Wednesday, Sep 11, 15:00-16:00, Zoom meeting: Link

Speaker: Giovanni Gravina
Title: Collision and self-contact for viscoelastic solids with Lipschitz boundaries
Abstract: In this talk, we will examine the time evolution of viscoelastic solids within a framework that allows for collisions and self-contact. In the static and quasi-static regimes, corresponding existence results have been shown through variational descriptions of the problem. For the fully dynamical case, however, collisions have so far either been ignored or handled using a simplified model (for example, using repulsive terms). In contrast to this, by employing a newly developed variational technique, we are able to prove the existence of solutions for arbitrary times. In the second part of the talk, we will delve into the latest developments concerning solids with Lipschitz boundaries and discuss technical and conceptual difficulties that arise due to the presence of corners.  —————————————————————————————————————————————–

Wednesday, Sep 25, 15:00-16:00, Zoom meeting: Link

Speaker: Jone Apraiz, Universidad del País Vasco
Title: Inverse problem for a one-dimensional fluid-solid interaction model
Abstract: In this talk we will first briefly review some geometric inverse problems
we have studied for the one-dimensional heat, wave and Burgers equations.
Then, we will consider a one-dimensional fluid-solid interaction model gov-
erned by the Burgers equation with a time varying interface. This is a
preliminary simplified version of other more complicate and more realistic
models in higher dimensions. We will see the results we have obtained for
the inverse problem of determining the shape of the interface from Dirichlet
and Neumann data at one end point of the spatial interval. In particular, we
will show uniqueness results and some conditional stability estimates. For
the proofs, it will be seen that some lateral estimates have been used and
adapted, which rely on appropriate Carleman and interpolation inequalities.
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Wednesday, Oct 09, 15:00-16:00, Zoom meeting: Link

Speaker: Silvia Villa, Università di Genova
Title: Structured stochastic zeroth order optimization
Abstract: In the framework of black-box optimization, I will present new algorithms, 
based on the stochastic estimation of the gradient via  finite differences with structured
directions. I will describe their convergence properties under various assumptions and
show some numerical results.
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Wednesday, Oct 16, 15:00-16:00, COX 306

Speaker: Ziad Musslimani, Florida State University
Title: Space-time nonlocal integrable systems
Abstract: n this talk I will review past and recent results pertaining to the emerging topic of integrable space-time nonlocal integrable nonlinear evolution equations. In particular, we will discuss blow-up in finite time for solitons and the physical derivations of many integrable nonlocal systems.
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Wednesday, Oct 23, 15:00-16:00, Zoom meeting: Link

Speaker: Emilio Vilches Gutiérrez, Universidad de O’Higgins, Chile
Title: Recent Developments in Moreau’s Sweeping Processes
Abstract: The sweeping process is a first-order differential inclusion involving the normal cone to a family of moving sets. It was introduced by J.J. Moreau in the early seventies to address an elastoplastic problem. Since then, it has been used to model constrained dynamical systems, nonsmooth electrical circuits, crowd motion, mechanical problems, and other applications. The aim of this talk is twofold. On the one hand, we will provide the main insights into the well-posedness of the sweeping process, and on the other hand, we will present the latest developments in the subject, such as optimal control and numerical approximation.
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Wednesday, Oct 30, 15:00-16:00, Room: SAS 4201

Speaker: Stefania Patrizi, The University of Texas at Austin
Title: The discrete dislocation dynamics of multiple dislocation loops
Abstract: We consider a nonlocal reaction-diffusion equation that physically arises from the classical Peierls–Nabarro model for dislocations in crystalline structures. Our initial configuration corresponds to multiple slip loop dislocations in $\mathbb R^n$, $n\geq 2$. After suitably rescaling the equation with a small phase parameter $\epsilon>0$, the rescaled solution solves a fractional Allen–Cahn equation. We show that, as $\epsilon \to 0$, the limiting solution exhibits multiple interfaces evolving independently and according to their mean curvature. 

This is a joint work with Mary Vaughan  (University of Western Australia).
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Wednesday, Jan 22, 15:00-16:00, Zoom meeting: Link

Speaker: David Salas, Universidad de O’Higgins, Chile
Title: The Bayesian approach for bilevel programming
Abstract: In this talk, we present the Bayesian approach for bilevel programming problems, first introduced by Mallozzi and Morgan [1] under the name of intermediate Stackelberg games, and rediscovered in [3]. The main idea of this approach is to treat the (possibly not single-valued) response set of the follower as an uncertainty set, and treat the actual response as a random variable following a decision-dependent distribution. This distribution models the belief of the leader, hence the name of the approach. In the first part of the talk we will explore the well-posedness of the model, focusing on sufficient conditions to guarantee the existence of optimal solutions. In particular, we will show that in the linear setting, the existence of solutions is guaranteed for a large family of beliefs derived from the natural adaptation of the uniform distribution. In the second part, we will study an application to stochastic linear bilevel optimization, where we leverage on the polyhedral structure of the problem which is revealed under the lens of the Bayesian approach. We will finish discussing some open problems and new directions of research in this topic. The talk is based on the recent contributions with A. Svensson and G. Muñoz  [2,3].

References:
[1] L. Mallozzi and J. Morgan. Hierarchical Systems with Weighted Reaction Set, pages 271–282. Springer US, Boston, MA, 1996.
[2] G. Muñoz, D. Salas, and A. Svensson. Exploiting the polyhedral geometry of stochastic linear bilevel programming. Math. Program., pages 1–36, 2024.
[3] D. Salas and A. Svensson. Existence of solutions for deterministic bilevel games under a general Bayesian approach. SIAM J. Optim., 33(3):2311–2340, 2023.
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Wednesday, Jan 29, 15:00-16:00, Zoom meeting: Link

Speaker: Michael Goldman, CMAP, Ecole Polytechnique.
Title: Recent progress on the optimal matching problem
Abstract: In this talk I will review some recent progress in the understanding of the (random) optimal matching problem. While the work of Ajtai-Komlos-Tusnady in the 80’s on this classical optimization problem attracted a lot of attention from the probability community (see the book by Talagrand), this problem has seen a renewed interest from the PDE community thanks to the ansatz proposed by Caracciolo Lucibello, Parisi and Sicuro in 2014. I will explain to which extent this ansatz can be rigorously justified and show how it leads to a deeper understanding of this problem.
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Wednesday, Feb 05, 15:00-16:00, SAS 4201

Speaker: Loc Nguyen, UNCC
Title:  Convexification-Based Approach for Solving the 3D Inverse Scattering Problem
Abstract: We present a globally convergent numerical method for solving the three-dimensional inverse scattering problem governed by the Helmholtz equation, with applications in medical imaging, nondestructive testing, geophysical exploration, and radar signal processing. Traditional least-squares optimization methods often struggle to find a minimizer without a reliable initial guess of the true solution. To overcome this challenge, we introduce a Carleman weight function into the nonconvex least-squares functional, transforming it into a strictly convex functional whose unique minimizer can be efficiently computed using the gradient descent method. We rigorously establish this convexification property and prove that the unique minimizer accurately reconstructs the true solution of the inverse scattering problem. The robustness and effectiveness of the proposed method are validated through numerical experiments using both synthetic and experimentally collected data.
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Wednesday, Feb 12, 15:00-16:00, SAS 4201

Speaker: Ryan Murray, NC State University
Title:  A variational approach to studying dimension reduction algorithms
Abstract: Dimension reduction algorithms, such as principal component analysis (PCA), multidimensional scaling (MDS), and stochastic neighbor embeddings (SNE and tSNE), are an important tool for data exploration, visualization, and subgroup identification. While these algorithms see broad application across many scientific fields, our theoretical understanding of non-linear dimension reduction algorithms remains limited. This talk will describe new results that identify large data limits for MDS and tSNE using tools from the Calculus of Variations. Along the way, we will showcase situations where standard libraries give outputs that are misleading, and propose new computational algorithms to mitigate these issues and improve efficiency. Connections with the celebrated Gromov-Wasserstein distance and manifold learning will also be highlighted. This talk will aim to be accessible to a broad mathematical audience.
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Wednesday, Feb 19, 15:00-16:00, Zoom meeting: Link

Speaker: Sorin PoP, Hasselt University
Title: Non-equilibrium models for flow in porous media
Abstract: Porous media flows appear in several fields of highest societal relevance, such as environmental engineering, energy resources management, and flow through biological tissues. The underlying mathematical models can be expressed as (systems of) evolution equations, having a nonlinear and possibly degenerate character.

Next to the underlying conservation laws (mass, momentum, etc.), one typically assumes that the unknowns in such models (e.g. the capillary pressure and the saturation) are connected through an algebraic relationship, obtained under equilibrium conditions. For example, for a given medium and at a given saturation of the, say, wetting phase, the capillary pressure will always take the same value. Such an assumption is, however, invalidated by several experimental results reported in the literature. More precisely, as the solution satisfies a maximum principle, and some stability estimates in various energy norms, such equilibrium-based porous media flow models rule out the formation of so-called saturation overshoots, or of finger-like patterns, which are observed experimentally. This motivates extending the models commonly used in the literature by incorporating non-equilibrium effects like dynamic capillarity or hysteresis.

After providing the application background, in this presentation we briefly present some qualitative properties for such models (including the existence and uniqueness of weak solutions), and analyze various discretization schemes and linear iterative methods that are used to approximate the solution.


This is a joint work with K. Mitra, C.J. van Duijn (Eindhoven), S. Lunowa (Munich) and X. Cao (Toronto)
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Wednesday, Feb 26, 15:00-16:00, Zoom meeting: Link

Speaker: Roberto Guglielmi, University of Waterloo
Title: Turnpike phenomena in optimal control
Abstract: We introduce the notion of turnpike in optimal control problems and provide a characterization of such property in terms of structural properties of the control system, such as exponential stabilizability and detectability. We discuss the role of dissipativity of the underlying optimal control problem and the hyperbolicity of the optimality system around optimal steady states in connection with the occurrence of turnpike phenomena.
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Wednesday, Mar 05, 15:00-16:00, Room: SAS 4201

Speaker: Kerrek Stinson, University of Utah
Title: Variational methods for fracture mechanics
Abstract: The Griffith criterion says that the energy to crack a brittle elastic material is proportional to the length of the crack. Understanding minimizers of the energy requires unraveling the complex interplay of bulk (elastic) and surface (crack) energies in the vectorial setting of linear elasticity. We discuss a compactness result based on concentration-compactness and existence of strong solutions. Further, in dimension 2, we prove that the crack of a minimizer is given by a Hölder surface outside of a singular set of points with dimension strictly less than 1, analogous to results for the scalar-valued Mumford-Shah functional.
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Wednesday, Mar 19, 15:00-16:00, Room: SAS 4201

Speaker: Piermarco Cannarsa, Tor Vergata University of Rome
Title: TBA
Abstract: TBA
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Wednesday, Mar 26, 15:00-16:00, 15:00-16:00, SAS 4201

Speaker: Wonjun Lee, University of Minnesota
Title: TBA
Abstract: TBA
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Wednesday, April 02, 15:00-16:00, Zoom meeting: Link

Speaker: TBA
Title: TBA
Abstract: TBA
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Wednesday, April 09, 15:00-16:00, Zoom meeting: Link

Speaker: Tarek M Elgindi, Duke University
Title: TBA
Abstract: TBA
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Wednesday, April 16, 15:00-16:00, Zoom meeting: Link

Speaker: Arunima Bhattacharya, UNC
Title: TBA
Abstract: TBA
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Monday, April 21, 1:45 PM – 2:45PM, SAS 4201

Speaker: Fabio Ancona, University of Padova
Title: TBA
Abstract: TBA