About

Matt Jacobs is an Assistant Professor in the Department of Mathematics at the University of California, Santa Barbara. He earned his Ph.D. in Mathematics from the University of Michigan in 2017. His research interests encompass a range of topics including calculus of variations, optimization, numerical methods, partial differential equations (PDEs), optimal transport, free boundary problems, and adversarial learning. His work integrates these areas to address complex mathematical problems and develop innovative computational techniques.

Research topics

• Computer Science
• Materials science
• Artificial Intelligence
• Mathematics
• Applied mathematics
• Chemical engineering
• Nanotechnology
• Discrete mathematics
• Mathematical physics
• Geology
• Mathematical analysis
• Metallurgy
• Geometry
• Pure mathematics
• Algorithm
• Library science
• Polymer chemistry
• Chemistry

Selected publications

• On the Singular Limit of Brinkman’s Law to Darcy’s Law

  Archive for Rational Mechanics and Analysis · 2026-05-13

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• Optimal transport of linear systems over equilibrium measures

  Automatica · 2025-02-26

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• Asymptotic Linear Convergence of ADMM for Isotropic TV Norm Compressed Sensing

  arXiv (Cornell University) · 2025-05-02

  preprintOpen access

  We prove an explicit local linear rate for ADMM solving the isotropic Total Variation (TV) norm compressed sensing problem in multiple dimensions, by analyzing the auxiliary variable in the equivalent Douglas-Rachford splitting on a dual problem. Numerical verification on large 3D problems and real MRI data will be shown. Though the proven rate is not sharp, it is close to the observed ones in numerical tests. The proven rate is not sharp, but it provides an explicit upper bound that appears close to the observed convergence rate in numerical experiments, although we do not claim this behavior holds in general.

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• Lagrangian Solutions to the Porous Media Equation and Reaction Diffusion Systems

  SIAM Journal on Mathematical Analysis · 2025-07-08

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• Regularity and Nondegeneracy for Tumor Growth with Nutrients

  Archive for Rational Mechanics and Analysis · 2025-01-11 · 1 citations

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• Guaranteeing Higher Order Convergence Rates for Accelerated Wasserstein Gradient Flow Schemes

  ArXiv.org · 2025-11-14

  preprintOpen accessSenior author

  In this paper, we study higher-order-accurate-in-time minimizing movements schemes for Wasserstein gradient flows. We introduce a novel accelerated second-order scheme, leveraging the differential structure of the Wasserstein space in both Eulerian and Lagrangian coordinates. For sufficiently smooth energy functionals, we show that our scheme provably achieves an optimal quadratic convergence rate. Under the weaker assumptions of Wasserstein differentiability and $λ$-displacement convexity (for any $λ\in \mathbb{R}$), we show that our scheme still achieves a first-order convergence rate and has strong numerical stability. In particular, we show that the energy is nearly monotone in general, while when the energy is $L$-smooth and $λ$-displacement convex (with $λ>0$), we prove the energy is non-increasing and the norm of the Wasserstein gradient is exponentially decreasing along the iterates. Taken together, our work provides the first fully rigorous proof of accelerated second-order convergence rates for smooth functionals and shows that the scheme performs no worse than the classical scheme JKO scheme for functionals that are $λ$-displacement convex and Wasserstein differentiable.

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• Nonlocal approximation of slow and fast diffusion

  Journal of Differential Equations · 2025-01-31 · 5 citations

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• On the existence of solutions to adversarial training in multiclass classification

  European Journal of Applied Mathematics · 2024-12-02 · 1 citations

  articleOpen access

  Abstract Adversarial training is a min-max optimization problem that is designed to construct robust classifiers against adversarial perturbations of data. We study three models of adversarial training in the multiclass agnostic-classifier setting. We prove the existence of Borel measurable robust classifiers in each model and provide a unified perspective of the adversarial training problem, expanding the connections with optimal transport initiated by the authors in their previous work [21]. In addition, we develop new connections between adversarial training in the multiclass setting and total variation regularization. As a corollary of our results, we provide an alternative proof of the existence of Borel measurable solutions to the agnostic adversarial training problem in the binary classification setting.

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• A Score-Based Deterministic Diffusion Algorithm with Smooth Scores for General Distributions

  Proceedings of the AAAI Conference on Artificial Intelligence · 2024-03-24 · 1 citations

  articleOpen access

  Score matching based diffusion has shown to achieve the state of art results in generation modeling. In the original score matching based diffusion algorithm, the forward equation is a differential equation for which the probability density equation evolves according to a linear partial differential equation, the Fokker-Planck equation. A drawback of this approach is that one needs the data distribution to have a Lipschitz logarithmic gradient. This excludes a large class of data distributions that have a compact support. We present a deterministic diffusion process for which the vector fields are always Lipschitz and hence the score does not explode for probability measures with compact support. This deterministic diffusion process can be seen as a regularization of the porous media equation equation, which enables one to guarantee long term convergence of the forward process to the noise distribution. Though the porous media equation is itself not always guaranteed to have a Lipschitz vector field, it can be used to understand the closeness of the output of the algorithm to the data distribution as a function of the the time horizon and score matching error. This analysis enables us to show that the algorithm has better dependence on the score matching error than approaches based on stochastic diffusions. Using numerical experiments we verify our theoretical results on example one and two dimensional data distributions which are compactly supported. Additionally, we validate the approach on a modified MNIST data set for which the distribution is concentrated on a compact set. In each of the experiments, the approach using deterministic diffusion performs better that the diffusion algorithm with stochastic forward process, when considering the FID scores of the generated samples.

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• An Optimal Transport Approach for Computing Adversarial Training Lower Bounds in Multiclass Classification

  arXiv (Cornell University) · 2024-01-17 · 1 citations

  preprintOpen access

  Despite the success of deep learning-based algorithms, it is widely known that neural networks may fail to be robust. A popular paradigm to enforce robustness is adversarial training (AT), however, this introduces many computational and theoretical difficulties. Recent works have developed a connection between AT in the multiclass classification setting and multimarginal optimal transport (MOT), unlocking a new set of tools to study this problem. In this paper, we leverage the MOT connection to propose computationally tractable numerical algorithms for computing universal lower bounds on the optimal adversarial risk and identifying optimal classifiers. We propose two main algorithms based on linear programming (LP) and entropic regularization (Sinkhorn). Our key insight is that one can harmlessly truncate the higher order interactions between classes, preventing the combinatorial run times typically encountered in MOT problems. We validate these results with experiments on MNIST and CIFAR-$10$, which demonstrate the tractability of our approach.

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Frequent coauthors

• Flavien Léger

  11 shared

• Stanley Osher

  8 shared

• Nicolás García Trillos

  University of Wisconsin–Madison

  6 shared

• Dino Di Carlo

  6 shared

• Jiajun Tong

  5 shared

• Wuchen Li

  5 shared

• Inwon C. Kim

  4 shared

• Andrea L. Bertozzi

  4 shared