talk, we discuss the development of a computational simulation environment to aid in the design of efficient production facilities which incorporate these novel adsorption media. We provide analysis of discretizations of the nonlinear adsorption model, and we demonstrate the use of the simulations to evaluate the impact of design changes on metrics of interest to the biopharmaceutical industry.

Talk title: Stabilized Integrating Factor Runge-Kutta Method and Unconditional Preservation of Maximum Bound Principle

Speaker: Lili Ju, University of South Carolina

Talk abstract: Maximum bound principle (MBP) is an important property for a large class of semilinear parabolic equations, in the sense that their solution preserves for all time a uniform pointwise bound in absolute value imposed by the initial and boundary conditions. It has been a challenging problem on how to design unconditionally MBP-preserving time stepping schemes for these equations, especially the ones with order greater than one. We combine the integrating factor Runge-Kutta (IFRK) method with the linear stabilization technique to develop a stabilized IFRK (sIFRK) method, and successfully derive the sufficient conditions for the proposed method to preserve MBP unconditionally in the discrete setting. We then elaborate some sIFRK schemes with up to the third order accuracy, which are proven to be unconditionally MBP-preserving by verifying these conditions. In addition, it is shown that many strong-stability-preserving sIFRK (SSP-sIFRK) schemes do not satisfy these conditions, except the first-order one. Various numerical experiments are also carried out to demonstrate the performance of the proposed method.

Talk title: A reduced basis method for radiative transfer equation

Speaker: Fengyan Li, Rensselaer Polytechnic Institute

Talk abstract: Leveraging the existence of a hidden low-rank structure hinted by the diffusive limit, in this work, we design and test an angular-space reduced order model for the linear radiative transfer equation based on reduced basis methods (RBMs). Our algorithm is built upon a high-fidelity solver employing the discrete ordinates method in the angular space, an upwind discontinuous Galerkin method for the physical space, with an efficient synthetic accelerated source iteration for the resulting linear system. Strategies are particularly proposed to tackle the challenges associated with the scattering operator within the RBM framework. This is a joint work with Z. Peng, Y. Chen, and Y. Cheng.

Talk title: Weighted least-squares finite element method for Poroelasticity

Speaker: Hyesuk Lee, Clemson University

Talk abstract: In this talk, we consider a weighted least-squares method for a poroelastic structure governed by Biot's consolidation model. The quasi-static model equations are converted to a first-order system of four-field, and the least-squares functional is defined for the time discretized system. We consider two different sets of weights for the functional and show coercivity and continuity properties of the least-squares functional, from which an a priori error estimate for the primal variables is derived. Numerical experiments are provided to illustrate the performance of the proposed method.

Talk title: Biological Flow Simulations: From IB to IIM, Algorithm & Theory

Speaker: Zhilin Li, North Carolina State University

Talk abstract: This talk aims at general audience for an introduction to Biological Flow Simulations using Immersed Boundary and Immersed Interface methods (IB and IIM).

Peskin’s Immersed Boundary (IB) method is well-known and has been extensively applied to vast biological flow simulations such as the blood flow is hearts, cell and tumor computation and detections. The Immersed Interface Method (IIM) was motivated by Peskin's Immersed Boundary (IB) method for better accuracy and to deal with discontinuous coefficients. I will explain the IB and IIM for Stokes and Navier Stokes equations with moving interfaces such as an elastic membrane motion in a fluid, and non-extensible membranes in an incompressible flow.

If time allows, I will also discuss some recent progress in the theoretical field on the convergence of the IB method with known velocity (Dirichlet) boundary condition.

We have proved that the IB method is inconsistent but first order accurate in the infinity norm while the pressure has half order convergence in the L-2 norm. The proof is based discrete Green’s functions and estimates on bounded lattices, new L-2 discrete delta functions and splitting the singularities. Numerical examples with non-trivial and general interfaces are also provided.

Talk title: Pressure Robust Scheme for Incompressible Flow

Speaker: Lin Mu, University of Georgia

Talk abstract: In this talk, we shall introduce the recent development regarding the pressure robust finite element method (FEM) for solving incompressible flow. We shall take weak Galerkin (WG) scheme as the example to demonstrate the proposed enhancement technique in designing the robust numerical schemes and then illustrate the extension to other finite element methods. Weak Galerkin (WG) Method is a natural extension of the classical Galerkin finite element method with advantages in many aspects. For example, due to its high structural flexibility, the weak Galerkin finite element method is well suited to most partial differential equations on the general meshing by providing the needed stability and accuracy. Due to the viscosity and pressure independence in the velocity approximation, our scheme is robust with small viscosity and/or large permeability, which tackles the crucial computational challenges in fluid simulation. We shall discuss the details in the implementation and theoretical analysis. Several numerical experiments will be tested to validate the theoretical conclusion.

Talk title: Multiscale Cardiovascular Dynamics Models

Speaker: Mette Olufsen, North Carolina Sate University

Talk abstract: This study discusses the use of one-dimensional fluid dynamics models for prediction of flow, pressure, shear stress, and wave propagation to assess changes in cardiovascular dynamics with disease. Focus is on systems-level predictions using computational framework merging data and modeling. We use a multiscale approach including the large arteries and veins, arterioles and venules, and capillaries. The large arteries and veins are represented by a directed graph extracted from computed tomography images, whereas the network of arterioles and venules are represented by structured trees with parameters informed by data. The capillary network modeled using a sheet approximation, is coupled to the network of arterioles and veins in a ladderlike figuration. In the large vessels, we solve the 1D Navier Stokes equations, while in the network of small vessels and capillaries we solve linearized equations, which are coupled to the large vessels via outflow boundary conditions. The model is calibrated to a healthy control, and we progressively increase disease severity via vessel stiffening and narrowing. This study will show examples of model predictions for patients with pulmonary hypertension, Fontan circulation, and porto-systemic shunt. We demonstrate the importance of sensitivity analysis and parameter inference to render the models patient specific and show how the calibrated models can be used to predict effects of treatment or challenges to the system (such as exercise).

Talk title: Exploring the impact of “Silent Spreaders” in COVID-19 Dynamics

Speaker: Omar Saucedo, Virginia Tech

Talk abstract: The dynamic nature of the COVID-19 pandemic has demanded a public health response that is constantly evolving due to the novelty of the virus. Many jurisdictions in the United States, Canada, and across the world have adopted social distancing and recommended the use of face masks to minimize disease spread. Considering these measures, it is important to understand the contributions of subpopulations – such as “silent spreaders” – to disease transmission dynamics in order to inform public health strategies. Thus, we develop a hybrid stochastic model that includes two classes of individuals: silent spreaders, who neither have symptoms nor disease-induced mortality; and symptomatic spreaders, who experience symptoms and a resultant mortality rate. Using previously established methodologies, we study the role of demographic and environmental variability on COVID-19 dynamics.

Talk title: Applications of Persistent Spectral Graphs in COVID-19

Speaker: Rui Wang, Michigan State University

Talk abstract: Persistent spectral graph is a new paradigm for the multiscale analysis of the topological invariants and geometric shapes of high-dimensional datasets. Motivated by the success of persistent homology and multiscale graphs in dealing with complex biomolecular data, we construct a family of spectral graphs. Specifically, families of persistent Laplacian matrices (PLMs) corresponding to various topological dimensions are constructed by continuously increasing a filtration parameter. The harmonic spectra from the null spaces of PLMs can capture the topological structures, whereas the non-harmonic spectra of PLMs characterize the additional geometric shape of a given high-dimensional data. In addition, we developed an open-source package called HERMES that can provide an efficient and robust tool to calculate the multidimensional harmonic and non-harmonic spectra of PLMs. This enables broad real-world applications in biology, medicine, and engineering. During the COVID-19 pandemic, we developed a Math-AI model by integrating persistent spectral graphs, genomics, biophysics, experimental data, and deep learning to forecast the mutational impacts on COVID-19 vaccines, anti-body drugs, and diagnostics. Our model has accurately forecasted the incoming dominance of Omicron, Omicron BA.2, and BA.4/BA.5 variants one or two months ahead.

Talk title: Introduction on conforming discontinuous Galerkin finite elements

Speaker: Shangyou Zhang, University of Delaware

Talk abstract: In all existing discontinuous Galerkin methods, the degree k polynomial solution converges at the optimal order only, i.e., order k plus 1 in L2 norm and order k in H1 norm. In the new conforming discontinuous Galerkin method, the numerical flux is no longer the average of two discontinuous functions on two sides of an edge or a triangle, but an average of four discontinuous functions nearby.

With properly reconstructed trace and properly chosen spaces for the gradient and the Hessian, the CDG degree k polynomial solution converges at two-order and four-order above the optimal order for second order elliptic equations and fourth order elliptical equations, respectively, i.e., order k plus 3 in L2 norm and order k plus 2 in H1 norm for second order elliptic equations, and order k plus 5 in L2 norm and order k plus 3 in H2 norm for fourth order elliptic equations. In other words, the CDG $C^{-1}$-$P_3$ solution is as good as the $C^1$-$P_7$ conforming finite solution in approximation.