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Date Time |
Location | Speaker |
Title – click for abstract |
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01/21 09:00am |
Zoom |
Cristiana De Filippis University of Parma |
mu- ellipticity and nonautonomous integrals
mu-ellipticity describes certain degenerate forms of ellipticity, typical of convex integrals at linear, or nearly linear growth such as the area integral, or the iterated logarithmic model. The regularity of solutions to autonomous or totally differentiable problems is classical after Bombieri and De Giorgi and Miranda, Ladyzhenskaya and Ural’tseva and Frehse and Seregin. The anisotropic case is a later achievement of Bildhauer, Fuchs and Mingione, Beck and Schmidt and Gmeineder and Kristensen, that provided a complete partial and full regularity theory. However, all the approaches developed so far break down in presence of nondifferentiable ingredients. In particular, Schauder theory for certain significant anisotropic, nonautonomous functionals with Hölder continuous coefficients was only recently obtained by Mingione and myself. I will give an overview of the latest progress on the validity of Schauder theory for anisotropic problems whose growth is arbitrarily close to linear within the maximal nonuniformity range, and discuss sharp results and deep insights on more general nonautonomous area type integrals. From recent, joint work with Filomena De Filippis (Parma), Giuseppe Mingione (Parma), and Mirco Piccinini (Pisa). |
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01/28 2:00pm |
BLOCKER 302 |
Thomas Chen University of Texas Austin |
Explicit construction of global minimizers and the interpretability problem in Deep Learning
Deep Learning (DL) as a core subfield of Machine Learning and Artificial Intelligence is at the center of extraordinary technological progress. However, despite of remarkable advances in applications and theoretical analysis, the conceptual and rigorous reasons for the functioning of DL networks are at present not clearly understood (the problem of “interpretability"). In this talk, we present some recent results aimed at the rigorous mathematical understanding of how and why supervised learning works. For underparametrized DL networks, we explicitly construct global, zero loss cost minimizers for sufficiently clustered data. In addition, we derive effective equations governing the cumulative biases and weights, and show that gradient descent corresponds to a dynamical process in the input layer, whereby clusters of data are progressively reduced in complexity ("truncated") at an exponential rate that increases with the number of data points that have already been truncated. For overparametrized DL networks, we prove that the gradient descent flow is homotopy equivalent to a geometrically adapted flow that induces a (constrained) Euclidean gradient flow in output space. If a certain rank condition holds, the latter is, upon reparametrization of the time variable, equivalent to simple linear interpolation. This in turn implies zero loss minimization and the phenomenon known as "neural collapse”. A majority of this work is joint with Patricia Munoz Ewald (UT Austin). |
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01/28 3:00pm |
Blocker 302 |
Abed ElRahman Hammoud Princeton University |
Artificial Intelligence for Downscaling: Application to Uncertain Chaotic Systems
Reliable high-resolution state estimates for forecasts and reanalyses are pivotal in environmental applications, particularly in ocean and atmospheric sciences. These are typically achieved by integrating observational data into dynamical models through processes such as data assimilation (DA), when enhancing the reliability of forecasts and reanalysis, or downscaling when bridging the gap between coarse-scale observations and fine-scale information. Current DA and downscaling techniques rely on limiting assumptions and tend to be computationally demanding, especially in the presence of observational and model uncertainties. Artificial intelligence (AI) emerges as a powerful avenue for developing efficient data-driven tools that enhance reliability and alleviate computational demands of conventional DA and downscaling algorithms.
This talk aims to present recent developments in AI tools that address challenges pertaining to downscaling with application to chaotic dynamical systems, and within an uncertain framework. The state-of-the-art dynamical downscaling algorithm, Continuous data assimilation (CDA), and its discrete-in-time counterpart (DDA) are first explored in the setting involving observational errors. Since CDA relies on an abstract lifting function called the determining form map, a physics-informed deep neural network (PI-DNN) named CDAnet is proposed to approximate this intractable mapping. CDAnet is then evaluated under observational and model uncertainties in application to the Rayleigh-Benard convection problem, validating and further extending upon the knowledge from theory. |
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02/04 09:00am |
Zoom |
Agnieszka Świerczewska-Gwiazda Warsaw University |
Cahn-Hillard and Keller-Segel systems as high-friction limits of gas dynamics
Several recent studies considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using the relative entropy method. Applying the recent result about relations between non-local and local Cahn-Hilliard, we also derive rigorously the large-friction nonlocal- to-local limit. The result is formulated for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation which are known to exist on arbitrary intervals of time. This approach provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation. During the talk I will also discuss the high-friction limit of the Euler-Poisson system. |
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02/04 10:00am |
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Lan-Anh Nguyen Rochester Institute of Technology and HNUE |
Inverse problems for evolutionary hemi-quasi -variational inequalities with applications
This talk addresses inverse problems associated with setvalued hemi-quasi-variational inequalities within the framework of reflexive Banach spaces. Initially, we establish solvability results and demonstrate the weak compactness of solution sets for specific classes of parametric generalized hemi-quasi-variational inequalities. Our approach involves utilizing variational selection to distinguish between monotone and pseudo-monotone components. Subsequently, we delve into the exploration of existence results for the inverse problem, employing a comprehensive regularization framework. Additionally, we offer applications of our theoretical results, particularly in the context of elliptic and parabolic hemi-quasi-variational inequalities encountered in nonlinear implicit obstacle problems and contact problems.
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02/11 3:00pm |
ZOOM |
Ian Tice Carnegie Mellon University |
Stationary and slowly traveling solutions to the free boundary Navier-Stokes equations
The stationary problem for the free boundary incompressible Navier-Stokes equations lies at the confluence of two distinct lines of inquiry in fluid mechanics. The first views the dynamic problem as an initial value problem. In this context, the stationary problem arises naturally as a special type of global-in-time solution with stationary sources of force and stress. One then expects solutions to the stationary problem to play an essential role in the study of long-time asymptotics or attractors for the dynamic problem. The second line of inquiry, which dates back essentially to the beginning of mathematical fluid mechanics, concerns the search for traveling wave solutions. In this context, a huge literature exists for the corresponding inviscid problem, but progress on the viscous problem was initiated much more recently in the work of the speaker and co-authors. For technical reasons, these results were only able to produce traveling solutions with nontrivial wave speed. In this talk we will discuss the well-posedness theory for the stationary problem and show that the solutions thus obtained lie along a one-parameter family of slowly traveling wave solutions. This is joint work with Noah Stevenson. |
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02/18 3:00pm |
Blocker 302 |
José Palacios University of Toronto |
Linearized dynamic stability for vortices of Ginzburg-Landau evolutions
We consider the problem of dynamical stability for the vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic "particle plus field'' description of interacting vortices. In this talk we focus on co-rotational perturbations of vortices and establish a variety of pointwise dispersive and decay estimates for their linearized evolution in the relativistic (or Klein-Gordon) case. One of the main ingredients is the construction of the distorted Fourier transform associated with the (two) linearized operators at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag in the context of stability of blow-up for wave maps and relies on the spectral analysis of Schrodinger operators with strongly singular potentials (see also Gezstesy-Zinchenko). However, since the vortex is not given by an explicit formula, and one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, the linear analysis requires some additional work. In particular, to construct the distorted Fourier basis and to control the spectral measure some additional arguments are needed, compared to previous works. This is joint work with Fabio Pusateri. |
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02/25 09:00am |
Zoom |
Aleksis Vuoksenmaa University of Helsinki |
TBA
TBA |
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02/26 3:00pm |
Zoom |
Ahmad Abassi University of California Berkeley |
Finite-depth standing water waves: theory, computational algorithms, and rational approximations
We generalize the semi-analytic standing-wave framework of Schwartz and Whitney (1981) and Amick and Toland (1987) to finite-depth standing gravity waves. We propose an appropriate Stokes-expansion ansatz and iterative algorithm to solve the system of differential equations governing the expansion coefficients. We then present a more efficient algorithm that allows us to compute the asymptotic solution to higher orders. Finally, we conclude with numerical simulations of the algorithms implemented in multiple-precision arithmetic on a supercomputer to study the effects of small divisors and the analytic properties of rational approximations of the computed solutions. *Joint work with Prof. Jon Wilkening, University of California, Berkeley. **This is joint seminar with the Numerical Analysis Seminar. |
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02/28 4:00pm |
Blocker 302 |
Hamza Ruzayqat King Abdullah University of Science and Technology (KAUST) |
BAYESIAN ANOMALY DETECTION IN VARIABLE-ORDER AND VARIABLE-DIFFUSIVITY FRACTIONAL MEDIA
Fractional diffusion equations (FDEs) are powerful tools for modeling anomalous diffusion in complex systems, such as fractured media and biological processes, where nonlocal dynamics and spatial heterogeneity are prominent. These equations provide a more accurate representation of such systems compared to classical models but pose significant computational challenges, particularly for spatially varying diffusivity and fractional orders. In this talk I will present a Bayesian inverse problem for FDEs in a 2-dimensional bounded domain with an anomaly of unknown geometric and physical properties, where the latter are the diffusivity and fractional order fields. To tackle the computational burden of solving dense and ill-conditioned systems, we employ an advanced finite-element scheme incorporating low-rank matrix representations and hierarchical matrices. For parameter estimation, we implement two surrogate-based approaches using polynomial chaos expansions: one constructs a 7-dimensional surrogate for simultaneous inference of geometrical and physical parameters, while the other leverages solution singularities to separately infer geometric features, then constructing a 2-dimensional surrogate to learn the physical parameters and hence reducing the computational cost immensely. |
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03/04 3:00pm |
Blocker 302 |
Animikh Biswas University of Maryland Baltimore County |
TBA
TBA |
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03/18 3:00pm |
BLOCKER 302 |
Tomasz Komorowski Polish Academy of Sciences |
TBA
TBA |
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03/25 3:00pm |
Blocker 302 |
Marita Thomas Freie Universitaet - Berlin |
TBA
TBA |
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04/01 3:00pm |
BLOCKER 302 |
Connor R Mooney University of California Irvine |
TBA
TBA |
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04/08 09:00am |
Zoom |
Borjan Geshkovski INRIA Paris |
TBA
TBA |
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04/08 3:00pm |
Blocker 302 |
Jinkai Li South China Normal University |
TBA
TBA |
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04/22 3:00pm |
Blocker 302 |
Ugur G. Abdulla Okinawa Institute of Science and Technology, Japan |
Kolmogorov Problem and Wiener-type Criteria for the Removability of the Fundamental Singularity for the Elliptic and Parabolic PDEs
This talk will address the major problem in the Analysis of PDEs on the nature of singularities reflecting the natural phenomena. I will present my solution of the "Kolmogorov's Problem" (1928) expressed in terms of the new Wiener-type criterion for the removability of the fundamental singularity for the heat equation. The new concept of regularity or irregularity of singularity point for the parabolic (or elliptic) PDEs is defined according to whether or not the caloric (or harmonic) measure of the singularity point is null or positive. The new Wiener-type criterion precisely characterizes the uniqueness of boundary value problems with singular data, reveal the nature of the harmonic or caloric measure of the singularity point, asymptotic laws for the conditional Brownian motion, and criteria for thinness in minimal-fine topology. The talk will end with the description of some outstanding open problems and perspectives of the development of the potential theory of nonlinear elliptic and parabolic PDEs. |