Portland State University
Applied and Computational Mathematics Seminar

Student presentations:

• Emily Bogle

• Gabriel Pinochet-Soto

• Austen Nelson

Faculty discussion

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.

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.

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.

Research Updates:

• Ester Toobian

• Quincy Robinson

• Anthony Kolshorn

• Jay Gopalakrishnan

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

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.

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.

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.

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.

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.

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.