Abstract: Computing high-dimensional PDEs remains an important, yet challenging problem in computational mathematics. We propose a scalable deep learning–based algorithm for solving PDEs that integrates preconditioned gradient methods with adversarial training. We reformulate the problem as an inf–sup saddle-point problem and apply the Primal–Dual Hybrid Gradient algorithm. To enhance efficiency and stability, we introduce suitable preconditioning operators into the proximal steps of the PDHG algorithm, which leads to a natural gradient ascent–descent scheme for updating neural network parameters. The resulting natural gradients are evaluated using the Krylov subspace method (MINRES), allowing the inversion of preconditioning matrices to be handled through matrix–vector products. Convergence guarantees are established for time-continuous formulation of the algorithm when applied to linear equations, particularly elliptic PDEs in divergence form. We further validate the proposed method on a broader range of linear and nonlinear PDEs across varying dimensions. Numerical results demonstrate that our approach is both efficient and robust, exhibiting more stable convergence and enhanced accuracy when compared with some of the existing benchmark solvers.
Speaker: Dave Levermore (U. Maryland) - When: Mon, September 14, 2026 - 3:00pm Where: Kirwan Hall 1313
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Abstract: The kinetic theory of gases was very controversial at the dawn of the twentieth century. Hilbert posed his Sixth Problem to spur the development of mathematics that might help resolve these controversies and advance the understanding of how macroscopic physics emerges from microscopic physics. In particular, he cited the work of Boltzmann, which was built upon the work of Clausius and Maxwell. We will describe the Sixth Problem as Hilbert saw it, through the lens of the work of Clausius, Maxwell, and Boltzmann. It is still an active area of research. Recent results by Deng, Hani and Ma helped earn a 2026 Fields Medal for Yu Deng. We will describe these and other significant recent advances, but show that the problem largely remains open.
Speaker: Todd Rowland (UMD) - When: Tue, September 15, 2026 - 2:30pm Where: Kirwan Hall 3206
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Abstract: This seminar brings people from around campus to talk about how they use AI. We will discuss the upcoming talks this semester, and openings left in fall and in the spring.
Speaker: Hannah Hoganson (UMD) - When: Tue, September 15, 2026 - 3:30pm Where: Kirwan Hall 1311
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Abstract: Countable Stone spaces—compact, Hausdorff, totally disconnected spaces familiar from Stone duality— are completely classified up to homeomorphism by countable ordinals. In this talk, we ask what the groups of homeomorphisms of these spaces look like from very far away.
The answer turns out to be surprisingly combinatorial. For most successor ordinals, their large-scale geometry is described by an infinite Hamming cube: vertices are binary sequences with finite support, and moving one step means changing a single coordinate. We'll explore how this Hamming cube arises naturally by looking at clopen partitions of a countable Stone space, and how the Cantor–Bendixson derivative lets us move between different ordinal ranks.
Speaker: Yaakov Malinovsky (University of Maryland, Baltimore County) - https://yaakovmalinovsky.github.io/ When: Wed, September 16, 2026 - 2:00pm Where: Kirwan Hall 1311
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Abstract: Negative dependence among participants’ outcomes arises naturally in probabilistic models of tournaments and plays an important role in various asymptotic results, including limit theorems, Poisson approximations, and the behavior of extreme scores. In particular, the property of negative orthant dependence has been shown in several works for different tournament models, usually requiring a separate proof in each case. We present a unified and more general approach by establishing the stronger property of negative association.
This stronger notion of dependence, together with Poisson approximation and large deviation methods, allows us to derive limit distributions for extreme order statistics, such as the maximum and the second highest scores, even though their exact distributions are not computationally tractable for general tournament sizes. These results are obtained for a general round-robin tournament model with equally strong players in which the possible match scores form a countable subset of the unit interval. Round-robin tournaments are also classical paired comparison models in statistics.
Abstract: Diffusion models generate samples from an unknown data distribution by learning to reverse an Ornstein–Uhlenbeck noising process, replacing the adversarial min–max game of GANs — whose Hamiltonian descent–ascent structure produces permanent oscillations rather than convergence — with a supervised regression problem for the score. In practice, one often wants to sample not from the prior
itself but from a posterior conditioned on a measurement (inpainting, super-resolution, guided generation). Diffusion Posterior Sampling (DPS) does this by recycling a pretrained score and passing the conditional expectation through the likelihood, a heuristic that works remarkably well but is not exact. We quantify the resulting error: the DPS surrogate path fails to satisfy the forward equation by a reaction term, and the induced bias admits an explicit Feynman–Kac representation, so that importance-weighting DPS samples recovers the true posterior. The same analysis reveals a hidden, extremely stiff annealing schedule in the standard implementation, which violates the explicit Euler stability criterion and explains why deployed fixes such as early stopping of the guidance and implicit integration work. Based on joint work with Sebastian Motsch, Advait Parulekar, Will Porteous, and Sanjay Shakkottai.
Speaker: Xianyang Zhang (Texas A&M University) - https://zhangxiany-tamu.github.io/ When: Thu, September 17, 2026 - 3:30pm Where: Kirwan Hall 1311
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Abstract: Demographic parity requires predictions to have the same distribution across sensitive groups. In regression, however, enforcing this requirement exactly can substantially reduce predictive accuracy. This talk studies fair regression under both exact and approximate demographic parity.
I will first discuss exact demographic parity. Using optimal transport, we characterize the optimal fair prediction distribution as a Wasserstein barycenter of the group-specific prediction distributions. This characterization also provides a practical way to construct the corresponding fair predictor.
I will then consider approximate demographic parity, which allows the group-specific prediction distributions to differ within a chosen tolerance. We impose direct pairwise Wasserstein constraints, making the trade-off between fairness and predictive accuracy explicit and tunable. Under suitable assumptions, the problem can be reformulated as an optimization over group-specific quantile functions. For squared loss with two groups, the solutions take three interpretable forms: soft thresholding under \(W_1\), shrinkage under \(W_2\), and clipping under \(W_\infty\). I will also discuss computation for multiple groups, statistical guarantees, simulations, and an application to a real dataset.