Portland State University
Applied and Computational Mathematics Seminar

Research Updates:

• Austen Nelson 

Title: Learning Rates, Momentum, and Randomized Kaczmarz

Abstract: In this talk, we consider learning rates and momentum in the context of the randomized Kaczmarz algorithm, which is an instance of stochastic gradient descent for a linear least squares loss function. First, we consider the problem of determining an optimal learning rate schedule for the Kaczmarz algorithm for noisy linear systems. Second, we consider how momentum affects how the randomized Kaczmarz algorithm converges in the direction of singular vectors of the matrix defining the linear loss function.

Title: Separable Shape Tensors: Emerging Methods in Pattern Recognition

Abstract: Scientists often leverage image segmentation to extract shape ensembles containing thousands of curves representing patterns in measurements. Shape ensembles facilitate inferences about important changes when comparing and contrasting images---e.g., examining material microstructures. We introduce novel pattern recognition formalisms combined with inference methods over ensembles containing thousands of segmented curves. This is accomplished by accurately approximating eigenspaces of composite integral operators to motivate discrete, dual representations collocated at quadrature nodes. Approximations are projected onto underlying matrix manifolds and the resulting separable shape tensors constitute rigid-invariant decompositions of curves into generalized (linear) scale variations and complementary (nonlinear) undulations. With thousands of curves segmented from pairs of images, we demonstrate how data-driven features of separable shape tensors inform explainable binary classification utilizing a product maximum mean discrepancy; absent labeled data, building interpretable feature spaces in seconds without high performance computation, and detecting discrepancies below cursory visual inspections.

Biography: Zach is a staff applied mathematician in the Applied and Computational Mathematics Division within the Information Technology Laboratory at NIST - Boulder, CO. His research involves model-based dimension reduction and manifold learning for facilitating novel explanations and interpretations related to artificial intelligence and machine learning. Specifically, he often works with data-driven applications involving imaging/signal processing, geometry, and optimization.

Research Updates:

• Nick Fischer

• Rebecca Bryant

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Title: Are my physics unstable???

Abstract: I've been exploring the fidelity of numerical approximations to continuous spectra of hyperbolic partial differential equation systems with variable coefficients. I'm particularly interested in the ability of discrete methods to accurately discover sources of physical instabilities. By focusing on the perturbed equations that arise in linearized problems, we apply high-order accurate summation-by-parts finite difference operators, with weak enforcement of boundary conditions through the simultaneous-approximation-term technique, which leads to a provably stable numerical discretization with formal order of accuracy given by 2,3,4 and 5. I derive analytic solutions using Laplace transform methods, which provide important ground truth for ensuring numerical convergence at the correct theoretical rate. I find that the continuous spectrum is better captured with mesh refinement, although dissipative strict stability (where the growth rate of the discrete problem is bounded above by the continuous) is not obtained. We also find that sole reliance on mesh refinement can be a problematic means for determining physical growth rates as some eigenvalues emerge (and persist with mesh refinement) based on spatial order of accuracy but are non-physical. This suggests that numerical methods be used to approximate discrete spectra when numerical stability is guaranteed and convergence of the discrete spectra is evident with both mesh refinement and increasing order of accuracy.

Title: A Stable Measure for Conditional Periodicity of Time Series using Persistent Homology

Abstract: Given a pair of time series, we study how the periodicity of one influences the periodicity of the other. Existing measures of similarity between two time series, e.g., cross-correlation, coherence, and cross-recurrence, have been shown to be experimentally robust to small changes in the inputs. But we have yet to find any measures with known theoretical stability results. 

We use persistence homology, a widely used method from computational topology, to define a conditional periodicity score that quantifies the periodicity of one univariate time series f1 given another f2, denoted score(f1 | f2), and derive theoretical stability results for the same. With the use of dimension reduction techniques in mind, we prove a new stability result for score(f1 | f2) when projections of the time series under principal component analysis (PCA) are used in the analysis. We also derive a lower bound on the minimum embedding dimension to use in our framework which guarantees that any two such embeddings give scores that are within \epsilon of each other. We experimentally demonstrate the increased accuracy and stability of our score on multiple pairs of signals.

This is joint work with Elizabeth Thompson. A preprint is available on arXiv (https://arxiv.org/abs/2501.02817).

Title: Reaction-diffusion equations in a heterogeneous environment.

Abstract:

We consider a model that describes the density of a population in a heterogenous, patchy environment. The density is governed by one reaction diffusion equation in one interval, and another reaction diffusion equation in another interval. At the interface where both intervals meet, we impose continuous density and flux conditions, and at the other endpoints of the two intervals, we impose Neumann boundary conditions. We shall discuss a well-posedness problem: Does the model exhibit a positive steady state solution, and if so, is it unique? We obtain affirmative answers to these questions, provided that the reaction terms have specific properties which are satisfied by some generalized logistic functional forms.

Based on joint work with Jane MacDonald and Swati Patel

CADES Group Meeting Agenda:

• Announcement of next RTG fellows

• Short research updates:

• Shivam Patel

• Ashlynn Crisp

• Emily Bogle

• Mason Spears

• Justin Valentine