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Topological edge states of continuous Hamiltonians

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arXiv 2025年

作者： Frazier, Matthew Bal, Guillaume Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Departments of Mathematics and Statistics Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

This paper concerns the topological classification of continuous Hamiltonians that find applications in biased cold plasmas and photonics. Besides a magnetic bias, the Hamiltonians are parametrized by a plasma frequency and a fixed vertical wavenumber. Eight distinct phases of matter are identified as these parameters vary. When insulating gaps are shared by two such phases, asymmetric edge modes propagate along interfaces separating the two phases. Here we apply the notion of a bulk difference invariant (BDI) to this Hamiltonian, and show by numerical diagonalizations of interface Hamiltonians that after an appropriate regularization our BDI correctly predicts edge transport as described by a bulk edge correspondence. We also derive theoretical tools to compute the BDI and show the limitations of the bulk edge correspondence (BEC) when the phase transition is too singular. Copyright © 2025, The Authors. All rights reserved.

关键词： Topology

GEOMETRIC PROGRAMMING FOR 3D CIRCUITS

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arXiv 2025年

作者： Wang, Rongbiao Lim, Lek-Heng Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Computational and Applied Mathematics Initiative Department of Statistics University of Chicago ChicagoIL60637-1514 United States

With the soaring demand for high-performing integrated circuits, 3D integrated circuits (ICs) have emerged as a promising alternative to traditional planar structures. Unlike existing 3D ICs that stack 2D layers, a full 3D IC features cubic circuit elements unrestricted by layers, offering greater design freedom. Design problems such as floorplanning, transistor sizing, and interconnect sizing are highly complex due to the 3D nature of the circuits and unavoidably require systematic approaches. We introduce geometric programming to solve these design optimization problems systematically and efficiently. © 2025, CC BY-NC-SA.

关键词： Three dimensional integrated circuits

A flow-kick model of dryland vegetation patterns: the impact of rainfall variability on resilience

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arXiv 2025年

作者： Gandhi, Punit Oline, Matthew Silber, Mary Department of Mathematics and Applied Mathematics Virginia Commonwealth University RichmondVA23284 United States Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Department of Statistics and Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

In many drylands around the globe, vegetation self-organizes into regular spatial patterns in response to aridity stress. We consider the regularly-spaced vegetation bands, on gentle hill-slopes, that survive low rainfall conditions by harvesting additional stormwater from upslope low-infiltration bare zones. We are interested in the robustness of this pattern formation survival mechanism to changes in rainfall variability. For this, we use a flow-kick modeling framework that treats storms as instantaneous kicks to the soil water. The positive feedbacks in the storm-level hydrology, that act to concentrate water within the vegetation bands, are captured through the spatial profiles of the soil water kicks. Between storms, the soil water and vegetation, modeled by a two-component reaction-diffusion system, evolve together. We use a combination of linear stability analysis and numerical simulation to compare predictions of idealized periodic rainfall, with no variability, to predictions when there is randomness in the timing and magnitude of water input from storms. We show that including these random elements leads to a decrease in the parameter range over which patterns appear. This suggests that an increase in storm variability, even with the same mean annual rainfall, may negatively impact the resilience of these pattern-forming dryland ecosystems. Copyright © 2025, The Authors. All rights reserved.

关键词： Vegetation

Batched Nonparametric Contextual Bandits

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IEEE Transactions on Information Theory 2025年

作者： Jiang, Rong Ma, Cong University of Chicago Committee on Computational and Applied Mathematics IL60637 United States University of Chicago Department of Statistics ChicagoIL60637 United States

We study nonparametric contextual bandits under batch constraints, where the expected reward for each action is modeled as a smooth function of covariates, and the policy updates are made at the end of each batch of observations. We establish a minimax regret lower bound for this setting and propose a novel batch learning algorithm that achieves the optimal regret (up to logarithmic factors). In essence, our procedure dynamically splits the covariate space into smaller bins, carefully aligning their widths with the batch size. Our theoretical results suggest that for nonparametric contextual bandits, a nearly constant number of policy updates can attain optimal regret in the fully online setting. © 1963-2012 IEEE.

关键词： batch learning contextual bandits dynamic binning minimax rate regret bounds

Sharp concentration of simple random tensors

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arXiv 2025年

作者： Al-Ghattas, Omar Chen, Jiaheng Sanz-Alonso, Daniel Department of Statistics University of Chicago IL60637 United States Committee on Computational and Applied Mathematics University of Chicago IL60637 United States

This paper establishes sharp dimension-free concentration inequalities and expectation bounds for the deviation of the sum of simple random tensors from its expectation. As part of our analysis, we use generic chaining techniques to obtain a sharp high-probability upper bound on the suprema of multi-product empirical processes. In so doing, we generalize classical results for quadratic and product empirical processes to higher-order settings. © 2025, CC BY.

关键词： Tensors

Integral equations for flexural-gravity waves: analysis and numerical methods

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arXiv 2025年

作者： Askham, Travis Hoskins, Jeremy G. Nekrasov, Peter Rachh, Manas Department of Mathematical Sciences New Jersey Institute of Technology NewarkNJ07102 United States Department of Statistics and CCAM University of Chicago ChicagoIL60637 United States Committee on Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States Center for Computational Mathematics Flatiron Institute New YorkNY10010 United States

In this work, we develop a fast and accurate method for the scattering of flexural-gravity waves by a thin plate of varying thickness overlying a fluid of infinite depth. This problem commonly arises in the study of sea ice and ice shelves, which can have complicated heterogeneities that include ridges and rolls. With certain natural assumptions on the thickness, we present an integral equation formulation for solving this class of problems and analyze its mathematical properties. The integral equation is then discretized and solved using a high-order-accurate, FFT-accelerated algorithm. The speed, accuracy, and scalability of this approach are demonstrated through a variety of illustrative examples. © 2025, CC BY.

关键词： Gravity waves

Solving Inverse Problems with Deep Linear Neural Networks: Global Convergence Guarantees for Gradient Descent with Weight Decay

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arXiv 2025年

作者： Laus, Hannah Parkinson, Suzanna Charisopoulos, Vasileios Krahmer, Felix Willett, Rebecca Department of Mathematics Technical University of Munich Germany Germany Committee on Computational and Applied Mathematics University of Chicago United States Data Science Institute University of Chicago United States Technical University of Munich Germany Department of Computer Science University of Chicago United States Department of Statistics University of Chicago United States

Machine learning methods are commonly used to solve inverse problems, wherein an unknown signal must be estimated from few measurements generated via a known acquisition procedure. In particular, neural networks perform well empirically but have limited theoretical guarantees. In this work, we study an underdetermined linear inverse problem that admits several possible solution mappings. A standard remedy (e.g., in compressed sensing) establishing uniqueness of the solution mapping is to assume knowledge of latent low-dimensional structure in the source signal. We ask the following question: do deep neural networks adapt to this low-dimensional structure when trained by gradient descent with weight decay regularization? We prove that mildly overparameterized deep linear networks trained in this manner converge to an approximate solution that accurately solves the inverse problem while implicitly encoding latent subspace structure. To our knowledge, this is the first result to rigorously show that deep linear networks trained with weight decay automatically adapt to latent subspace structure in the data under practical stepsize and weight initialization schemes. Our work highlights that regularization and overparameterization improve generalization, while overparameterization also accelerates convergence during training. © 2025, CC BY.

关键词： Deep neural networks

A FOLIATED AND REVERSIBLE FINSLER MANIFOLD IS DETERMINED BY ITS BROKEN SCATTERING RELATION

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Pure and applied Analysis 2021年第4期3卷 789-811页

作者： DE HOOP, Maarten V. Ilmavirta, Joonas Lassas, Matti Saksala, Teemu Departments of Computational and Applied Mathematics Earth Science Rice University Houston TX United States Department of Mathematics and Statistics University of Jyväskylä Jyväskylä Finland Department of Mathematics and Statistics University of Helsinki Helsinki Finland Department of Mathematics North Carolina State University Raleigh NC United States

The broken scattering relation consists of the total lengths of broken geodesics that start from the boundary, change direction once inside the manifold, and propagate to the boundary. We show that if two reversible Finsler manifolds satisfying a convex foliation condition have the same broken scattering relation, then they are isometric. This implies that some anisotropic material parameters of the earth can be in principle reconstructed from single scattering measurements at the surface. © 2021, Mathematical Sciences Publishers. All rights reserved.

关键词： anisotropic elasticity distance functions Finsler manifold scattering relation

Long distance propagation of wave beams in paraxial regime

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arXiv 2024年

This paper concerns the propagation of high frequency wave-beams in highly turbulent atmospheres. Using a paraxial model of wave propagation, we show in the long-distance weak-coupling regime that the wavefields are approximately described by a complex Gaussian field whose scintillation index is unity. This provides a model of the speckle formation observed in many practical settings. The main step of the derivation consists in showing that closed-form moment equations in the Itô-Schrödinger regime are still approximately satisfied in the paraxial regime. The rest of the proof is then an extension of results derived in [5]. Copyright © 2024, The Authors. All rights reserved.

关键词： Wave propagation