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GLOBAL SOLUTIONS TO THE SUPERCOOLED STEFAN PROBLEM WITH BLOW-UPS: REGULARITY AND UNIQUENESS

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Probability and Mathematical Physics

Probability and Mathematical Physics 2022年第1期3卷 171-213页

作者： Delarue, François Nadtochiy, Sergey Shkolnikov, Mykhaylo Laboratoire J.A.Dieudonné Université de Nice Sophia-Antipolis UMR CNRS-UNS No. 7351 Nice France Department of Applied Mathematics Illinois Institute of Technology Rettaliata Engineering Center Chicago IL United States ORFE Department Bendheim Center for Finance Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

We consider the supercooled Stefan problem, which captures the freezing of a supercooled liquid, in one space dimension. A probabilistic reformulation of the problem allows us to define global solutions, even in the presence of blow-ups of the freezing rate. We provide a complete description of such solutions, by relating the temperature distribution in the liquid to the regularity of the ice growth process. The latter is shown to transition between (i) continuous differentiability, (ii) Hölder continuity, and (iii) discontinuity. In particular, in the second regime we rediscover the square root behavior of the growth process pointed out by Stefan in his seminal 1889 paper for the ordinary Stefan problem. In our second main theorem, we establish the uniqueness of the global solutions, a first result of this kind in the context of growth processes with singular self-excitation when blow-ups are present. © 2022 Mathematical Sciences Publishers.

关键词： blow-ups free boundary problem heat equation interacting particle systems mean-field interaction physical solutions probabilistic reformulation self-excitation supercooled Stefan problem zero set

Method of moments for 3-D single particle ab initio modeling with non-uniform distribution of viewing angles

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arXiv 2019年

作者： Sharon, Nir Kileel, Joe Khoo, Yuehaw Landa, Boris Singer, Amit School of Mathematical Sciences Tel Aviv University Israel Program in Applied and Computational Mathematics Princeton University United States Department of Statistics and the College University of Chicago United States Applied Mathematics Program Yale University United States Program in Applied and Computational Mathematics Department of Mathematics Princeton University United States

Single-particle reconstruction in cryo-electron microscopy (cryo-EM) is an increasingly popular technique for determining the 3-D structure of a molecule from several noisy 2-D projections images taken at unknown viewing angles. Most reconstruction algorithms require a low-resolution initialization for the 3-D structure, which is the goal of ab initio modeling. Suggested by Zvi Kam in 1980, the method of moments (MoM) offers one approach, wherein low-order statistics of the 2-D images are computed and a 3-D structure is estimated by solving a system of polynomial equations. Unfortunately, Kam’s method suffers from restrictive assumptions, most notably that viewing angles should be distributed uniformly. Often unrealistic, uniformity entails the computation of higher-order correlations, as in this case first and second moments fail to determine the 3-D structure. In the present paper, we remove this hypothesis, by permitting an unknown, non-uniform distribution of viewing angles in MoM. Perhaps surprisingly, we show that this case is statistically easier than the uniform case, as now first and second moments generically suffice to determine low-resolution expansions of the molecule. In the idealized setting of a known, non-uniform distribution, we find an efficient provable algorithm inverting first and second moments. For unknown, non-uniform distributions, we use non-convex optimization methods to solve for both the molecule and distribution. Copyright © 2019, The Authors. All rights reserved.

关键词： Method of moments

Global weak solutions for generalized SQG in bounded domains

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arXiv 2017年

作者： Nguyen, Huy Quang PROGRAM IN APPLIED AND COMPUTATIONAL MATHEMATICS PRINCETON UNIVERSITY PRINCETONNJ08544 United States

We prove the existence of global L2 weak solutions for a family of generalized inviscid surface-quasi geostrophic (SQG) equations in bounded domains of the plane. In these equations, the active scalar is transported by a velocity field which is determined by the scalar through a more singular nonlocal operator compared to the SQG equation. The result is obtained by establishing appropriate commutator representations for the weak formulation together with good bounds for them in bounded *** Classification: 35Q35, 35Q86.35Q35, 35Q86 Copyright © 2017, The Authors. All rights reserved.

关键词： Velocity

Long Time Dynamics of Nonequilibrium Electroconvection

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arXiv 2023年

作者： Lee, Fizay-Noah Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States

The Nernst-Planck-Stokes (NPS) system models electroconvection of ions in a fluid. We consider the system, for two oppositely charged ionic species, on three dimensional bounded domains with Dirichlet boundary conditions for the ionic concentrations (modelling ion selectivity), Dirichlet boundary conditions for the electrical potential (modelling an applied potential), and no-slip boundary conditions for the fluid velocity. In this paper, we obtain quantitative bounds on solutions of the NPS system in the long time limit, which we use to prove 1) the existence of a compact global attractor with finite fractal (box-counting) dimension and 2) space-time averaged electroneutrality ρ ≈ 0 in the singular limit of Debye length going to zero, Ε → *** Codes 35Q30, 35Q35, 35Q92 © 2023, CC BY.

关键词： Navier Stokes equations

Improving the robustness of the immersed interface method through regularized velocity reconstruction

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arXiv 2025年

作者： Sun, Qi Kolahdouz, Ebrahim M. Griffith, Boyce E. Department of Mathematics University of North Carolina Chapel HillNC United States Department of Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

Robust, broadly applicable fluid-structure interaction (FSI) algorithms remain a challenge for computational mechanics. Efforts in this area are driven by the need to enhance predictive accuracy and efficiency in FSI simulations, align with experimental observations, and unravel complex multiscale and multiphysics phenomena, while addressing challenges in developing more robust and efficient methodologies. In previous work, we introduced an immersed interface method (IIM) for discrete surfaces and an extension based on an immersed Lagrangia-Eulerian (ILE) coupling strategy for modeling FSI involving complex geometries. The ability of the method to sharply resolve stress discontinuities induced by singular immersed boundary forces in the presence of low-regularity geometrical representations makes it a compelling choice for three-dimensional modeling of complex geometries in diverse engineering applications. Although the IIM we previously introduced offers many desirable advantages, it also imposes a restrictive mesh factor ratio, requiring the surface mesh to be coarser than the fluid grid to ensure stability. This is because when the mesh factor ratio constraint is not satisfied, parts of the structure motion are not controlled by the discrete FSI system. This constraint can significantly increase computational costs, particularly in applications involving multiscale geometries with highly localized complexity or fine-scale features. To address this limitation, we devise a stabilization strategy for the velocity restriction operator inspired by Tikhonov regularization. This study demonstrates that using a stabilized velocity restriction operator in IIM enables a broader range of structure-to-fluid grid-size ratios without compromising accuracy or altering the flow dynamics. This advancement significantly broadens the applicability of the method to real-world FSI problems involving complex geometries and dynamic conditions, offering a robust and computationally effi

关键词： Mesh generation

Three-dimensional dynamics and transition to turbulence in the wake of bluff objects

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Journal of Fluid Mechanics 1992年第1期238卷 1-30页

作者： Karniadakis, George Em Triantafyllou, George S. Department of Mechanical and Aerospace Engineering and Program of Applied and Computational Mathematics Princeton University Princeton NJ 08544 United States The Levich Institute and Department of Mechanical Engineering The City College of New York New York NY 10031 United States

The wakes of bluff objects and in particular of circular cylinders are known to undergo a ‘fast ’ transition, from a laminar two-dimensional state a t Reynolds number 200 to a turbulent state a t Reynolds number 400. The process has been documented in several eXperimental mvestigations, but the underlying physical mechanisms have remained largely unknown so far. In this paper, the transition process is investigated numerically, through direct simulation of the NavierStokes equations at representative Reynolds numbers, up to 500. A high-order timeaccurate, miXed spectral/spectral element technique is used. It is shown that the wake first becomes three-dimensional, as a result of a secondary instability of the two-dimensional vorteX street. This secondary instability appears at a Reynolds number close to 200. For slightly supercritical Reynolds numbers, a harmonic state develops, in which the flow oscillates at its fundamental frequency (Strouhal number) around a spanwise modulated time-average flow. In the near wake the modulation wavelength of the time-average flow is half of the spanwise wavelength of the perturbation flow, consistently with linear instability theory. The vorteX filaments have a spanwise wavy shape in the near wake, and form rib-like structures further downstream. At higher Reynolds numbers the three-dimensional flow oscillation undergoes a period-doubling bifurcation, in which the flow alternates between two different states. Phase-space analysis of the flow shows that the basic limit cycle has branched into two connected limit cycles. In physical space the period doubling appears as the shedding of two distinct types of vorteX filaments. Further increases of the Reynolds number result in a cascade of period-doubling bifurcations, which create a chaotic state in the flow at a Reynolds number of about 500. The flow is characterized by broadband power spectra, and the appearance intermittent phenomena. It is concluded that the wake undergoes transit

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Topological Classification of Insulators: II. Quasi-Two-Dimensional Locality

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arXiv 2024年

作者： Chung, Jui-Hui Shapiro, Jacob Program in Applied and Computational Mathematics Princeton University United States Department of Mathematics Princeton University United States

We provide an alternative characterization of two-dimensional locality (necessary e.g. to define the Hall conductivity of a Fermi projection) using the spectral projections of the Laughlin flux operator. Using this abstract characterization, we define generalizations of this locality, which we term quasi-2D. We go on to calculate the path-connected components of spaces of unitaries or orthogonal projections which are quasi-2D-local and find a starkly different behavior compared with the actual 2D column of the Kitaev table, exhibiting e.g., in the unitary chiral case, infinitely many Z-valued indices. Copyright © 2024, The Authors. All rights reserved.

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Robust Group Synchronization via Quadratic programming

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arXiv 2022年

作者： Shi, Yunpeng Wyeth, Cole Lerman, Gilad Program in Applied and Computational Mathematics Princeton University United States School of Mathematics University of Minnesota United States

We propose a novel quadratic programming formulation for estimating the corruption levels in group synchronization, and use these estimates to solve this problem. Our objective function exploits the cycle consistency of the group and we thus refer to our method as detection and estimation of structural consistency (DESC). This general framework can be extended to other algebraic and geometric structures. Our formulation has the following advantages: it can tolerate corruption as high as the information-theoretic bound, it does not require a good initialization for the estimates of group elements, it has a simple interpretation, and under some mild conditions the global minimum of our objective function exactly recovers the corruption levels. We demonstrate the competitive accuracy of our approach on both synthetic and real data experiments of rotation *** Codes 90C26, 90C17, 68Q87, 65C20, 90-08, 60-08 Copyright © 2022, The Authors. All rights reserved.

关键词： Quadratic programming

The quenching-activation behavior of the gradient descent dynamics for two-layer neural network models

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arXiv 2020年

作者： Ma, Chao Wu, Lei Weinan, E. Department of Mathematics Princeton University Program in Applied and Computational Mathematics Princeton University Institute of Big Data Research

A numerical and phenomenological study of the gradient descent (GD) algorithm for training two-layer neural network models is carried out for different parameter regimes when the target function can be accurately approximated by a relatively small number of neurons. It is found that for Xavier-like initialization, there are two distinctive phases in the dynamic behavior of GD in the under-parametrized regime: An early phase in which the GD dynamics follows closely that of the corresponding random feature model and the neurons are effectively quenched, followed by a late phase in which the neurons are divided into two groups: a group of a few "activated" neurons that dominate the dynamics and a group of background (or "quenched") neurons that support the continued activation and deactivation process. This neural network-like behavior is continued into the mildly over-parametrized regime, where it undergoes a transition to a random feature-like behavior. The quenching-activation process seems to provide a clear mechanism for "implicit regularization". This is qualitatively different from the dynamics associated with the "mean-field" scaling where all neurons participate equally and there does not appear to be qualitative changes when the network parameters are changed. Copyright © 2020, The Authors. All rights reserved.

关键词： Gradient methods