Applied mathematics

Permanent URI for this collectionhttps://digital.lib.washington.edu/handle/1773/4891

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Data-Driven Methods for Reduced Order Models, System Identification, and Feedback Control
(2026-02-05) Griss Salas, Isaac Wenceslao; Kutz, J. Nathan
Data-driven modeling is an essential tool for understanding and controlling complex physical systems, particularly when first-principles models are incomplete or unavailable. We develop three distinct methodologies for learning dynamical systems from limited, noisy, and real-world data. We introduce a reduced-order modeling framework to capture the initial ejection dynamics of turbulent plumes directly from video. By combining Otsu's thresholding with a concentric-circle search to extract plume geometries, we construct interpretable low-dimensional representations of formation dynamics. Validated on unseen video data, this approach accurately recovers rapid transients that are unaddressed by classical Gaussian models and intractable for real-time computational fluid dynamics. We present an all-at-once methodology for learning systems of ordinary differential equations (ODEs) from scarce, partial, and noisy observations. This formulation utilizes a sparse recovery strategy over a function library while jointly employing reproducing kernel Hilbert space (RKHS) theory for state estimation and discretization. The approach demonstrates high sampling efficiency and robustness to noise, outperforming existing algorithms in equation discovery. We develop a stability-constrained Neural Ordinary Differential Equation (NODE) framework for learning asymptotically stable systems exhibiting multi-attractor and hysteretic dynamics. The learned model guarantees trajectory stability throughout the state space, enabling tractable feedback control policies capable of navigating nontrivial bifurcations and hysteresis loops. Collectively, these contributions advance data-efficient modeling by integrating reduced-order methods, sparse equation discovery, and stability-enforced neural networks for real-time inference and control.
From Oncogenesis to Immunotherapy: Mathematical Modeling of Heterogeneous Cancers
(2026-02-05) Sholokhova, Alanna Pauline; Bozic, Ivana
We utilize mathematical modeling to study two types of cancer with substantial mutational heterogeneity: chronic lymphocytic leukemia (CLL) and mismatch-repair deficient colorectal cancer (MMR-D CRC). First, to study the progression of CLL into an aggressive lymphoma, Richter’s Syndrome (RS), we analyze data from a recent mouse model and utilize a Bayesian modeling approach to show that growth patterns present in human disease are recapitulated in murine CLL/RS. Next, we use a stochastic branching process model to simulate the acquisition of tumor-specific neoantigens in MMR-D CRC. By using these in-silico tumors as initial conditions in a dynamical systems model of tumor-immune interactions, parameterized using clinical trial data, we characterize features associated with a durable response to immune checkpoint inhibitor (ICI) immunotherapy in MMR-D CRC.
Computations and Analysis with Non-normal Matrices
(2025-10-02) Wellen, Natalie; Greenbaum, Anne
The goal of this dissertation is to introduce some new iterative methods for solving problems involving non-Hermitian matrices and to add to our understanding of the convergence of these algorithms when applied to highly non-normal matrices. In the first part, the Arnoldi-OR algorithm [23] is introduced. Given an n by n matrix A and a rational function N (z)/D(z), where D(A) is nonsingular, this algorithm finds the approximations x_k, k = 1, 2, . . ., from successive Krylov subspaces, span{b, Ab, . . . , A^(k−1)b}, that minimize the 2-norm of the residual, ∥D(A)x_k −N(A)b∥_2 . Convergence of the Arnoldi-OR algorithm can be bounded based on the eigenvalues of A and the condition number of the best-conditioned matrix of eigenvectors, assuming that A is diagonalizable. This may be a large overestimate, however, if the best-conditioned matrix of eigenvectors is still very ill-conditioned. Starting in Chapter 4, the second part of this dissertation explores bounds on the norm of a function of A that can be applied when A is highly non-normal; i.e., either A is not diagonalizable or it is diagonalizable but the eigenvalues are very ill-conditioned. These bounds involve the ∞-norm of the function, not just on the eigenvalues of A, but on a larger set in the complex plane containing the eigenvalues. In the second part of the dissertation, we expand on some known results from Crouzeix and Palencia (2017) and Crouzeix and Greenbaum(2019) about K-spectral sets—sets Ω ⊂ C satisfying ∥f(A)∥ 2 ≤ K sup z∈Ω |f (z)| for all functions f analytic in Ω, (i.e. functions that can be arbitrarily well-approximated on Ω by rational functions with no poles in Ω). This work is described in Greenbaum and Wellen (2025). Here, we use it to give alternative bounds on the 2-norm of the residual in the Arnoldi-OR algorithm. We also consider a different way of solving non-Hermitian linear systems, which is to convert the problem to a Hermitian one, in Section 2.2. If one can find Hermitian positive definite matrices M and Y such that A = M −1 Y, then one can solve Ax = b using the conjugate gradient method (CG) applied to the system Yx = Mb with M as a preconditioner. We apply this technique to a problem involving the graph Laplacian that is of interest to Sandia National Laboratory, where I worked with Richard Lehoucq over the summer of 2023.
Spectral-in-time methods for time-evolution partial differential equations
(2025-08-01) Zhang, Yiting; Trogdon, Tom T.T.
In this thesis, we develop and analyze spectral-in-time methods to solve time-evolution partial differential equations (PDEs). To improve the computational complexity of solving the resulting linear system, we develop an algorithm using QR factorization in conjunction with the Woodbury matrix identity \cite{max1950inverting}. Taking advantage of the intrinsic structure of the spectral discretizations of PDE problems, this algorithm can reduce the computational complexity compared to conventional direct solvers. However, as the dimension of the problem increases, solving the problem by applying the spectral-in-time method globally in time has a long execution time and requires more storage availability. To deal with these challenges, we propose an approach that embeds the spectral-in-time method within a time-stepping framework, which is similar to the classical time-stepping methods. This approach preserves high precision while considerably improving the computational efficiency by decomposing the entire temporal domain into manageable subintervals. Moreover, we extend our method beyond linear PDEs by using Newton's method within the spectral-in-time framework for nonlinear PDEs. Instead of computing and storing the exact Jacobian matrices, we develop a final-time Jacobian matrix approach that generates much sparser Jacobian matrices. Through comprehensive numerical experiments, we show that this novel approach produces results indistinguishable in accuracy from the use of the exact Jacobian-based method with a slight reduction in the convergence of Newton's method.
Open-Source Dynamical Systems Research, with a Side of (Francis) Bacon
(2025-08-01) Stevens-Haas, Jacob; Aravkin, Aleksandr; Kutz, Nathan
Sparse Identification of Nonlinear dynamics (SINDy) is a family of methods for explicitly identifying differential equations from data. The open-source Python package pysindyprovides the engineering to support ongoing SINDy research. I discuss original and community innovations in smoothing and sparse optimization, as well as a colocation approach to simultaneous estimation of states and sparse coefficients. The compatibility of these methods through the pysindy API has lessons for mathematics as an experimental field. My contribution to the state of the art includes both original innovations and ongoing support for research contributions from the community. These innovations include smoothing methods such as kernel and Kalman, and sparse regression approaches involving Monte Carlo estimation, physics constraints, or mixed-integer optimization. The pysindy changes have also allowed a principled approach to simultaneous optimization of states and coefficients. Across these projects and more, the requirement for a consistent API has given rise to a common experimental language. This defense codifies that language in additional pacakges and suggests useful lessons for the open-source, numerical lab.
Rational Approximation and Coefficient Recovery
(2025-08-01) Dou, ShiLin; Wilber, Heather
We present a new framework for recovering Chebyshev coefficients of non-periodic functions using rational approximation. Building on the AAA algorithm, we construct a type (m−1, m) rational approximant whose poles and residues can be efficiently computed. Using the change of variable x = cos θ, the approximant is transformed into a rational trigonometric function in θ, whose Fourier coefficients can be expressed as short exponential sums via Fourier inversion. Mapping back, these exponentials provide a reconstruction of the original function’s Chebyshev expansion.We demonstrate that our method (i) achieves high-accuracy Chebyshev coefficient recovery from both equally spaced and clustered data, (ii) outperforms classical polynomial-based approaches for functions exhibiting slow coefficient decay or singular behavior, and (iii) supports fundamental arithmetic operations and differentiation directly in the coefficient domain.
Modeling and Algorithms for Nonconvex TrajectoryGeneration Problems: From Constraint Reformulations and First-Order Proximal Methods to Structure-Exploiting Convex Solvers
(2025-08-01) Luo, Dayou; AÃ§ıkmeşe, BehÃ§et; Aleksandr, Aleksandr
Trajectory generation plays a central role in modern Guidance, Navigation, and Control (GNC) systems by converting high-level mission objectives into reference trajectories that comply with system dynamics and task constraints. Among various methods, optimization-based formulation is favored for its ability to integrate dynamics, constraints, and performance objectives within a unified framework. Although various optimization-based trajectory generation methods have been successfully deployed in practice, many suffer from fundamental limitations that undermine their reliability. In particular, existing approaches often lack rigorous theoretical guarantees for convergence and constraint satisfaction, especially when implemented in discretized form. Furthermore, real-time deployment is frequently hindered by high computational costs and a heavy reliance on manual parameter tuning. To address these challenges, this dissertation presents three core contributions that improve the theoretical reliability and computational efficiency for trajectory generation. The first part of the dissertation focuses on a structured form of nonconvexity arising from thrust lower bounds, a setting where a method known as Lossless Convexification (LCvx) has been widely adopted in applications due to its empirical effectiveness. LCvx addresses this challenge by relaxing the original nonconvex problem as a convex optimization problem whose solution, under certain conditions, is also optimal for the nonconvex formulation. However, existing LCvx theory provides guarantees only for continuous-time optimal control problems, where trajectories and controls are modeled as functions over a continuous time domain. In contrast, practical implementations rely on discrete-time formulations, where the optimization variables are finite-dimensional vectors defined over temporal grids. This gap raises concerns about the theoretical soundness of applying LCvx in practical applications. In this dissertation, we extend LCvx theory to the discrete-time setting and establish formal guarantees that support its use in realistic implementations. In particular, we show that the solution to the convex problem resulting from LCvx satisfies the original nonconvex constraints up to a number of violations bounded by a linear function of the state dimension~$n_x$, where the exact form of the bound may vary across different problems. The second part addresses trajectory generation under general nonconvex constraints. We first introduce a unified modeling and algorithmic framework that integrates prox-linear methods with exact penalty formulations. Moreover, due to discretization, classical trajectory generation algorithms typically guarantee constraint satisfaction only at grid points, and violations may inevitably occur between grid points. To address this issue, we propose a novel approach that incorporates an integral reformulation of the constraints into the optimization procedure, thereby ensuring constraint satisfaction over the entire time horizon. Lastly, to reduce the manual effort commonly required for parameter tuning, we design a proportional-integral (PI)-inspired autotuning scheme within this framework, which introduces a vectorized exact penalty comprising both linear and quadratic terms. After each prox-linear subproblem is solved, the penalty weights are adaptively updated: the linear term accumulates constraint violations across iterations, analogous to the integral part of PI, while the quadratic term responds directly to the current violation, corresponding to the proportional part. Theoretical guarantees and convergence analysis are provided for all methods introduced in this part. Finally, we propose Newton-PIPG, an efficient method for solving quadratic programming (QP) problems arising in optimal control, subject to additional set constraints. Such problems can serve as subproblems from general nonconvex trajectory generation algorithms. Newton-PIPG integrates the Proportional-Integral Projected Gradient (PIPG) method with the Newton method, thereby achieving both global convergence and local quadratic convergence. The PIPG method, an operator-splitting algorithm, seeks a fixed point of the PIPG operator. Under mild assumptions, we demonstrate that this operator is locally smooth, which enables the application of the Newton method to solve the corresponding nonlinear fixed-point equation. Furthermore, we prove that the linear system associated with the Newton method is locally nonsingular under strict complementarity conditions. To enhance computational efficiency, we developed a specialized matrix factorization technique that exploits the typical sparsity structure of optimal control problems and makes use of block Cholesky decomposition. Numerical experiments demonstrate that Newton-PIPG achieves high accuracy and reduces computation time, particularly in settings where feasibility is easily guaranteed.
Weaving order from uncertainty: Design, Analysis, and Applications of Transport-based Generative Models
(2025-01-23) Pandey, Biraj; Hosseini, Bamdad
Generative machine learning algorithms are pivotal for advancing artificial intelligence and for gaining insights into biological neural systems. In this thesis, we present a comprehensive study that integrates theoretical analysis, efficient algorithm development, and biological applications in generative modeling. We first establish a general theoretical framework for minimum divergence transport estimators, a prominent class of generative models. We derive a priori error bounds that quantify the generalization performance of these models in terms of model and sample complexity. Building upon this theory, we introduce a flow-based transport algorithm for generative modeling that utilizes kernel methods to minimize Maximum Mean Discrepancy (MMD). Our method offers an efficient alternative to existing neural network approaches, achieving comparable performance with fewer parameters and reduced training time. We apply the theoretical results from the first part to derive generalization bounds for this algorithm. Finally, we explore the intersection of generative modeling and neuroscience by developing a generative model for receptive fields in sensory neuronal systems using Gaussian processes. This model elucidates how sensory neurons transform inputs to create robust representations. Our biological insights inspire an initialization strategy that improves the efficiency of neural network training.
From Mechanism to Practice: Evolutionary Forecasting for SARS-CoV-2
(2025-01-23) Figgins, Marlin; Bozic, Ivana; Bedford, Trevor
Novel genetic variants often arise due to mutations in circulating viral populations. These mutations can sometimes provide fitness advantages to members of the population allowing them to out-compete other variants through mechanisms such as partial immune escape and increased transmissibility. This interplay of mutation, transmission, and selection leads to evolution in the population. Therefore, understanding the genetic composition of viral populations and its relation to virus phenotype can be useful for understanding the current and future epidemic potential of viral variants.This dissertation develops several theoretical ideas, statistical methods, and software tools that enable evolutionary forecasting for SARS-CoV-2 and other rapidly evolving pathogens using concepts from population genetics, mathematical epidemiology, and statistics. We begin by developing a Bayesian method for estimating the effective reproduction number of genetic variants using counts of variant sequences and measures of incidence such as case counts. To evaluate this method among others, we develop a workflow to compare frequency-based forecast models in a live forecasting environment, quantifying the short-term accuracy of such methods and suggesting a minimal sequencing capacity to ensure high quality forecasts. Next, we develop a larger theory for how mechanistic models of transmission constrain how variant frequencies change over time. This leads to theoretical results for the trade-off between immune escape and increased transmissibility and suggests new methods for modeling variant fitness using approximate Gaussian processes as well as latent pseudo-immune factors. We then apply these ideas to incorporate molecular data on immune escape and phylogenetic structure into relative fitness estimates to enable out-of-sample prediction of relative fitness from sequence-level predictors. Our focus then shifts to the operational problem of evolutionary forecasting, where we develop open-source software and visualization tools that can be used to implement, automate, and interpret evolutionary forecasts.
Explicit solutions to linear, second-order, initial and boundary value problems with variable coefficients
(2024-10-16) Farkas, Matthew; Deconinck, Bernard
I derive explicit solution representations for linear, second-order Initial-Boundary Value Problems (IBVPs) with coefficients that are spatially varying, with linear, constant-coefficient, two-point boundary conditions. I accomplish this by considering the variable-coefficient problem as the limit of a constant-coefficient interface problem, previously solved using the Unified Transform Method of Fokas. Our method produces an explicit representation of the solution, allowing us to determine properties of the solution directly. I prove that these representations are solutions to fully and partially dissipative problems under general conditions. As explicit examples, I demonstrate the solution procedure for different IBVPs of variations of the heat equation, and the linearized complex Ginzburg-Landau (CGL) equation (with periodic boundary conditions). The solution can be used to find the eigenvalues of second-order linear operators (including non-self-adjoint ones) as roots of a transcendental function, and their eigenfunctions may be written explicitly in terms of the eigenvalues.
Asymptotic and non-asymptotic model reduction for kinetic descriptions of plasma
(2024-10-16) Coughlin, John; Shumlak, Uri; Hu, Jingwei
Plasma dynamics are coupled across microscopic and macroscopic scales by a variety of nonlinear mechanisms. These include repartition of energy due to kinetic microinstabilities, suppression of fluid instabilities by kinetic stabilization effects, and other mechanisms. At the macroscopic scale plasmas are well-described by fluid equations, whose formal validity depends on the long-time regularization of the phase space distribution function by collisions and magnetic gyrotropization. However, accurately capturing multiscale coupling requires multiscale reduced models which are both efficient and accurate in transition regimes. These regimes, where either collisional or magnetic gyrotropic regularization are marginal, are characterized by the ratio of the (collisional or magnetic) mean free path to a characteristic gradient scale length. This work studies two families of reduced plasma models for transition regimes in depth. The first is an asymptotic expansion for Braginskii-type transport coefficients in the so-called drift ordering for low-beta plasmas. The expansion captures leading-order finite Larmor radius effects for arbitrary collisionality. We present a new derivation of this expansion, evaluate its performance numerically, and provide a numerically feasible approximation. The second family of methods is dynamical low-rank (DLR) methods, which are not based on an asymptotic expansion and have the potential to overcome the curse of dimensionality for kinetic equations. We present two novel DLR schemes for plasma kinetic equations with a focus on fluid-kinetic coupling. One is a DLR method that retains low rank in the highly collisional asymptotic limit. The other is a fully locally conservative DLR method for the Vlasov-Dougherty-Fokker-Planck equation which achieves second-order accuracy in time. All discretizations are described in detail and accompanied by numerical results demonstrating the merit of the proposed approach.
Data-Driven Methods for Sparse Sensor Problems in Spatiotemporal Systems
(2024-09-09) Mei, Jiazhong; Kutz, J. Nathan; Brunton, Steven L.
Spatiotemporal systems across various fields are often complex and high-dimensional. This thesis addresses several key problems related to sparse sensing, focusing on the exploitation of low-dimensional structures within high-dimensional data to optimize limited sensor usage for system understanding. We first examine the power grid system and optimize PMU sensor placements such that the dynamical modes and their properties inferred from measurements can be used to characterize faults at different locations. Then, we shift our attention to mobile sensor applications and their associated challenges. Leveraging system observability as a critical metric for path planning, we employ a Kalman filter estimator to optimize sensor trajectories. We introduce a greedy path planning method that enhances the conditioning of system observability along the path, assuming no constraints on sensor movement. We further extend the exploration of mobile sensor path planning by considering the additional complexity of sensor control and movement introduced by background flows. We design an end-to-end model using deep reinforcement learning to simultaneously optimize path decisions and sensor control for mobile sensors. Lastly, we delve into nonlinear reconstruction for mobile sensors using decoder networks and address the challenges of long time dependency and sensitivity to measurement noise. Our proposed robust state space decoder model, with intricately designed parameter initialization, demonstrates improved performance compared to existing estimation models.
Deep learning frameworks for modeling how neural circuits learn
(2024-09-09) Liu, Yuhan Helena; Shea-Brown, Eric
The brain's prowess in learning and adapting remains an enigma, particularly in its approach to the 'temporal credit assignment' problem. How do neural circuits determine which specific states and connections contribute to future outcomes, and subsequently adjust these for enhanced learning? My thesis addresses this by combining insights from the latest large-scale neuroscience data and recent deep learning theoretical tools. The first two projects introduce novel learning rules inspired by the Allen Institute's transcriptomics data, which revealed widespread and intricate cell-type-specific interactions among neuromodulatory molecules. This rule enables neurons to propagate credit information efficiently, enhancing learning performance beyond that of biologically plausible predecessors. Extensive computational experiments confirm the significant role of local neuromodulatory signals in learning, offering new perspectives on neural information processing. My third project assesses the generalization capabilities of bio-plausible learning rules through the lens of deep learning theory, particularly focusing on the curvature of the loss landscape via the loss’ Hessian eigenspectrum. Our findings reveal that these rules often settle in high-curvature regions of the loss landscape, indicating suboptimal generalization. This analysis led to a mathematical theorem linking synaptic weight update dynamics to landscape curvature, proposing neuromodulator-driven adjustments as a potential enhancement for learning rule performance. Given how initial conditions can greatly influence a system’s future trajectory, the fourth project delves into the impact of initial connectivity structures on learning dynamics in neural circuits. By examining various connectivity patterns derived from neuroscience data, including recent electron microscopy data, we analyze how these structures influence learning regimes, implicating metabolic costs and risks of catastrophic forgetting. Our findings suggest that high-rank initializations utilize pre-existing high-dimensional input expansion to facilitate input decoding, leading to minimal changes post-training and increasing the propensity for lazy learning. These specific initializations thus predispose networks toward certain learning behaviors, critically affecting their ability to adapt and generalize.
Interest Rate Problems: Implied Volatility of Options on Bonds and Forward Rates, and Optimal Times to Buy and Sell a Home
(2024-09-09) Suaysom, Natchanon; Lorig, Matthew
In this Thesis, we examine problems in financial mathematics whose characteristics are significantly influenced by the dynamics of the interest rate.In the first part of the Thesis, we derive an explicit asymptotic approximation for the implied volatilities of Call options written on bonds and forward rates assuming the instantaneous short-rates of interest are described by Affine Term-Structure (ATS) models and Quadratic Term-Structure (QTS) models, respectively. For specific short-rate models, we perform numerical experiments in order to gauge the accuracy of our approximation. In the second part of the Thesis, we derive the optimal stopping times to buy and sell a home. We begin by assuming that home prices are set by a representative home-buyer, who can afford to pay only a fixed cash-flow per unit time for housing. The cash-flow is a fraction of their salary, which grows at a rate that is proportional to the risk-free rate of interest. The mortgage rate paid by the home-buyer is fixed at the time of purchase and equal to the risk-free rate of interest plus a positive constant. In this setting, we consider an investor who wishes to buy and then sell a home in order to maximize his discounted expected profit. This leads to a nested optimal stopping problem, which can be solved using nonnegative concave majorant approach. Additionally, we provide a detailed analytic and numerical study of the case in which the risk-free rate of interest is modeled by a Cox-Ingersoll-Ross (CIR) process.
Thermodynamic behavior of living systems: a biophysical approach to stochastic single-cell gene expression dynamics
(2024-04-26) Angelini, Erin; Qian, Hong
Representing the state of a cell through its gene expression profile, we consider observable phenotypes as (metastable) attractor states of the underlying biochemical gene expression kinetics. The traditional deterministic dynamics, however, does not capture the possibility of spontaneous transitions between attractors (i.e., phenotype switching). In this work, we frame this picture in terms of stochastic dynamical systems and models: a quasi-potential ``landscape" for an arbitrary dynamical system emerges as a large deviations rate function for the density of a singularly perturbed stochastic differential equation (SDE). This quasi-potential, which exists even for nonequilibrium biochemical dynamics in living cells, admits the most probable path between any two attractors and the characteristic time scale on which these transitions occur. We discuss the implications of this framework for the population dynamics at the level of cell cultures or tissues, specifically its applications to cancer population dynamics. We also consider two different frameworks for analyzing data from single-cell experiments through the lens of phenotypic attractors and state transitions. The first draws on the mathematics of thermodynamics, namely convex analysis and the Legendre-Fenchel transform, to derive an energy-like representation of an ergodic system from statistical measurements of the system. The latter is a maximum likelihood framework for inferring cell proliferation and phenotype switching rates from single-cell data, which we extend from a previous work to novel single-cell experiments using DNA-barcode lineage tracing. In addition to this work, we outline possible directions suitable for future research projects.
Optimization of Infectious-Disease Mitigation Strategies with Economic or Equity Perspectives
(2024-02-12) Stafford, Erin; Kot, Mark
Infectious-disease outbreaks in human, livestock, and plant populations continue to be a problem that can affect our day-to-day lives and have broader societal implications. There- fore, the need to prevent the spread of infectious disease is of great importance. Which disease-mitigation strategies are best depends on which factors are most important to decision makers. In this dissertation, I focus on the use of optimization with compartmental models to determine the best mitigation strategies for infectious-disease outbreaks. First, I describe the use of compartmental models to study infectious-disease dynamics. I also provide back- ground on optimal control theory and give examples of how optimal control theory and other optimization methods are used to give insight into the effectiveness of different disease mitigation strategies. Next, I use two optimal-control models to determine if contact-reducing disease-mitigation strategies can be economically advantageous when used to control the spread of Staphylococcus aureus in dairy cows. Both models use SIS models to describe the dynamics of S. aureus transmission. Moreover, both models consider revenue from healthy cows producing saleable milk, a cost from sick cows, and a loss of revenue when implementing mitigation strategies. The second model, however, also takes into consideration mild infections of S. aureus where infected cows may still produce saleable milk of lesser quality. Using these models, I found that using costly mitigation strategies to reduce contacts between infective and susceptible cows is economically beneficial. The dynamics of the second optimal-control model, where severity is considered, are more interesting as multiple candidate solutions satisfying the necessary conditions of Pontryagin’s maximum principle may coexist. The behaviors of these candidate solutions may be very different, but they may also produce similar economic outputs. I then study the effects of a very different type of disease-mitigation strategy, vaccination, on COVID-19 outcomes. In this chapter, I find which vaccination strategies minimize either overall disease burden, inequity in disease outcomes between racial groups, or a combination of measures. I find that, when vaccine is limited, there is a trade-off between minimizing disease burden and minimizing inequity. Allocation strategies that minimize combinations of measures can similarly improve both disease burden and inequity, but not to the same extent as when minimizing either measure alone. By increasing the vaccine supply, however, the trade-off greatly lessens.
A Mathematical Theory for Optimal Marital Interactions
(2024-02-12) Henson, Micah; Tung, Ka-Kit; Kot, Mark
The study of marriage dynamics and of strategies to reduce the likelihood of divorce has beenan important research area for decades. Gottman’s [15] research on successful marriages revealed three interaction styles: conflict-avoiding, validating, and volatile. There has not been progress in explaining how couples evolve into these styles of interactions and why failure to do so leads to failed marriages. The first chapter shows that the ubiquitous conflict-avoider style naturally arises through a couple maximizing a goal in their marriage when they do not consider an emotional cost. This leads to a mathematical optimal-control problem. In the second chapter, we present a differential-game-theory model where we explore what happens when spouses have different marriage goals. We also show that validating interaction-styles arise from the psychological cost of interacting with and/or ignoring one’s spouse. In the final chapter, we present strategies for marriage repair.
Conservative discontinuous Galerkin methods for the kinetic Fokker-Planck equation
(2024-02-12) Vaes, Wietse; Hu, Jingwei
We consider the kinetic Fokker-Planck equation, a simplified model of the Vlasov-Landau equation, that describes collisions in plasma. This diffusion-type equation exhibits numerous noteworthy properties. One such property is the conservation of mass, momentum and energy. The numerical methods in this thesis, namely the local and recovery discontinuous Galerkin methods for diffusion-type equations, maintain this over large and truncated domains. Employing these methods results in stability results that fall in line with theoretical expectations. However, the findings also include unexpected convergence and asymptotic behaviors, which prompt further investigation.
Reduced Order Model for Global Atmospheric Chemistry Data
(2023-09-27) Velegar, Meghana; Kutz, J. Nathan
Global atmospheric chemistry is an exceptionally high-dimensional problem as it involves hundreds of chemical species that are coupled with each other via a set of ordinary differential equations. Models of atmospheric chemistry that are used to simulate the spatio-temporal evolution of these chemical constituents need to keep track of each chemical species on a global scale (longitude, latitude, elevation) and at each point in time. This data can be exceptionally high-dimensional so as to be not computationally tractable. Thus computationally scalable methods are required for the analysis, reproduction and forecasting of atmospheric chemistry dynamics. First, we introduce a new set of algorithmic tools capable of producing scalable, low-rank decompositions of global spatio-temporal atmospheric chemistry data. By exploiting emerging {\em randomized linear algebra} algorithms, a suite of decompositions are proposed that extract the dominant features from {\em big data} sets with improved interpretability. Importantly, our proposed algorithms scale with the intrinsic rank of the global chemistry space rather than measurement space, thus allowing for efficient representation and compression of the data. Next, we introduce the optimized dynamic mode decomposition algorithm for constructing an adaptive and computationally efficient reduced order model of global atmospheric chemistry dynamics. Forecasting is also achieved with a low-rank linear model that uses a linear superposition of the dominant spatio-temporal features. Bagging OPtimized DMD or BOP-DMD produces an ensemble of DMD models, thereby quantifying uncertainty, reducing model variance and suppressing over-fitting by design. We compute the temporal uncertainty metrics for the optDMD forecasts using the BOP-DMD architecture. Lastly, we explore a data-driven scalable sparse sensor placement architecture for monitoring and reproduction of global atmospheric chemistry dynamics. By combining 1) machine learning, i.e. the POD dimensionality reduction technique, which learns and extracts a set of tailored library of features in the training data to produce low-dimensional representations of the full state, and 2) sparse sampling, i.e. designing highly specialized optimal sensors using the tailored features and QR pivoting, we reconstruct the full signal in the POD basis from a small subset of sensor or point measurements instantaneously. We also discover correlation between different chemical species, indicating that the chemical space can also be compressed.
Interpretation and Optimization of Recurrent Neural Network Performance Through Lyapunov Exponents Methodology
(2023-08-14) Vogt, Ryan Andrew; Shlizerman, Eli
Common deep learning models for learning multivariate time series data are Recurrent Neural Networks (RNN). These models are ubiquitous computing systems which have been studied for decades. The propagation of gradients over long time-sequences can make training RNNs particularly challenging and difficult to interpret. The hidden states of RNNs can be viewed as non-autonomous dynamical systems which can be analyzed using dynamical systems tools. In this work, we leverage Lyapunov Exponents, a dynamical systems tool which measures the rate at which nearby trajectories expand or contract over time to analyze the propagation of information in RNNs and relate these properties to RNN training and performance. We show that several statistics of the Lyapunov spectrum have moderate correlation with network loss on both classification and regression tasks, and emerge early in training. We also train an autoencoder to learn the relation between the full Lyapunov spectrum and an RNN's loss on given tasks. The latent representation of the autoencoder distinguishes between high- and low-accuracy networks across a variety of network hyperparameters, including initialization parameter, network size, and network architecture more effectively than direct statistics of the Lyapunov spectrum. From a theoretical perspective to further analyze Lyapunov Exponents of RNNs, we derive a direct expression for gradient in terms of the components of RNNs' Lyapunov Exponents which measure directions (vectors Q) and factors (scalars R) of expansion and contraction over a sequence. We find that the Q vectors associated with the greatest degree of expansion become increasingly aligned with the dominant directions of the gradient extracted by singular value decomposition. Furthermore, we show that the predictions generated by RNN are maximally affected by input perturbations at moments which the R values are maximal. These results showcase correlation between dynamical systems stability theory for RNNs, network performance, and loss gradients. This may open the way to design hyperparameter optimization algorithms and adaptive training methods that account for state-space dynamics as measured by Lyapunov Exponents to improve computations. It may also provide a unifying dynamical systems framework to study RNN performance across network architectures and tasks.

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