Applied Mathematics seminars

Yousef Saad examines relaxed gradient descent for large-scale optimization. Relaxing the optimal step length in Cauchy's steepest descent avoids its characteristic zigzag behavior and can bring the search direction close to an eigenvector of the Hessian. Once that alignment is sufficiently accurate, properties of the Lanczos method can accelerate convergence. The talk analyzes several such strategies and illustrates them in global minimization of strictly convex functions, retaining the simplicity and low memory requirements that make gradient methods attractive for machine learning.

Daniel Szyld extends randomized point and block Gauss-Seidel and Gauss-Southwell convergence results from Hermitian positive-definite matrices to certain non-Hermitian classes, including overlapping variables in domain decomposition. The analysis treats a range of sampling probabilities and greedy selection strategies and identifies choices that optimize the bounds. The best expected convergence bounds for randomized methods match those of more expensive deterministic Gauss-Southwell algorithms. These results establish a convergence rate for asynchronous iterations. Joint work with Andreas Frommer.

Erik Boman discusses preconditioning for asynchronous linear solvers. Inner products create synchronization requirements in Krylov methods, while preconditioners can also improve iterations such as Richardson's method. The talk focuses on asynchronous incomplete factorizations and introduces ATS-ILU, an iterative incomplete LU method with synchronous and asynchronous versions that performs competitively with ParILU.

Edmond Chow examines how asynchronous updates improve parallel iterative computation. The first part covers asynchronous versions of classical first- and second-order linear iterations, Chebyshev methods, and multigrid, with attention to efficiency and fault tolerance. The second introduces reinforcement learning and asynchronous state-value estimation for finding optimal policies. When the state space is too large to enumerate, these updates focus computational effort on frequently visited regions.

Optimal transport is a mathematical method to define a distance between probability distributions. This is particularly useful in various domains, including physics, biology, machine learning, and economics, among others. After introducing the Optimal Transport (OT) problem at finite temperature, we show how it can be formulated as a statistical physics problem. This approach allows us to derive very efficient algorithms to effectively compute the distance between two probability distributions. The a priori unrelated Schrödinger bridge (SB) problem is presented, and it is shown to be a dynamical version of the optimal transport problem. Indeed, the Schrodinger bridge looks for the most probable path in probability distribution space, which connects two given probabilities. The Schrodinger bridge problem, originally devised for freely diffusing particles, can be generalized to the case of interacting particles. It can be formulated in terms of functional integrals over bosonic fields, which allows us to derive partial differential equations that characterize the most probable paths in probability space. CARL VAN VREESWIJK MEMORIAL LECTURE 2026. Presented in the van Vreeswijk Theoretical Neuroscience Seminar series (formerly WWTNS) on 2026-04-15. Recording duration: 00:55:05.

Davide Palitta introduces the S^T S-SVD, a decomposition of A derived from the SVD of its sketch SA. It is exact with high probability, preserves singular values probabilistically, and makes left singular vectors orthonormal in the sketch-induced seminorm, with lower computational cost. The talk relates this perspective to subspace embeddings and least-squares residuals, assesses sketch quality, and bounds departures from ordinary orthogonality in randomized QR. A further application extends the nearest-orthogonal-matrix problem to S^T S-orthogonality. The work builds on Gilbert, Park, and Wakin and is joint with Valeria Simoncini.

Joel Tropp studies structured dimension reduction through the injectivity of random maps, motivated by fast low-rank approximation and least-squares regression. This viewpoint sharpens guarantees for sparse maps and gives exponential improvements for tensor-product dimension reduction. Experiments assess the resulting structured random matrices on synthetic problems and scientific applications. Joint work with Chris Camaño, Ethan Epperly, and Raphael Meyer, available as arXiv:2508.21189.

Sherry Xiaoye Li surveys randomized construction of hierarchically low-rank matrices, including H/H2, HODLR, HSS, and butterfly formats with different off-diagonal structures. Applications include integral equations, boundary elements, discretized PDEs, and statistical or machine-learning kernel matrices, using either iterative matrix-vector products or direct factorization and solves. Constructing these representations from an implicit dense operator is often the main cost. The talk offers a unified view of sketch distributions, sketch sizes, approximation error bounds, high-performance implementation, applications, and open questions.

Matrix multiplication underpins scientific computing and artificial intelligence, yet practical numerical libraries and hardware accelerators commonly retain the classical cubic-time algorithm despite decades of subcubic theoretical advances. This talk reviews the effort to make faster multiplication algorithms useful in practice. It examines why arithmetic complexity alone does not determine performance: some algorithms require enormous matrices or incur large hidden constants, while communication costs, numerical stability and the match between software and hardware create additional obstacles. The historical perspective connects asymptotic algorithm design to actual performance and power consumption.

Pharmacokinetics in the eye is an important factor for the success of ocular drug delivery and treatment. Pharmacokinetic features determine the feasible routes of drug administration, dosing levels and intervals, and it has impact on eventual drug responses. Several physical, biochemical, and flow-related barriers limit drug exposure of anterior and posterior ocular target tissues during treatment during local (topical, subconjunctival, intravitreal) and systemic administration (intravenous, per oral). Mathematical models integrate joint impact of various barriers on ocular pharmacokinetics (PKs) thereby helping drug development. The models are useful in describing (top-down) and predicting (bottom-up) pharmacokinetics of ocular drugs. This is useful also in the design and development of new drug molecules and drug delivery systems. Furthermore, the models can be used for interspecies translation and probing of disease effects on pharmacokinetics. In this lecture, ocular pharmacokinetics and current modelling methods (noncompartmental analyses, compartmental, physiologically based, and finite element models) are introduced. Future challenges are also highlighted (e.g. intra-tissue distribution, prediction of drug responses, active transport).

Mikaela Iacobelli introduces kinetic theory and the mathematical description of charged particles in plasmas through Vlasov-type equations. The colloquium examines stability and instability, well-posedness, and how solutions behave in singular limits. It also presents a new family of Wasserstein-type distances that offers tools for studying the stability of kinetic equations. The exposition is intended for a broad mathematical audience.

Nearest-neighbor indexes usually rely on a single distance function, but accurate comparisons can be expensive. This talk proposes a bi-metric framework: a cheap proxy metric builds the index, while the query procedure uses a limited number of evaluations of both the proxy and an expensive ground-truth metric. The theory applies to DiskANN and Cover Tree. When the proxy approximates the ground-truth metric within a bounded factor, the resulting structure can achieve arbitrarily good approximation guarantees under the accurate metric. Experiments on text retrieval using models with very different computational costs show improved accuracy-efficiency tradeoffs on almost all MTEB datasets compared with alternatives such as reranking. Joint work with Haike Xu and Sandeep Silwal.

Changes in ocular hemodynamics may be indicative of pathological conditions in the eye (e.g. glaucoma, age-related macular degeneration), but also elsewhere in the body (e.g. systemic hypertension, diabetes, neurodegenerative disorders). Thanks to its transparent fluids and structures that allow the light to go through, the eye offers a unique window on the circulation from large to small vessels, and from arteries to veins. Deciphering the causes that lead to changes in ocular hemodynamics in a specific individual could help prevent vision loss as well as aid in the diagnosis and management of diseases beyond the eye. In this talk, we will discuss how mathematical and computational modelling can help in this regard. We will focus on two main factors, namely blood pressure (BP), which drives the blood flow through the vessels, and intraocular pressure (IOP), which compresses the vessels and may impede the flow. Mechanism-driven models translates fundamental principles of physics and physiology into computable equations that allow for identification of cause-to-effect relationships among interplaying factors (e.g. BP, IOP, blood flow). While invaluable for causality, mechanism-driven models are often based on simplifying assumptions to make them tractable for analysis and simulation; however, this often brings into question their relevance beyond theoretical explorations. Data-driven models offer a natural remedy to address these short-comings. Data-driven methods may be supervised (based on labelled training data) or unsupervised (clustering and other data analytics) and they include models based on statistics, machine learning, deep learning and neural networks. Data-driven models naturally thrive on large datasets, making them scalable to a plethora of applications. While invaluable for scalability, data-driven models are often perceived as black- boxes, as their outcomes are difficult to explain in terms of fundamental principles of physics and physiology and this limits the delivery of actionable insights. The combination of mechanism-driven and data-driven models allows us to harness the advantages of both, as mechanism-driven models excel at interpretability but suffer from a lack of scalability, while data-driven models are excellent at scale but suffer in terms of generalizability and insights for hypothesis generation. This combined, integrative approach represents the pillar of the interdisciplinary approach to data science that will be discussed in this talk, with application to ocular hemodynamics and specific examples in glaucoma research.

Nassim Nicholas Taleb examines three technical problems in applied probability. First, he considers how errors in probabilities derived from fat-tailed survival functions can translate into disproportionately large, potentially unbounded errors in thresholds. He relates this to uncertainty in disease growth rates and forecasting. Second, he discusses what he regards as fundamental probability errors in psychology papers. Third, he examines how correlation and relative distances can mislead in the geometry of information, proposing heuristics for applying entropy-based methods to genetic distances. The lecture connects these examples through the consequences of using probabilistic quantities outside the conditions under which they are informative.

Myopia is predicted to affect 50% of all people worldwide by 2050, and is a risk factor for significant, potentially blinding ocular pathologies, such as retinal detachment and glaucoma. Thus, there is significant motivation to better understand the process of myopigenesis and to develop effective anti-myopigenic treatments. In nearly all cases of human myopia, scleral remodeling is an obligate step in the axial elongation that characterizes the condition. Here I will describe the development of a biomechanical assay based on transient unconfined compression of scleral samples. By treating the scleral as a poroelastic material, one can determine scleral biomechanical properties from extremely small samples, such as obtained from the mouse eye. These properties provide proxy measures of scleral remodeling, and have allowed us to identify all-trans retinoic acid (atRA) as a myopigenic stimulus in mice. I will also describe nascent collaborative work on modeling the transport of atRA in the eye.

In this presentation, we will discuss adaptations of historical examples of mathematical research to bring out some of the intuitive judgments that accompany the working practice of mathematicians when reasoning by analogy. The main epistemological claim that we will aim to illustrate is that a central part of mathematical training consists in developing a quasi-perceptual capacity to distinguish superficial from deep analogies. We think of this capacity as an instance of Hadamard’s (1954) discriminating faculty of the mathematical mind, whereby one is led to distinguish between mere “hookings” (77) and “relay-results” (80): on the one hand, suggestions or ‘hints’, useful to raise questions but not to back up conjectures; on the other, more significant discoveries, which can be used as an evidentiary source in further mathematical inquiry. In the second part of the presentation, we will present some recent applications of this epistemological framework to mathematics education projects for middle and high schools in Italy.

The affinity between statistical physics and machine learning has a long history. I will describe the main lines of this long-lasting friendship in the context of current theoretical challenges and open questions about deep learning. Theoretical physics often proceeds in terms of solvable synthetic models, I will describe the related line of work on solvable models of simple feed-forward neural networks. I will highlight a path forward to capture the subtle interplay between the structure of the data, the architecture of the network, and the optimization algorithms commonly used for learning.

Neural activity in many sensory systems is organized on low-dimensional manifolds by means of convex receptive fields. Neural codes in these areas are constrained by this organization, as not every neural code is compatible with convex receptive fields. The same codes are also constrained by the structure of the underlying neural network. In my talk I will attempt to provide answers to the following natural questions: (i) How do recurrent circuits generate codes that are compatible with the convexity of receptive fields? (ii) How can we utilize the constraints imposed by the convex receptive field to understand the underlying stimulus space. To answer question (i), we describe the combinatorics of the steady states and fixed points of recurrent networks that satisfy the Dale’s law. It turns out the combinatorics of the fixed points are completely determined by two distinct conditions: (a) the connectivity graph of the network and (b) a spectral condition on the synaptic matrix. We give a characterization of exactly which features of connectivity determine the combinatorics of the fixed points. We also find that a generic recurrent network that satisfies Dale's law outputs convex combinatorial codes. To address question (ii), I will describe methods based on ideas from topology and geometry that take advantage of the convex receptive field properties to infer the dimension of (non-linear) neural representations. I will illustrate the first method by inferring basic features of the neural representations in the mouse olfactory bulb.

In this talk, we propose to decipher the activity of neural networks via a “multiply and conquer” approach. This approach considers limit networks made of infinitely many replicas with the same basic neural structure. The key point is that these so-called replica-mean-field networks are in fact simplified, tractable versions of neural networks that retain important features of the finite network structure of interest. The finite size of neuronal populations and synaptic interactions is a core determinant of neural dynamics, being responsible for non-zero correlation in the spiking activity and for finite transition rates between metastable neural states. Theoretically, we develop our replica framework by expanding on ideas from the theory of communication networks rather than from statistical physics to establish Poissonian mean-field limits for spiking networks. Computationally, we leverage our original replica approach to characterize the stationary spiking activity of various network models via reduction to tractable functional equations. We conclude by discussing perspectives about how to use our replica framework to probe nontrivial regimes of spiking correlations and transition rates between metastable neural states.

A major challenge for causal inference from time-series data is the trade-off between computational feasibility and accuracy. Motivated by process motifs for lagged covariance in an autoregressive model with slow mean-reversion, we propose to infer networks of causal relations via pairwise edge measure (PEMs) that one can easily compute from lagged correlation matrices. Motivated by contributions of process motifs to covariance and lagged variance, we formulate two PEMs that correct for confounding factors and for reverse causation. To demonstrate the performance of our PEMs, we consider network interference from simulations of linear stochastic processes, and we show that our proposed PEMs can infer networks accurately and efficiently. Specifically, for slightly autocorrelated time-series data, our approach achieves accuracies higher than or similar to Granger causality, transfer entropy, and convergent crossmapping -- but with much shorter computation time than possible with any of these methods. Our fast and accurate PEMs are easy-to-implement methods for network inference with a clear theoretical underpinning. They provide promising alternatives to current paradigms for the inference of linear models from time-series data, including Granger causality, vector-autoregression, and sparse inverse covariance estimation.