Past Events

January 10 (1:30pm): RTG Group Meeting 

Research Updates:

• Michael Wells 

• Victor Rielly

• Michael Neunteufel

• Will Pazner

January 24 (1:30pm): Jeffrey Ovall, Portland State University

Title: Landscape Refinement for (Large) Invariant Subspaces

Abstract: When approximating a collection of eigenvalue-eigenvector pairs (eigenpairs) of a differential operator via finite element methods, the prevailing wisdom is that the finite element space should be well-suited to approximating the entire invariant subspace (the span of the eigenvectors), as opposed to just being well-suited to approximating some particular basis of this space.  Over the past 15 years or so, there have been a variety of approaches proposed for achieving this goal as part of an adaptive algorithm.  Each approach uses the current computed approximation of the eigenvector basis in some (sophisticated) way to estimate subspace approximation error and determine how to refine the current finite element space to improve the next approximation.  When the collection of desired eigenpairs is large, these approaches become increasingly costly.  Additionally, if the desired eigenvalues are higher in the spectrum and the current space is "too coarse", it is unclear that refinement based on eigenvector information from the current space is meaningful---if it leads to improvements in the next approximation, that may just be on accident.  We propose an alternative means of driving adaptive refinement, based on the solution of a single source problem---the so-called landscape function.

After providing some theoretical and heuristic justification that landscape refinement is not a ridiculous proposition, we illustrate its practical performance on a variety of examples, primarily in the hp-adaptive setting, where we think it can provide a particularly attractive alternative to current approaches.  Some experiments are also done in the h-adpative setting, where landscape refinement for modest clusters higher in the spectrum is also competitive, but might instead be used to guide refinement until the space is "fine enough" for eigenvector-based refinement to become meaningful.  

This is joint work with Stefano Giani (Durham University) and Gabriel Pinochet-Soto (Portland State University)

February 7 (1:30pm): RTG Group Meeting

Research Updates:

• Julie Zhu

• Gabriel Pinochet-Soto

• Justin Valentin

• Austen Nelson

February 7 (3:15pm, Maseeh Colloquium): Laurent Younes, Johns Hopkins University

Title: Shape Alignment via Allen-Cahn Nonlinear-Convection

Abstract: We present a phase field method on shape registration to align shapes of possibly different topology.  A soft end-point optimal control problem is introduced whose minimum measures the minimal control norm required to align an initial shape to a final shape, up to a small error term.  Inspired by level-set methods and large diffeomorphic metric mapping, the controls spaces are integrable scalar functions to serve as a normal velocity and smooth reproducing kernel Hilbert spaces to serve as velocity vector fields.  The paths in control spaces then follow an evolution equation which is a generalized convective Allen-Cahn.  The existence of mild solutions to the evolution equation is proved, the minimums of the time disctretized optimal control problem are characterized, and numerical simulations of minimums to the fully discretized optimal control problem are provided.

March 7 (1:30pm): RTG Group Meeting 

Research Updates:

• Bryttani Nieves

• Tamara Gratcheva

• Esther Toobian

• Emily Bogle

• Rebecca Bryant

March 7 (3:15pm, Maseeh Colloquium): David Watkins, Washington State University

Title: Bulge Chasing is Pole Swapping

Abstract: At the beginning of the era of electronic computing there

was a big effort to produce software to make the newly constructed

hardware useful.  In the area of scientific computing, one need that

was recognized early on was for efficient and reliable methods to

compute the eigenvalues of a matrix.  This need was met around 1960 by

the so-called QR algorithm, especially the implicitly-shifted variant

due to John Francis.  For the generalized eigenvalue problem, Moler

and Stewart introduced a variant of Francis's algorithm called the QZ

algorithm.   These algorithms, with various bells and whistles added

over the years, are still the dominant algorithms today.   These are

bulge-chasing algorithms.   They create bulges at one end of a

(Hessenberg) matrix or pencil and chase them to the other end.  A few

years ago a new class of algorithms, pole-swapping algorithms, was

introduced by Camps, Meerbergen, Vandebril, and others.   It turns out

that pole swapping is a generalization of bulge chasing.   It might

happen that new pole-swapping codes will supplant the current QR and

QZ codes in the major software packages.   Whether this turns out to

be true or not, the pole-swapping viewpoint is extremely valuable for

a detailed understanding of what makes this class of algorithms, both

bulge-chasing and pole-swapping, work.

March 14 (1:30pm): Matthew Zahr, University of Notre Dame

Title: High-order implicit shock tracking for shock-dominated flows

Abstract: Shock-dominated flows are notoriously challenging to simulate accurately and efficiently using modern numerical methods. Strong propagating features such as shock waves and contact discontinuities often require delicate stabilization, small time steps, tight grid spacing, and sometimes additional requirements on the mesh (e.g., well-conditioned quad/hex elements aligned with shocks). In this talk, I will present recent advances to implicit shock tracking, a class of numerical methods that aim to overcome these challenges by deforming the mesh such that element faces align with flow features. This ensures discontinuous features (contacts and inviscid shocks) are represented perfectly by inter-element jumps in the solution basis without the need for stabilization.  Viscous features (viscous shocks and boundary layers) are accurately represented without stabilization by compressing elements into the thickness of the feature and locally increasing the polynomial degree. Unlike traditional shock fitting methods, the optimal grid is computed as the solution of a PDE-constrained optimization problem. Numerical demonstrations of the method for supersonic and hypersonic flows show implicit shock tracking delivers accurate solutions on traditionally coarse grids.

October 4 (1:30pm): RTG Group Meeting 

Student presentations:

• Emily Bogle

• Gabriel Pinochet-Soto

• Austen Nelson

Faculty discussion

October 11 (1:30pm): Jeffrey Ovall, Portland State University

Title: Computational tools for resolving eigenvectors of interest

Abstract: When computing eigenpairs of (differential) operators it is sometimes desired to determine only those eigenvectors having some specified property of interest, such as strong spatial localization in a designated part of the domain, nearness to a given subspace, approximate symmetries, etc.  We will focus on second-order linear elliptic operators for our discussion---more specifically, reaction-diffusion operators and magnetic Schroedinger operators.  In the first part of this talk, we describe an algorithm designed to "see" only those eigenvectors of interest, and illustrate it on several examples.  This algorithm is a natural extension of one we derived for detecting spatial localization, and the key theoretical results supporting that algorithm carry over in the more general setting.  The second part of this talk concerns what might be considered a pre-conditioning of the eigenvalue problem for the magnetic Schroedinger operator.   We propose a "canonical" gauge transform that replaces the given operator with one that has the same spectrum, but whose eigenvectors can be more efficiently and stably approximated.

October 18 (1:30pm): Luka Marohnic, University of Zagreb

Title: Detecting Near Resonances in Acoustic Scattering Using Randomized Subspace Embeddings and Hierarchical LU Factorization

Abstract: In this talk, we investigate the scattering of time-harmonic acoustic waves by a bounded, penetrable homogeneous object within the framework of the Helmholtz transmission problem. Our primary focus is on identifying parameter values that lead to near-resonant behavior in the setting which guarantees the unique solvability of the system. To address this task numerically, we utilize a randomized singular value decomposition (SVD) algorithm to analyze the inverse of the solution operator matrix arising in the Galerkin discretization method. Additionally, we discuss employing a hierarchical approach for sampling the inverse matrix and its benefits in terms of both storage efficiency and computational performance.

October 25 (1:30pm): Luka Grubisic, University of Zagreb

Title: Monotone convergence of domain truncation based finite element schemes

Abstract: Starting from the notion of the generalized strong resolvent condition we present the monotone form convergence theorem of Kato and Weidmann. We then show the convergence results for finite element approximations of the reaction diffusion operators posed in unbounded domains (Schrödinger Hamilton operators). Finally, we address the approximation algorithms based on filtered iteration and the contour integration for these problems. An example of the Schrödinger Hamiltonian with a short-range potential will be used as a benchmark example.

November 1 (1:30pm): RTG Group Meeting

Research Updates:

• Ester Toobian

• Quincy Robinson

• Anthony Kolshorn

• Jay Gopalakrishnan

Faculty discussion (RTG PhD students are invited to stick around for this)

November 8 (1:30pm): Ricardo Baptista, California Institute of Technology

Title: Dynamics and Memorization Behaviour of Score-Based Diffusion Models

Abstract: Diffusion models have emerged as a powerful framework for generative modeling that relies on score matching to learn gradients of the data distribution's log-density. A key element for the success of diffusion models is that the optimal score function is not identified when solving the denoising score matching problem. In fact, the optimal score in both unconditioned and conditioned settings leads to a diffusion model that returns to the training samples and effectively memorizes the data distribution. In this presentation, we study the dynamical system associated with the optimal score and describe its long-term behavior relative to the training samples. Lastly, we show the effect of two forms of score function regularization on avoiding memorization: restricting the score's approximation space and early stopping of the training process. These results are numerically validated using distributions with and without densities including image-based problems.

November 8 (3:15pm, Maseeh Colloquium): Ricardo Baptista, California Institute of Technology

Title: Towards large-scale data assimilation with structured generative models

Abstract: Accurate state estimation, also known as data assimilation, is essential for geophysical forecasts, ranging from numerical weather prediction to long-term climate studies. While ensemble Kalman methods are widely adopted for this task in high dimensions, these methods are inconsistent at capturing the true uncertainty in non-Gaussian settings. In this presentation, I will introduce a scalable framework for consistent data assimilation. First, I will demonstrate how inference methods based on conditional generative models generalize ensemble Kalman methods and correctly characterize the probability distributions in nonlinear filtering problems. Second, I will present a dimension reduction approach for limited data settings by identifying and encoding low-dimensional structure in generative models with guarantees on the approximation error. The benefits of this framework will be showcased in applications from fluid mechanics with chaotic dynamics, where classic methods are unstable in small sample regimes.

Brief Bio: Ricardo is an incoming Assistant Professor at the University of Toronto in the Department of Statistical Sciences and is currently a visitor at Caltech. Ricardo received his PhD from MIT, where he was a member of the Uncertainty Quantification group. The core focus of his work is on developing the methodological foundations of probabilistic modeling and inference. He is broadly interested in using machine learning methods to better understand and improve the accuracy of generative models for applications in science and engineering.

November 15 (1:30pm): Bamdad Hosseini, University of Washington

Title: Computational Measure Transport for Scientific Computing: Theory and Algorithms

Abstract:  Many recent generative models in machine learning and data science can be understood from the lens of measure transport. This approach allows us to provide theory for such algorithms but also inform the design of new methods. In this talk I will present such a formalization for the particular class of minimum divergence generative models. I will show simple theoretical results that allow us to obtain quantitative error bounds for such methods and present various parameterizations that can be understood using classic results from approximation theory and statistical analysis. I will further discuss applications of minimum divergence models to scientific computing problems with a particular focus on inverse problems and conditional sampling, which hints at an entirely new paradigm for statistical inference.

November 22 (1:30pm): Noel Walkington, Carnegie Mellon University

Title: Discontinuous Galerkin Schemes and the Convection Diffusion Equation

Abstract: This presentation will begin with a review of the good, bad, and ugly properties of various finite element schemes for elliptic, convection, and convection-diffusion equations. Subsequently, properties of the hybridized discontinuous Galerkin (HDG) method will be developed. Stability and convergence of solutions computed using HDG finite element schemes for both elliptic and convection diffusion equations will be considered. We show that convergence of numerical solutions to the convection diffusion equation are "asymptotic robust"; that is, solutions converge independently of how the diffusion constant and mesh parameters tend to zero.

November 22 (3:15pm, Maseeh Colloquium): Noel Walkington, Carnegie Mellon University

Title: Modeling Multiphase Porous Flows

Abstract: In order to model many geological flows of contemporary interest it is necessary to include the thermodynamics of the underlying processes. Examples include CO_2 sequestration and the release of greenhouse gasses dissolved in melting permafrost.  Tractable models of such problems can only involve gross (macroscopic) properties, since a precise description of the physical system is neither available nor computationally tractable.

This talk will first briefly review the role thermodynamics plays in classical continuum mechanics; in particular, how dissipation            principles give rise to bounds above and beyond the natural conservation properties.  The development of macroscopic models of      geological flows involving multiple components undergoing changes of phase will then be considered. These models involve an          amalgamation of classical and continuum thermodynamics to yield systems of conservation laws which inherit natural dissipation     principles. The convexity (concavity) of the free energy (entropy) functions is essential for the development of stability estimates        of solutions to these equations, and the development of stable numerical schemes.

December 6 (1:30pm): RTG Group Meeting (canceled)

April 5 (1:30pm): RTG Group Meeting

• Discussion of RAIN Meeting at PSU (Will Pazner)

• Inputs for RTG annual report

April 12 (1:30pm): Panayot Vassilevski, Portland State University

Title: Interpolation Spaces, Fractional Order Operators, and Scales of Multilevel Decompositions

Abstract: In applications involving Monte Carlo simulations based on solving PDEs with stochastic coefficients, one needs efficient ways of sampling these coefficients from certain distributions. The sampling process itself can be formulated as solving an elliptic PDE with a random r.h.s. (so-called white noise). There is a scale of PDEs that produce random coefficients from a scale of distributions. In this presentation, we will provide an application of the interpolation theory of spaces and operators to define such scales. Secondly, in the finite element setting, we will consider computationally feasible alternatives involving computable hierarchical decomposition of fractional order operators. Some preliminary numerical illustration will be presented.

April 19 (1:30pm): Safa Mote, Portland State University

Title: Improved subseasonal prediction of South Asian monsoon rainfall using data-driven forecasts of oscillatory modes

Abstract: Predicting the temporal and spatial patterns of South Asian monsoon rainfall within a season is of critical importance due to its impact on agriculture, water availability, and flooding. The monsoon intraseasonal oscillation (MISO) is a robust northward-propagating mode which determines the active and break phases of the monsoon, and much of the regional distribution of rainfall. However, dynamical atmospheric forecast models predict this mode poorly. Data-driven methods for MISO prediction have shown more skill, but only predict the rainfall portion corresponding to MISO. Here, we combine state-of-the-art ensemble precipitation forecasts from a high-resolution atmospheric model with data-driven forecasts of MISO using a novel method. We thereby achieve improvements in rainfall forecasts over India, as well as the broader monsoon region, at 10–30 day lead times, an interval that is generally considered to be a predictability gap. Our results demonstrate the potential of leveraging the predictability of intraseasonal oscillations to improve extended-range forecasts; more generally, they point towards a future of combining dynamical and data-driven forecasts for Earth system prediction.

April 26 (1:30pm): Guoshen Fu, University of Notre Dame

Title: Finite elements for optimal transport, Schrodinger bridge, and mean-field control

Abstract: We first give a brief introduction of the Monge-Kantorovich optimal transport problem and the closely related Schrödinger bridge problem. The problems can be casted into a (fluid) dynamic formulation through Benamou and Brenier's celebrated work in 2000. It is a linear (PDE) constrained (convex) optimization problem. We show how finite element methods can be naturally used to discretize this optimization problem, which leads to a discrete saddle point optimization problem. Then a first-order optimization solver, namely, preconditioned Primal-Dual Hybrid Gradient, is used to solve this optimization problem. We further extend our discretization approach to more general mean-field control problems for reaction-diffusion systems including control of droplet dynamics in thin-film lubrication theory.

April 27: Cascade RAIN meeting

May 3: RTG Group Meeting (cancelled)

May 10 (1:30pm): Patrick De Leenheer, Oregon State University

Title: The many proofs of the reduction phenomenon

Abstract: In this talk we consider several proofs, old and new, of the reduction phenomenon. Many mathematical models of biological systems that incorporate dispersal and reproduction exhibit the phenomenon that increased dispersal reduces overall growth. Karlin first proved this for basic, parameterized linear discrete-time models in 1975, and many other proofs have been discovered since then. We review some of these proofs, indicate how they are connected and also provide streamlined, novel proofs of this celebrated biological principle.

May 17 (1:30pm): Patrick De Leenheer, Oregon State University

Title: Power domination with random sensor failure 

Abstract: The power domination problem seeks to find the placement of the minimum number of sensors called phasor measurement units (PMUs) needed to monitor an electric power network. Vertices with a PMU and vertices adjacent to PMUs are observed. Then, for any observed vertex with exactly one unobserved neighbor, we can use conservation of energy laws to observe the unobserved neighbor. This process continues until no more vertices can be observed. We study the challenges that arise when PMUs are allowed to fail with probability q, as well as developing sufficient and necessary conditions for observing the network within different tolerances.

May 24 (1:30pm): RTG Group Meeting

• Student Research Updates

  • Julie Zhu

  • Vicky Haney

  • Tamara Gratcheva

  • Acamaro Cutcher

• RTG Faculty Discussion and Planning

May 31 (1:30pm): Richard Mills, Argonne National Laboratory

Title: An Introduction to Nonlinear Solver Composition and Preconditioning with PETSc

Abstract: The Portable, Extensible Toolkit for Scientific Computation (PETSc) is a widely-used open-source framework that provides scalable parallel data structures and solvers for problems that commonly arise in the context of partial differential equation-based simulations. In this talk, I will introduce PETSc by example, focusing on features that enable nonlinear solver composition and preconditioning. Most high-performance linear solvers in use today are composed of several algorithmic components, generally a Krylov subspace method combined with one or more preconditioners (typically multigrid or domain-decomposition methods). It is possible to generalize these concepts to solvers for nonlinear algebraic systems, sometimes with dramatically improved performance. I will introduce basic solvers for nonlinear algebraic systems and do some live demonstrations of using them with PETSc. After demonstrating some failure cases for the basic solvers, I will introduce the concepts behind nonlinear solver composition and preconditioning, and then demonstrate how sophisticated nonlinear solvers can be composed at run-time using PETSc to tackle some extremely nonlinear problems. Experiments with such solvers indicate that they can be extremely powerful, though they can also be frustratingly fragile, and they are not yet well-explored, experimentally or theoretically -- and, therefore, may present interesting research opportunities!

(Time and audience interest permitting, we may also explore some "bonus" content on using GPUs with PETSc, or composing multi-physics solvers using PETSc's FieldSplit preconditioning capabilities towards the end of the seminar.)

NOTE: Those interested may wish to try following along with the live demonstrations. Those who want to do so should come to the seminar with a laptop on which a recent release of PETSc is already installed. See https://petsc.org/release/install/ for installation instructions. If you run into difficulty, you can contact the speaker (rtmills@anl.gov) or petsc-maint@mcs.anl.gov (probably best for quickest response, as all PETSc maintainers see messages to this address).

The talk slides can be found here: https://climatemodeling.org/~rmills/talks/PSU-ACM-seminar-2024.pdf

The online exercises related to the demos can be found here: 

https://xsdk-project.github.io/MathPackagesTraining2022/lessons/nonlinear_solvers_petsc/

June 7 (1:30pm): Pedram Hassanzadeh, University of Chicago

Title: Integrating the spectral analyses of neural networks and nonlinear physics for explainability, generalizability, and stability

Abstract: In recent years, there has been substantial interest in using deep neural networks (NNs) to improve the modeling and prediction of complex, multiscale, nonlinear dynamical systems such as turbulent flows and Earth’s climate. In idealized settings, there has been some progress for a wide range of applications from data-driven spatio-temporal forecasting to long-term emulation to subgrid-scale modeling. However, to make these approaches practical and operational, i.e., scalable to real-world problems, a number of major questions and challenges need to be addressed. These include 1) instabilities and the emergence of unphysical behavior, e.g., due to how errors amplify through NNs, 2) learning in the small-data regime, 3) interpretability based on physics, and 4) out-of-distribution generalization (e.g., extrapolation to different parameters, forcings, and regimes) which is essential for applications to non-stationary systems such as a changing climate. While some progress has been made in addressing (1)-(4), e.g., doing transfer learning for generalization, these approaches have been often ad-hoc, as currently there is no rigorous framework to analyze deep NNs and develop systematic and general solutions to (1)-(4). In this talk, I will discuss some of the approaches to address (1)-(4), for example, once we identify spectral bias as the cause of instabilities in state-of-the-art weather models like Pangu-weather, GraphCast, and FourCastNet. Then I will introduce a new framework that combines the spectral (Fourier) analyses of NNs and nonlinear physics, and leverages recent advances in theory and applications of deep learning, to move toward rigorous analysis of deep NNs for applications involving dynamical systems. For example, this approach can guide and explain transfer learning and pruning in such applications. I will use examples from turbulence modeling and weather/climate prediction to discuss these methods and ideas.

June 7 (3:15pm, Maseeh Colloquium): Pedram Hassanzadeh, University of Chicago

Title: Artificial Intelligence and the 2nd Revolution in Weather and Climate Prediction

Abstract: Accurate weather and climate predictions are critical for many applications, from early warnings for extreme events to improving resiliency and planning adaptation and mitigation. The current state of the art of weather and climate prediction relies on numerical solutions of the governing equations of the atmosphere, ocean, and other components of the Earth system, and is a result of a slow 50-year scientific revolution. However, the enormous computational cost of the current numerical weather and climate models hinders efforts on reducing the uncertainties in these predictions. In recent years, artificial intelligence (AI) techniques have received significant attention as tools that can help with improving weather and climate prediction and reducing these uncertainties. In fact, for 1-15 day weather forecasting, the AI-based models have shown substantial success in outperforming the numerical models at a fraction of the computational cost. Dubbed the second revolution in weather forecasting, this success suggests that AI can potentially transform the state of the art of climate prediction too, once a number of major challenges are addressed. I will discuss these challenges, and particularly how integrating fundamental concepts and tools from math, climate physics, and computer science need to be integrated to make progress.

January 12 (1:30pm): Andreas Rupp, Lappeenranta-Lahti University of Technology

Title: Partial differential equations on hypergraphs and networks of surfaces

Abstract: Albeit many physical, sociological, engineering, and economic processes have been described by partial differential equations posed on domains which cannot be described as subsets of linear spaces or smooth manifolds, there is still a lack of mathematical tools and general purpose software specifically addressing the challenges arising from the discretization of these models.

This presentation establishes a general approach to formulate partial differential equations (PDEs) on networks of (hyper)surfaces, referred to as hypergraphs. Such PDEs consist of differential expressions with respect to all hyperedges (surfaces) and compatibility conditions on the hypernodes (joints, intersections of surfaces). These compatibility conditions ensure conservation properties (in case of continuity equations) or incorporate other properties motivated by physical or mathematical modeling. We illuminate how to discretize such equations numerically using hybrid discontinuous Galerkin (HDG) methods. These methods consist of local solvers (encoding the differential expressions on hyperedges) and a global compatibility condition (related to our hypernode conditions). We complement the physically motivated compatibility conditions by a derivation through a singular limit analysis of thinning structures yielding the same results.

January 19 (1:30pm): RTG Group Meeting (cancelled due to weather)

January 26 (1:30pm): No meeting this week

February 2 (1:30pm): Jeffrey Ovall, Portland State University

Title: Understanding eigenvector localization phenomena for (magnetic) Schrödinger operators

Abstract: We discuss a variety of theoretical results and computational tools that can shed light on the strong spatial localization of eigenvectors of a class of elliptic differential operators.  Such localized eigenvectors imply similar localization of waves for physical or mechanical systems that are stimulated at "frequencies" associated with the eigenvalues of localized eigenvectors.  A better understanding of this phenomenon can guide the design of devices/systems that exhibit desired acoustic or electromagnetic properties, for example.  This topic has drawn significant attention over the past decade from mathematicians, physicists and engineers.  We will focus on the magnetic Schrödinger operator, stating results obtained by our group over the past couple of years, and highlighting a few recent results and directions of further investigation.

February 9 (1:30pm): RTG Group Meeting

• Student Research Updates

  • Michael Wells

  • Victor Reilly 

  • Austen Nelson

  • Madison Ruff 

  • Henry Hillebrandt

  • Gabriel Pinochet

• RTG Faculty Discussion and Planning

February 9 (3:15pm, Maseeh Colloquium): Chen Grief, University of British Columbia

Title: Numerical solution of double saddle-point problems

Abstract: Double saddle-point systems are drawing increasing attention in the past few years, due to the importance of relevant applications and the challenge in developing efficient numerical solvers.  In this talk we describe some of their numerica properties.  We derive eigenvalue bounds, expressed in terms of extremal eigenvalues and singular values of block submatrices.  We also analyze the spectrum of preconditioned matrices based on block diagonal preconditioners using Schur complements, and it is shown in this case that the eigenvalues are clustered within a few intervals bounded away from zero, giving rise to rapid convergence of Krylov subspace solvers.  A few numerical experiments illustrate our findings.

February 16 (1:30pm): No meeting this week

February 23 (1:30pm): Hannah Kravitz, Portland State University

Title: Some Interesting Problems in Metric Graph Theory

Abstract: The field of metric graphs is an exciting new area of mathematics. Currently only a small community of mathematicians study these objects, but the topic is becoming more popular due to the abundance of interesting still-unsolved problems. In this talk (a research overview with a not-so-secret goal to recruit students to work on these projects), I will present some of the main problems I work on.

I study structures called ``metric graphs,” a type of network with a distance metric defined on the edges. This creates a 1D structure on which to solve partial differential equations (PDEs), with boundary conditions in the form of coupling conditions at the vertices. This seemingly simple setup (it is in one dimension after all), leads to many interesting questions like: 1) What causes waves to get trapped in certain parts of the graph (localization)?, 2) What determines which frequencies can exist on the graph (the eigenvalue problem)?, 3) How can we apply this work to the real-world (optical science, the spread of epidemics, and beyond)?, and 4) How can we utilize the structure of the graph to develop efficient algorithms?

March 1 (1:30pm): Amanda Howard, Pacific Northwest National Laboratory

Title: More of a good thing: multifidelity and stacking networks for physics-informed training

Abstract: Physics-informed neural networks and operator networks have shown promise for effectively solving equations modeling physical systems. However, these networks can be difficult or impossible to train accurately for some systems of equations. One way to improve training is through using a small amount of data in addition to physics. However, such data is expensive to produce. We will discuss multifidelity deep operator networks, which use data and physics simultaneously to increase the training accuracy. We will then introduce our novel multifidelity framework for stacking physics-informed neural networks and operator networks that facilitates training by progressively reducing the errors in our predictions for when no data is available. In stacking networks, we successively build a chain of networks, where the output at one step can act as a low-fidelity input for training the next step, gradually increasing the expressivity of the learned model.

March 8 (1:30pm): Heather Wilber, University of Washington

Title: Designing specialized low rank methods

Abstract: Many of the structured matrices we know and love (Toeplitz, Hankel, Cauchy, Vandermonde) have special properties that make it efficient to represent and store them computationally using low rank approximation methods. This may be surprising at first, since not all of these matrices are directly compressible. In this talk, we use rational approximation theory to explore why and how we can compress structured matrices such as these, the ways in which their structures can be exploited for fast computations, and how related structures arise more broadly in numerical linear algebra.

March 15 (1:30pm): RTG Group Meeting

• Student Research Updates

  • Gabriel Pinochet Soto

  • Micron Jenkins

  • Robyn Reid

  • Emily Bogle

• RTG Faculty Discussion and Planning

September 29 (1:30pm): RTG Group Meeting

• Student Research Updates

  • Vicky Haney

  • Victor Rielly

  • Austen Nelson

  • Rebecca Bryant

• RTG Faculty Discussion and Planning

October 6 (1:30pm): Julie Zhu, Portland State University

Title: Analysis and application for Ziolkowski's PML

Abstract:  Perfectly Matched Layer (PML) technique is an effective tool proposed by Berenger to solve a wave propagation problem in an unbounded domain without reflections. Here we are interested in the physically inspired PML model proposed by Ziolkowski in 1999. I will introduce a high-order finite difference time domain (FDTD) method and a few finite element time domain (FETD) methods for solving this PML model. Stability and convergence analysis are established. Numerical results are presented to support our analysis and demonstrate the wave absorption effectiveness of this PML.

October 13-15: 4th SIAM PNW Biennial Meeting

October 20 (1:30pm): Juergen Kritschgau, Portland State University

Title: Zero forcing and related parameters 

Abstract: Given a graph with blue and white vertices, the zero forcing color change rule states that a blue vertex $u$ turns a white neighbor $w$  blue if $w$ is the unique white neighbor of $u$. The zero forcing number of a graph is the minimum number of initial blue vertices required to turn a graph blue by iteratively applying the zero forcing color change rule. Zero forcing number and related parameters arise in the study of the maximum nullity of real symmetric matrices related to graphs, strong structural controllably of networks, phasor measurement unit placement in the electrical grid, and graph searching. In this talk, I will give an overview of zero forcing and discuss a few related parameters.

October 27 (1:30pm): Nicholas Fisher, Portland State University

Title: Divergence Free Kernel Methods: From Interpolation to Quasi-interpolation

Abstract: Divergence-free interpolation has been extensively studied and widely used in approximating vector-valued functions that are divergence-free as well as for the numerical computation of the Helmholtz-Hodge decomposition of vector fields. However, until recently, the literature contained no treatment of any corresponding divergence-free quasi-interpolation schemes. The aims of talk are two-fold: we give a brief introduction to divergence free kernel methods from the interpolation perspective before discussing a new analytically divergence-free quasi-interpolation scheme, derive its simultaneous approximation orders to both the approximated function and its derivatives, and apply the quasi-interpolant to the Helmholtz-Hodge decomposition of vector fields.

November 3 (1:30pm): RTG Group Meeting

• Student Research Updates

  • Samuel Reynolds

  • Melayne Barker

• RTG Faculty Discussion and Planning

November 16 (11:00am): Marko Hajba, University of Zagreb

Title: Tensorial Neural Networks for Eigenvalue Problems

Abstract: We study eigenmode localization for a class of elliptic reaction-diffusion operators. As the prototype model problem we use a family of Schrödinger Hamiltonians parametrized by random potentials.  Our computational model is posed in the truncated finite domain and this is an approximation of the standard Schrödinger Hamiltonian. Our chosen task is to compute localized bounded states at the lower end of the spectrum. Tensorial neural networks (TNN) are used as surrogate models which represent dependence of the ground state or landscape function on the localizing potential, depending on a problem we are solving.  Tensor neural networks contain large compress dense layers, reducing the number of parameters in a neural network through the use of truncated tensor factorizations. Further, we will also demonstrate the use of Variational Physics Informed Neural Network, together with the residual type error estimates, to obtain the ground state of the eigenvalue problem. Error estimators will be introduced to monitor the performance of the model. We present a host of numerical experiments to benchmark the accuracy and performance of the proposed algorithms.

Nomember 17 (1:30pm): Ari Stern, Washington University in St. Louis

Title: Hybridization and postprocessing in finite element exterior calculus

Abstract: Finite element exterior calculus (FEEC) unifies several families of conforming finite element methods for Laplace-type problems, including the scalar and vector Poisson equations. This talk presents a framework for hybridization of FEEC, which recovers known hybrid methods for the scalar Poisson equation and gives new hybrid methods for the vector Poisson equation using H(curl) elements. We also generalize Stenberg postprocessing, proving new superconvergence estimates. (Based on joint work with Gerard Awanou, Maurice Fabien, and Johnny Guzman.)

December 1 (1:30pm): RTG Group Meeting

• Student Research Updates

  • Gabriel Pinochet Soto

  • Barry Fadness

  • Tamara Gratcheva

• RTG Faculty Discussion and Planning