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se completan las sesiones hasta Análisis Funcional y Geometría incluida

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- nombre: Juan Pablo Agnelli
mail: agnelli@famaf.unc.edu.ar
charlas:
- titulo: ''
abstract: ''
- titulo: Synergistic multi-spectral CT reconstruction with directional total variation
abstract: This work considers synergistic multi-spectral CT reconstruction where information from all available energy channels is combined to improve the reconstruction of each individual channel. We propose to fuse these available data (represented by a single sinogram) to obtain a polyenergetic image that keeps structural information shared by the energy channels with an increased signal-to-noise ratio. This new image is used as prior information during a channel-by-channel minimization process through the directional total variation. We analyze the use of directional total variation within variational regularization and iterative regularization. Our numerical results on simulated and experimental data show improvements in terms of image quality and in computational speed.
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- titulo: ''
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speaker: Evelyn Cueva (Escuela Politécnica Nacional, Ecuador)
- titulo: Bayesian Approach Helmholtz Inverse Problem
abstract: This work present an implementation for solving the Helmholtz inverse problem via Finite Element Methods and Bayesian inference theory. The solution of the inverse problem is a posterior distribution based on a number observation data sets, the forward problem governing the state solution, and a statistical prior distribution which describes the behavior of the uncertainty linked to the parameters.
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- titulo: ''
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speaker: Lilí Guadarrama (Centro de Investigación en Matemáticas, México)
- titulo: Partial data Photoacoustic Tomography in unbounded domains
abstract: |
Photoacoustic Tomography is a promising hybrid medical imaging modality that is able to generate high-resolution and high-contrast images, by exploiting the coupling of electromagnetic pulses (in the visible region) and ultrasound waves via de photoacoustic effect.
In this talk I will introduce the modality and focus on the ultrasound propagation part which is mathematically modeled as an inverse initial source problem for the wave equation. We will pose this problem in a setting consisting of an unbounded domain with boundary, with observation taking place in a subset of it. We will address the question of uniqueness, stability and reconstruction of the initial source of acoustic waves.
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- titulo: ''
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speaker: Benjamín Palacios (Pontificia Universidad Católica de Chile, Chile)
- titulo: Particle Source Estimation via Analytical Formulations of the Adjoint Flux
abstract: Numerous applications of research interest may benefit from the study and development of particle sources estimation methods, such as non-destructive material identification and nuclear safety. In such problems, we are concerned with estimating the spatial distribution and intensity of internal sources of neutral particles. In this context, we use the adjoint to the transport operator to derive a linear model that relates the absorption rate measurements of a series of internal particle detectors with the coefficients of the source expansion on a given basis. We consider both one-dimensional energy-dependent and two-dimensional (\(XY\)-geometry) monoenergetic transport problems. We use the adjoint version of the Analytical Discrete Ordinates (ADO) method to write spatially explicit solutions for the adjoint flux for one-dimensional problems. We consider a nodal version of the adjoint ADO method to writing similar explicit expressions for the \(x\) and \(y\) averaged adjoint fluxes for the two-dimensional case. In both cases, we use explicit expressions to derive closed-form formulas to the absorption rate. Then, we apply the iterated Tikhonov algorithm to estimate the sources from noisy measurements and a Bayesian approach through the Metropolis-Hastings algorithm. We successfully estimated one-dimensional polynomial and piecewise constant localized energy dependent sources and two-dimensional piecewise constant localized sources from noise measurements. In either case, the use of closed-form expressions allowed a relevant reduction in computational time.
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speaker: Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul, Brasil)
- titulo: Inverse scattering for the Jacobi system using input data containing transmission eigenvalues
abstract: |
The Jacobi system with the Dirichlet boundary condition is considered on a half-line lattice with real-valued coefficients including weight factors. The inverse problem of recovery of the coefficients is analyzed by using certain input data sets containing the transmission eigenvalues.
This is joint work with T. Aktosun of University of Texas at Arlington and V. G. Papanicolaou of National Technical University of Athens.
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speaker: Abdon Choque Rivero (Universidad Michoacana de San Nicolás de Hidalgo, México)
- titulo: Identificación del coeficiente de velocidad local en un modelo bi-direccional para la propagación de ondas en la superficie de un canal con fondo variable
abstract: |
Un problema inverso es aquel donde se requiere determinar el valor de algunos parámetros de un modelo a partir de datos observados. El estudio de este tipo de problemas es de mucha importancia y aparece en diversas ramas de la ciencia y de las Matemáticas. En esta charla consideramos el problema inverso de identificación del coeficiente de velocidad lineal local en un sistema de tipo Boussinesq linealizado de tipo dispersivo deducido por Quintero y Muñoz (2004), a partir de las mediciones \(V_T(x), N_T(x)\) en un tiempo final \(T\), de la velocidad de las partículas en una profundidad fija y la amplitud de la onda, que se propaga en la superficie de un canal raso con fondo variable.
El problema inverso es reformulado como un problema de optimización con restricciones, donde la funcional objetivo es de tipo usado en métodos de mínimos cuadrados junto con una regularización del tipo introducido por Tikhonov (1963). El proceso de minimización del funcional Tikhonov adoptado se realiza utilizando el método iterativo quasi-Newton L-BFGS-B, introducido por Byrd et al. (1995). Se presentan simulaciones numéricas para mostrar la convergencia y robustez del método numérico propuesto, a\'un en el caso en que el coeficiente posee discontinuidades, o existe ruido en la medici\'on de los datos de entrada \(V_T(x), N_T(x)\).
Investigación realizada con el apoyo de la Universidad del Valle mediante proyecto C.I. 71235.
Referencias:
Tikhonov, A.N. (1963): Solution of incorrectly formulated problems and the regularization method. Soviet Math. Dokl. 4, 1035-1038.
Byrd, R.H., Lu, P., Nocedal, J. (1995): A limited memory algorithm for Bound Constrained Optimization. SIAM J. Sci. Stat. Comp. 16 (5): 1190-1208.
Quintero, J.R. , Muñoz, J.C. (2004): Existence and uniqueness for a Boussinesq system with a disordered forcing. Methods Appl. Anal., 11 (1): 015-032.
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speaker: Juan Carlos Muñoz Grajales (Universidad del Valle, Colombia)
- titulo: Joint curvature-driven diffusion and weighted anisotropic total variation regularization for local inpainting
abstract: An image inpainting problem consists of restoring an image from a (possibly noisy) observation, in which data from one or more regions is missing. Several inpainting models to perform this task have been developed, and although some of them perform reasonably well in certain types of images, quite a few issues are yet to be sorted out. For instance, if the original image is expected to be smooth, inpainting can be performed with reasonably good results by modeling the solution as the result of a diffusion process using the heat equation. For non-smooth images, however, such an approach is far from being satisfactory. On the other hand, Total Variation (TV) inpainting models based on high order PDE diffusion equations can be used whenever edge restoration is a priority. More recently, the introduction of spatially variant conductivity coefficients on these models, such as in the case of Curvature-Driven Diffusions (CDD), has allowed inpainted images with well defined edges and enhanced object connectivity. The CDD approach, nonetheless, is not suitable wherever the image is smooth, as it tends to produce piecewise constant solutions. Based upon this, we propose using CDD to gather a-priori information used at a second step in a weighted anisotropic mixed Tikhonov plus total- variation model that allows for both edge preservation and object connectivity while precluding the staircasing effect that all pure TV-based methods entail. Comparisons between the results of the implemented models will be illustrated by several computed examples, along with performance measures.
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speaker: Ruben D. Spies (Instituto de Matemática Aplicada del Litoral, Argentina)
- titulo: A Splitting Strategy for the Calibration of Jump-Diffusion Models
abstract: We present a splitting strategy to identify simultaneously the local-volatility surface and the jump-size distribution from quoted European prices. The underlying is a jump-diffusion driven asset with time and price dependent volatility. Our approach uses a forward partial-integro-differential equation for the option prices to produce a parameter-to-solution map. The corresponding inverse problem is then solved by Tikhonov-type regularization combined with a splitting strategy.
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speaker: Vinicus V. Albani (Universidade Federal de Santa Catarina, Brasil)
- titulo: Lipschitz Stability for the Backward Heat Equation and its application to Lightsheet Fluorescence Microscopy
abstract: |
In this talk we will present a result about Lipschitz stability for the reconstruction of a compactly supported initial temperature for the heat equation in \(\mathbb{R}^n\), from measurements along a positive time interval and over an open set containing its support. We apply these results to deduce a Lipschitz stability inequality for the backwards heat equation problem in \(\mathbb{R}\) with measurements on a curve contained in \( \mathbb{R}\times[0,T]\), leading to stability estimates for an inverse problem arising in 2D Fluorescence Microscopy.
This is joint work with Pablo Arratia, Evelyn Cueva, Axel Osses and Benjamin Palacios.
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speaker: Liliane Basso Barichello (Universidade Federal do Rio Grande do Sul, Brasil)
- titulo: Anatomical atlas of the upper part of the human head for electroencephalography and bioimpedance applications
abstract: |
Electrophysiology is the branch of physiology that investigates the electrical properties of biological tissues. Volume conductor problems in cerebral electrophysiology and bioimpedance do not have analytical solutions for nontrivial geometries and require a 3D model of the head and its electrical properties for solving the associated PDEs numerically.
Ideally, the model should be made with patient-specific information. In clinical practice, this is not always the case and an average head model is often used. Also, the electrical properties of the tissues might not be completely known due to natural variability.
A 4D (3D+T) statistical anatomical atlas of the electrical properties of the upper part of the human head for cerebral electrophysiology and bioimpedance applications is presented. The atlas is constructed based on MRI images of human individuals and comprises the electrical properties of the main internal structures and can be adjusted for specific electrical frequencies. The atlas also comprises a time-varying model of arterial brain circulation, based on the solution of the Navier-Stokes equations.
The atlas is an important tool for in silico studies on cerebral circulation and electrophysiology that require statistically consistent data, e.g., machine learning, sensitivity analyses, and as a benchmark to test inverse problem solvers. The atlas can also be used as statistical prior information for inverse problems in electrophysiology.
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speaker: Moura da Silva (Universidade Federal do ABC, Brasil y University of Helsinki, Finlandia)
- sesion: Estadística Matemática
organizadores:
- nombre: Florencia Leonardi
@ -279,38 +314,48 @@
- nombre: Daniela Rodriguez
mail: drodrig@dm.uba.ar
charlas:
- titulo: ''
abstract: ''
- titulo: Differentially private inference via noisy optimization
abstract: We propose a general optimization-based framework for computing differentially private M-estimators and a new method for the construction of differentially private confidence regions. Firstly, we show that robust statistics can be used in conjunction with noisy gradient descent and noisy Newton methods in order to obtain optimal private estimators with global linear or quadratic convergence, respectively. We establish global convergence guarantees, under both local strong convexity and self-concordance, showing that our private estimators converge with high probability to a neighborhood of the non-private M-estimators. The radius of this neighborhood is nearly optimal in the sense it corresponds to the statistical minimax cost of differential privacy up to a logarithmic term. Secondly, we tackle the problem of parametric inference by constructing differentially private estimators of the asymptotic variance of our private M-estimators. This naturally leads to the use of approximate pivotal statistics for the construction of confidence regions and hypothesis testing. We demonstrate the effectiveness of a bias correction that leads to enhanced small-sample empirical performance in simulations.
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speaker: Marco Avella Medina (Columbia University, Estados Unidos)
- titulo: Adaptive regression with Brownian path covariate
abstract: In this talk, we will study how to obtain optimal estimators in problems of non-parametric estimation. More specifically, we will present the Goldenshluger-Lepski (2011) method that allows one to obtain estimators that adapt to the smoothness of the function to be estimated. We will show how to extend this statistical procedure in regression with functional data when the regressor variable is a Wiener process \(W\). Using the Wiener-Ito decomposition of m(W), where \(m\) is the regression function, we will define a family of estimators that satisfy an oracle inequality, are proved to be adaptive and converge at polynomial rates over specific classes of functions.
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- titulo: ''
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speaker: Karine Bertin (Universidad de Valparaíso, Chile)
- titulo: 'Adjusting ROC curves for covariates: a robust approach'
abstract: |
ROC curves are a popular tool to describe the discriminating power of a binary classifier based on a continuous marker as the threshold is varied. They become an interesting strategy to evaluate how well an assignment rule based on a diagnostic test distinguishes one population from the other. Under certain circumstances the marker's discriminatory ability may be affected by certain covariates. In this situation, it seems sensible to include this information in the ROC analysis. This task can be accomplished either by the induced or the direct method.
In this talk we will focus on ROC curves in presence of covariates. We will show the impact of outliers on the conditional ROC curves and we will introduce a robust proposal. We follow a semiparametric approach where we combine robust parametric estimators with weighted empirical distribution estimators based on an adaptive procedure that downweights outliers.
We will discuss some aspects concerning consistency and through a Monte Carlo study we will compare the performance of the proposed estimators with the classical ones both, in clean and contaminated samples.
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- titulo: ''
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speaker: Ana M. Bianco (Universidad de Buenos Aires, Argentina)
- titulo: Least trimmed squares estimators for functional principal component analysis
abstract: 'Classical functional principal component analysis can yield erroneous approximations in presence of outliers. To reduce the influence of atypical data we propose two methods based on trimming: a multivariate least trimmed squares (LTS) estimator and its coordinatewise variant. The multivariate LTS minimizes the multivariate scale corresponding to \(h-\)subsets of curves while the coordinatewise version uses univariate LTS scale estimators. Consider a general setup in which observations are realizations of a random element on a separable Hilbert space \(\mathcal{H}\). For a fixed dimension \(q\), we aim to robustly estimate the \(q\) dimensional linear space in \(\mathcal{H}\) that gives the best approximation to the functional data. Our estimators use smoothing to first represent irregularly spaced curves in a high-dimensional space and then calculate the LTS solution on these multivariate data. The solution of the multivariate data is subsequently mapped back onto \(\mathcal{H}\). Poorly fitted observations can therefore be flagged as outliers. Simulations and real data applications show that our estimators yield competitive results when compared to existing methods when a minority of observations is contaminated. When a majority of the curves is contaminated at some positions along its trajectory coordinatewise methods like Coordinatewise LTS are preferred over multivariate LTS and other multivariate methods since they break down in this case.'
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speaker: Holger Cevallos-Valdiviezo (Escuela Superior Politécnica del Litoral, Ecuador y Ghent University, Bélgica)
- titulo: A non-asymptotic analysis of certain high-dimensional estimators for the mean
abstract: Recent work in Statistics and Computer Science has considered the following problem. Given a distribution \(P\) over \({\bf R}^d\) and a fixed sample size \(n\), how well can one estimate the mean \(\mu = \bf E_{X\sim P} X\) from a sample \(X_1,\dots,X_n\stackrel{i.i.d.}{\sim}P\) while only requiring finite second moments and allowing for sample contamination? It turns out that the best estimators are not related to the sample mean. In this talk we present a new analysis of certain approaches to this problem, and reproduce or improve previous results by several authors.
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speaker: Roberto Imbuzeiro Oliveira (Instituto de Matemática Pura e Aplicada, Brasil)
- titulo: Stick-breaking priors via dependent length variables
abstract: In this talk, we present new classes of Bayesian nonparametric prior distributions. By allowing length random variables, in stick-breaking constructions, to be exchangeable or Markovian, appealing models for discrete random probability measures appear. As a result, by tuning the stochastic dependence in such length variables allows to recover extreme families of random probability measures, i.e. Dirichlet and Geometric processes. As a byproduct, the ordering of the weights, in the species sampling representation, can be controlled and thus tuned for efficient MCMC implementations in density estimation or unsupervised classification problems. Various theoretical properties and illustrations will be presented.
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speaker: Ramsés Mena Chávez (Universidad Nacional Autónoma de México, México)
- titulo: 'Modelling in pandemic times: using smart watch data for early detection of COVID-19'
abstract: |
The COVID-19 pandemic brought many challengers to statisticians and modelers across all quantitative disciplines. From accelerated clinical trials to modelling of epidemiological interventions at a feverish pace, to the study of the impact of the pandemic in mental health outcomes and racial disparities. In this talk, we would like to share our experience using classical statistical modelling and machine learning to use data obtained from wearable devices as digital biomarkers of COVID-19 infection.
Early in the pandemic, health care workers in the Mount Sinai Health System (New York city) were prospectively followed in an observational study using the custom Warrior Watch Study app, to collect weekly information about stress, symptoms and COVID-19 infection. Participants wore an Apple Watch for the duration of the study, measuring heart rate variability (HRV), a digital biomarker previously associated with infection in other settings, throughout the follow-up period.
The HRV data collected through the Apple Watch was characterized by a circadian pattern, with sparse sampling over a 24-hour period, and non-uniform timing across days and participants. These characteristics preclude us from using easily derived features (ie mean, maximum, CV etc) with Machine learning methods to develop a diagnostic tool. As such, suitable modelling of the non-uniform, sparsely sampled circadian rhythm data derived from wearable devices are an important step to advance the use of integrated wearable data for prediction of health outcomes.
To circumvent such limitations, we introduced the mixed-effects COSINOR model, where the daily circadian rhythm is express as a non-linear function with three rhythm characteristics: the rhythm-adjusted mean (MESOR), half the extent of variation within a cycle (amplitude), and an angle relating to the time at which peak values recur in each cycle (acrophase). The longitudinal changes in the circadian patterns can then be evaluated extending the COSINOR model to a mixed-effect model framework, allowing for random effects and interaction between COSINOR parameters and time-varying covariates. In this talk, we will discuss our model framework, boostrapped-based hypothesis testing and prediction approaches, as well as our evaluation of HRV measures as early biomarkers of COVID-19 diagnosis. To facilitate the future use of the mixed-effect COSINOR model, we implemented in an R package cosinoRmixedeffects.
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speaker: Mayte Suarez-Farinas (Icahn School of Medicine at Mount Sinai, Estados Unidos)
- sesion: Teorı́a de bordismo y acciones de grupos finitos
organizadores:
- nombre: Andrés Ángel
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- nombre: Carlos Segovia
mail: csegovia@matem.unam.mx
charlas:
- titulo: ''
abstract: ''
- titulo: On the evenness conjecture for equivariant unitary bordism
abstract: The evenness conjecture for equivariant unitary bordism states that these homology groups are free modules of even degree with respect to the unitary bordism ring. This conjecture is known to be true for all finite abelian groups and some semidirect products of abelian groups. In this talk I will talk about some progress done with the group of quaternions of order 8, together with an approach of Eric Samperton and Carlos Segovia to construct a free action on a surface which does not bound equivariantly. This last construction might provide a counterexample of the evenness conjecture in dimension 2.
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speaker: Bernardo Uribe (Universidad del Norte, Colombia)
- titulo: Extending free group action on surfaces
abstract: We are interested in the question of when a free action of a finite group on a closed oriented surface extends to a non-necessarily free action on a 3-manifold. We show the answer to this question is affirmative for abelian, dihedral, symmetric and alternating groups. We present also the proof for finite Coxeter groups.
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- titulo: ''
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speaker: Carlos Segovia (Universidad Nacional Autónoma de México, México)
- titulo: When does a free action of a finite group on a surface extend to a (possibly non-free) action on a 3-manifold?
abstract: Dominguez and Segovia recently asked the question in the title, and showed that the answer is “always” for many examples of finite groups, including symmetric groups, alternating groups and abelian. Surprisingly, in joint work with Segovia, we have found the first examples of finite groups that admit free actions on surfaces that do NOT extend to actions on 3-manifolds. Even more surprisingly, these groups are already known in algebraic geometry as counterexamples to the Noether conjecture over the complex numbers. In this talk, I will explain how to find these groups, and, more generally, how to decide algorithmically if any fixed finite group admits a non-extending action.
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- titulo: ''
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speaker: Eric G. Samperton (University of Illinois, Estados Unidos)
- titulo: The Loch Ness Monster as Homology Covers
abstract: The Loch Ness Monster (LNM) is, up to homeomorphisms, the unique orientable, second countable, connected and Hausdorff surface of infinite genus and exactly one end. In this talk I would like to discuss some properties on the LNM. In particular, we note that LNM is the homology cover of most of the surfaces and also it is the derived cover of uniformizations of Riemann surfaces (with some few exceptions). This is a joint work, in progress, with Ara Basmajian.
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speaker: Rubén A. Hidalgo (Universidad de la Frontera, Chile)
- titulo: Diffeomorphisms of reducible three manifolds and bordisms of group actions on torus
abstract: I first talk about the joint work with K. Mann on certain rigidity results on group actions on torus. In particular, we show that if the torus action on itself extends to a \(C^0\) action on a three manifold \(M\) that bounds the torus, then \(M\) is homeomorphic to the solid torus. This also leads to the first example of a smooth action on the torus via diffeomorphisms that isotopic to the identity that is nontrivial in the bordisms of group actions. Time permitting, I will also talk about certain finiteness results about classifying space of reducible three manifolds which came out of the cohomological aspect of the above project.
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speaker: Sam Nariman (Purdue University, Estados Unidos)
- titulo: Stolz' Positive scalar curvature surgery exact sequence and low dimensional group homology
abstract: We will show how positive knowledge about the Baum-Connes Conjecture for a group, together with a Pontrjagyn character and knowledge about the conjugacy classes of finite subgroups and their low dimensional homology provide an estimation of the degree of non rigidity of positive scalar curvature metrics of spin high dimensional manifolds with the given fundamental group.
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speaker: Noé Bárcenas (Universidad Nacional Autónoma de México, México)
- titulo: Signature of manifold with single fixed point set of an abelian normal group
abstract: In this talk we will report progress made in the task of reducing the computation of the signature of a (possibly non compact) oriented smooth dimensional manifold with an orientation preserving co-compact smooth proper action of a discrete group with a single non empty fixed point submanifold of fixed dimension with respect to the action of a (finite) abelian normal subgroup. The aim of this reduction is to give a formula for the signature in terms of the signature of a manifold with free action and the signature of a disc neighborhood of its fixed-point set with quasi-free action. A necessary task in this reduction is the generalization of Novikov’s additivity to this context.
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speaker: Quitzeh Morales (Universidad Pedagógica de Oaxaca, México)
- titulo: \(Z_p\)-bordism and the mod(\(p\))-Borsuk-Ulam Theorem
abstract: |
Crabb-Gonçalves-Libardi-Pergher classified for given integers \(m,n, \geq 1\) the bordism class of a closed smooth \(m\)-manifold \(X^m\) with a free smooth involution \(\tau\) with respect to the validity of the Borsuk-Ulam property that for every continuous map \(\varphi: X^m \to R^n\), there exists a point \(x \in X^m\) such that \(\varphi(x)= \varphi(\tau(x))\).
In this work together with Barbaresco-de Mattos-dos Santos-da Silva, we are considering the same problem for free \(Z_p\) action.
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speaker: Alice Kimie Miwa Libardi (Universidad Estatal Paulista, Brasil)
- sesion: Ecuaciones de evolución no-lineales y su Dinámica
organizadores:
- nombre: Jaime Angulo Pava
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- nombre: Claudio Muñoz
mail: cmunoz@dim.uchile.cl
charlas:
- titulo: ''
abstract: ''
- titulo: Regularity criteria for weak solutions to the three-dimensional MHD system
abstract: |
In this talk we will first review various known regularity criteria and partial regularity theory for 3D incompressible Navier-Stokes equations. I will then present two generalizations of partial regularity theory of Caffarelli, Kohn and Nirenberg to the weak solutions of MHD equations. The first one is based on the framework of parabolic Morrey spaces. We will show parabolic Hölder regularity for the "suitable weak solutions" to the MHD system in small neighborhoods. This type of parabolic generalization using Morrey spaces appears to be crucial when studying the role of the pressure in the regularity theory and makes it possible to weaken the hypotheses on the pressure. The second one is a regularity result relying on the notion of "dissipative solutions". By making use of the first result, we will show the regularity of the dissipative solutions to the MHD system with a weaker hypothesis on the pressure.
This is a joint work with Diego Chamorro (Université Paris-Saclay, site Evry).
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speaker: Jiao He (Université Paris-Saclay, Francia)
- titulo: On the infinite energy weak solutions for the MHD equations
abstract: |
In this talk, we will consider the magneto-hydrodynamics (MHD) equations placed on the whole three-dimensional space. These equations write down as a nonlinear coupled system of two Navier-Stokes type equations, where the unknowns are the velocity field, the magnetic field and the pressure term. In the first part of this talk, within the framework of a kind of weighted L^2 spaces, we expose some new energy controls which allow us to prove the existence of global in time weak solutions. These solutions are also called the infinite energy weak solutions, in contrast with the classical theory of finite energy weak solutions in the L^2 space. The uniqueness of both finite energy and infinite energy weak solutions remains a very challenging open question. Thus, in the second part of this talk, we study a a priori condition, known as weak-strong uniqueness criterion, to ensure the uniqueness of the infinite energy weak solutions. This result is given in a fairly general multipliers type space, which contains some well-known functional spaces previously used to prove some weak-strong uniqueness criteria.
This is a joint work with Pedro Fernandez-Dalgo (Université Paris-Saclay).
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- titulo: ''
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speaker: Oscar Jarrín (Universidad de las Américas, Ecuador)
- titulo: Long-time asymptotics for a damped Navier-Stokes-Bardina model
abstract: We consider finite energy solutions for a damped Navier-Stokes-Bardina’s model and show that the corresponding dynamical system possesses a global attractor. We obtain upper bounds for its fractal dimension. Furthermore, we examine the long-term behavior of solutions of the damped Navier-Stokes-Bardina’s equation in the energy space.
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speaker: Fernando Cortez (Escuela Politécnica Nacional, Ecuador)
- titulo: Compact embeddings of p-Sobolev-like cones of nuclear operators
abstract: We prove that a cone of nuclear operators, whose eigenfunctions belong to a p-Sobolev space and that have finite total energy, is compactly embedded in the trace norm. This result is analogous to the classical Sobolev embeddings but at operators level. In the path we prove regularity properties for the density function of any operator living in the cone. Also, departing from Lieb-Thirring type conditions, we obtain some Gagliardo-Nirenberg inequalities. By using the compactness property, several free-energy functionals for operators are shown to have a minimizer. The entropy term of these free-energy functionals is generated by a Casimir-class function related to the eigenvalue problem of the Schrödinger operator.
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speaker: Juan Mayorga (Yachay Tech, Ecuador)
- titulo: Scattering for quadratic-type Schrödinger systems in dimension five without mass-resonance
abstract: In this talk we will discuss the scattering of non-radial solutions in the energy space to coupled system of nonlinear Schrödinger equations with quadratic-type growth interactions in dimension five without the mass-resonance condition. Our approach is based on the recent ideas introduced by Dodson and Murphy, which relies on an interaction Morawetz estimate.
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speaker: Ademir Pastor (Universidade Estadual de Campinas, Brasil)
- titulo: Stability of smooth periodic traveling waves in the Camassa-Holm equation
abstract: |
Smooth periodic travelling waves in the Camassa–Holm (CH) equation are revisited in this talk. We show that these periodic waves can be characterized in two different ways by using two different Hamiltonian structures. The standard formulation, common to the Korteweg–de Vries (KdV) equation, has several disadvantages, e.g., the period function is not monotone and the quadratic energy form may have two rather than one negative eigenvalues. We explore the nonstandard formulation common to evolution equations of CH type and prove that the period function is monotone and the quadratic energy form has only one simple negative eigenvalue. We deduce a precise condition for the spectral and orbital stability of the smooth periodic travelling waves and show numerically that this condition is satisfied in the open region of three parameters where the smooth periodic waves exist.
This is a joint work with Dmitry E. Pelinovsky (McMaster University), AnnaGeyer (Delft University of Technology) and Renan H. Martins (State University of Maringa).
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speaker: Fábio Natali (Universidade Estadual de Maringá, Brasil)
- titulo: Spectral stability of monotone traveling fronts for reaction diffusion-degenerate Nagumo equations
abstract: This talk addresses the spectral stability of monotone traveling front solutions for reaction diffusion equations where the reaction function is of Nagumo (or bistable) type and with diffusivities which are density dependent and degenerate at zero (one of the equilibrium points of the reaction). Spectral stability is understood as the property that the spectrum of the linearized operator around the wave, acting on an exponentially weighted space, is contained in the complex half plane with non-positive real part. The degenerate fronts studied in this paper travel with positive speed above a threshold value and connect the(diffusion-degenerate) zero state with the unstable equilibrium point of the reaction function. In this case, the degeneracy of the diffusion coefficient is responsible of the loss of hyperbolicity of the asymptotic coefficient matrices of the spectral problem at one of the end points, precluding the application of standard techniques to locate the essential spectrum. This difficulty is overcome with a suitable partition of the spectrum, a generalized convergence of operators technique, the analysis of singular (or Weyl) sequences and the use of energy estimates. The monotonicity of the fronts, as well as detailed descriptions of the decay structure of eigenfunctions on a case by case basis, are key ingredients to show that all traveling fronts under consideration are spectrally stable in a suitably chosen exponentially weighted L2 energy space.
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speaker: Ramón G. Plaza (Universidad Nacional Autónoma de México, México)
- titulo: Blow-up solutions of the intercritical inhomogeneous NLS equation
abstract: |
We consider the inhomogeneous nonlinear Schrödinger (INLS) equation
$$i u_t +\Delta u+|x|^{-b}|u|^{2\sigma} u = 0, \,\,\, x\in \mathbb{R}^N,$$
with \(N\geq 3\) and \(0<b<\min\{\frac{N}{2},2\}\). We focus on the intercritical case, where the scaling invariant Sobolev index \(s_c=\frac{N}{2}-\frac{2-b}{2\sigma}\) satisfies \(0<s_c<1\). In this talk, for initial data in \(\dot H^{s_c}\cap \dot H^1\), we discuss the existence of blow-up solutions and also a lower bound for the blow-up rate in the radial and non-radial settings.
This is a joint work with Mykael Cardoso (Universidade Federal do Piauí, Brasil).
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speaker: Luiz Gustavo Farah (Universidade Federal de Minas Gerais, Brasil).
- titulo: A sufficient condition for asymptotic stability of kinks in general (1+1)-scalar field models
abstract: |
In this talk I will discuss stability properties of kinks for the (1+1)-dimensional nonlinear scalar field theory models \(\partial_t2\phi -\partial_x2\phi + W'(\phi) = 0, \quad (t,x)\in\R\times\R\). The orbital stability of kinks under general assumptions on the potential \(W\) is a consequence of energy arguments. The main result I will present is the derivation of a simple and explicit sufficient condition on the potential \(W\) for the asymptotic stability of a given kink. This condition applies to any static or moving kink, in particular no symmetry assumption is required. Applications of the criterion to the \(P(\phi)_2\) theories and the double sine-Gordon theory will be discussed.
This is a joint work with Y. Martel, C. Muñoz and H. Van Den Bosch.
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speaker: Michal Kowalczyk (Universidad de Chile, Chile)
- titulo: Korteweg de-Vries limit for the Fermi-Pasta-Ulam System
abstract: In this talk, we are going to discuss dispersive properties for the Fermi-Pasta-Ulam (FPU) system with infinitely many oscillators. Precisely, we see that FPU systems are reformulated and their solutions satisfies Strichartz, local smoothing, and maximal function estimates in comparison with linear Korteweg–de Vries (KdV) flows. With these properties, we finally show that the infinite FPU system can be approximated by counter-propagating waves governed by the KdV equation as the lattice spacing approaches zero.
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speaker: Chulkwang Kwak (Ehwa Womans University, Corea del Sur)
- titulo: TBA
abstract: TBA
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speaker: Hanne Van Den Bosch (Universidad de Chile, Chile)
- sesion: Combinatoria
organizadores:
- nombre: Celina Miraglia Herrera de Figueiredo
@ -404,38 +494,93 @@
- nombre: Lucia Moura
mail: lmoura@uottawa.ca
charlas:
- titulo: ''
abstract: ''
- titulo: Complete colorings on circulant graphs and digraphs
abstract: |
A <em>complete</em> \(k\)-vertex-coloring of a graph \(G\) is a vertex-coloring of \(G\) using \(k\) colors such that for every pair of colors there is at least two incident vertices in \(G\) colored with this pair of colors. The \emph{chromatic} \(\chi(G)\) and \emph{achromatic} \(\alpha(G)\) numbers of \(G\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(G\), therefore \(\chi(G)\leq\alpha(G)\).
The dichromatic number and the diachromatic number, generalice the concepts of chromatic number and achromatic number. An <em>acyclic</em> \(k\)-vertex-coloring of a digraph \(D\) is vertex coloring using \(k\) colors such that \(D\) has no monochromatic cycles and a \emph{complete} \(k\)-vertex-coloring of a digraph \(D\) is a vertex coloring using \(k\) colors such that for every ordered pair \((i,j)\) of different colors, there is at least one arc \((u,v)\) such that \(u\) has color \(i\) and \(v\) has color \(j\). The dichromatic number \(dc(D)\) and diachromatic number \(dac(D)\) of \(D\) are the smallest and the largest number of colors in a complete proper \(k\)-vertex-coloring of \(D\), therefore \(dc(D)\leq dac(D)\).
We determine the achromatic and diachromatic numbers of some specific circulant graphs and digraphs and give general bounds for these two parameters on these graphs and digraphs. Also, we determine the achromatic index for circulant graphs of order \(q^2+q+1\) using projective planes.
Joint work with: Juan José Montellano-Ballesteros, Mika Olsen and Christian Rubio-Montiel.
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- titulo: ''
abstract: ''
speaker: Gabriela Araujo-Pardo (Universidad Nacional Autónoma de México, México)
- titulo: Fault Localization via Combinatorial Testing
abstract: |
In this talk, we explore combinatorial arrays that are useful in identifying faulty interactions in complex engineered systems. A {\sl separating hash family} ${\sf SHF}_\lambda(N; k,v,\{w_1,\dots,w_s\})$ is an $N \times k$ array on $v$ symbols, with the property that no matter how disjoint sets $C_1, \dots, C_s$ of columns with $|C_i| = w_i$ are chosen, there are at least $\lambda$ rows in which, for every $1 \leq i < j \leq s$, no entry in a column of $C_i$ equals that in a column of $C_j$. (That is, there are $\lambda$ rows in which sets $\{C_1,\dots,C_s\}$ are {\sl separated}.) Separating hash families have numerous applications in combinatorial cryptography and in the construction of various combinatorial arrays; typically, one only considers whether two symbols are the same or different. We instead employ symbols that have algebraic significance.
We consider an ${\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})$ whose symbols are column vectors from ${\mathbb F}_q^s$. The entry in row $r$ and column $c$ of the ${\sf SHF}$ is denoted by ${\bf v}_{r,c}$. Suppose that $C_1, \dots, C_s$ is a set of disjoint sets of columns. Row $r$ is {\sl covering} for $\{C_1, \dots, C_s\}$ if, whenever we choose $s$ columns $\{ \gamma_i \in C_i : 1 \leq i \leq s\}$, the $s \times s$ matrix $[ {\bf v}_{r,\gamma_1} \cdots {\bf v}_{r,\gamma_s} ]$ is nonsingular over ${\mathbb F}_q$. Then the ${\sf SHF}_\lambda(N; k,q^s,\{w_1,\dots,w_s\})$ is {\sl covering} if, for every way to choose $\{C_1, \dots, C_s\}$, there are at least $\lambda$ covering rows.
We establish that covering separating hash families of type $1^t d^1$ give an effective construction for detecting arrays, which are useful in screening complex systems to find interactions among $t$ or fewer factors without being masked by $d$ or fewer other interactions. This connection easily accommodates outlier and missing responses in the screening. We explore asymptotic existence results and explicit constructions using finite geometries for covering separating hash families. We develop randomized and derandomized construction algorithms and discuss consequences for detecting arrays. This is joint work with Violet R. Syrotiuk (ASU).
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- titulo: ''
abstract: ''
speaker: Charles J. Colbourn (Arizona State University, Estados Unidos)
- titulo: On the \(\Delta\)-interval and \(\Delta\)-convexity numbers of graphs and graph products
abstract: |
Given a graph \(G\) and a set \(S \subseteq V(G)\), the \(\Delta\)-interval of \(S\), \([S]_{\Delta}\), is the set formed by the vertices of \(S\) and every \(w \in V(G)\) forming a triangle with two vertices of \(S\). If \([S]_\Delta = S\), then \(S\) is \(\Delta\)-convex of \(G\); if \([S]_{\Delta} = V(G)\), then \(S\) is a \(\Delta\)-interval set of \(G\). The \(\Delta\)-interval number of \(G\) is the minimum cardinality of a \(\Delta\)-interval set and the \(\Delta\)-convexity number of \(G\) is the maximum cardinality of a proper \(\Delta\)-convex subset of \(V(G)\). In this work, we show that the problem of computing the \(\Delta\)-convexity number is {\sf W}[1]-hard and {\sf NP}-hard to approximate within a factor \(O(n^{1-\varepsilon})\) for any constant \(\varepsilon > 0\) even for graphs with diameter \(2\) and that the problem of computing the \(\Delta\)-interval number is {\sf NP}-complete for general graphs. For the positive side, we present characterizations that lead to polynomial-time algorithms for computing the \(\Delta\)-convexity number of chordal graphs and for computing the \(\Delta\)-interval number of block graphs. We also present results on the \(\Delta\)-hull, \(\Delta\)-interval and \(\Delta\)-convexity numbers concerning the three standard graph products, namely, the Cartesian, strong and lexicographic products, in function of these and well-studied parameters of the operands.
Joint work with: Bijo S. Anand, Prasanth G. Narasimha-Shenoi, and Sabeer S. Ramla.
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- titulo: ''
abstract: ''
speaker: Mitre Dourado (Universidade Federal do Rio de Janeiro, Brasil)
- titulo: Unavoidable patterns
abstract: Ramsey's theorem states that, for any integer \(t\), if \(n\) is a sufficiently large integer, then every \(2\)-edge-coloring of a complete graph \(K_n\) contains a monochromatic \(K_t\). On the other hand, Turán's theorem says that, if a graph has sufficiently many edges, then it contains a \(K_t\) as a subgraph. In relation to these two types of problems, we study which \(2\)-colored patterns are forced to appear in differently saturated \(2\)-edge-colorings of the complete graph \(K_n\) provided \(n\) is large enough. This is a joint work with Yair Caro and Amanda Montejano.
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speaker: Adriana Hansberg (Universidad Nacional Autónoma de México, México)
- titulo: Approximating graph eigenvalues using tree decompositions
abstract: |
Spectral graph theory aims to extract structural information about graphs from spectra of different types of matrices associated with them, such as the adjacency matrix, Laplacian matrices, and many others. A basic step in any such application consists of computing or approximating the spectrum or some prescribed subset of eigenvalues. One strategy for this is to compute diagonal matrices that are congruent to the original matrices, which can be done quite efficiently if we have a graph decompositions such as the tree decomposition with bounded width. To be precise, let \(M=(m_{ij})\) be a symmetric matrix of order \(n\) whose elements lie in an arbitrary field \(\mathbb{F}\), and let \(G\) be the graph with vertex set \(\{1,\ldots,n\}\) such that distinct vertices \(i\) and \(j\) are adjacent if and only if \(m_{ij} \neq 0\). We present an algorithm that finds a diagonal matrix that is congruent to \(M\) in time \(O(k^2 n)\), provided that we are given a suitable tree decomposition of \(G\) with width \(k\). In addition to the above applications, this allows one to compute the determinant and to find the rank of a symmetric matrix in time \(O(k^2 n)\).
This is joint work with M. FŸrer (Pennsylvannia State University) and Vilmar Trevisan (Universidade Federal do Rio Grande do Sul).
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speaker: Carlos Hoppen (Universidade Federal do Rio Grande do Sul, Brasil)
- titulo: Fault tolerance in cryptography using cover-free families
abstract: |
Cover-free families (CFFs) have been investigated under different names and as a solution to many problems related to combinatorial group testing. A \(d\)-cover-free family \(d\)-CFF\((t, n)\) is a set system with n subsets of a \(t\)-set, where the union of any d subsets does not contain any other. A \(d\)-CFF\((t, n)\) allows for the identification of up to \(d\) defective elements in a set of \(n\) elements by performing only \(t\) tests (typically \(t \ll n\)).
We explore different aspects of cover-free families in order to achieve fault tolerance in cryptography. For instance, while CFFs are used as a solution to many problems in this area, we note that some of those problems require CFFs with increasing \(n\). In this context, we investigate new infinite families and their constructions in order to better approach fault tolerance in cryptography. This is joint work with Lucia Moura.
[1] IDALINO, T. B.; MOURA, L., Nested cover-free families for unbounded fault-tolerant aggregate signatures. Theoretical Computer Science, 854 (2021), 116--130.
[2] IDALINO, T. B.; MOURA, L., Embedding cover-free families and cryptographical applications. Advances in Mathematics of Communications, 13 (2019), 629--643.
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speaker: Thais Bardini Idalino (Universidade Federal de Santa Catarina, Brasil)
- titulo: Perfect sequence covering arrays
abstract: |
An \((n,k,\lambda)\) perfect sequence covering array is a subset of the \(n!\) permutations of the sequence \((1, 2, \dots, n)\) whose elements collectively contain each ordered \(k\)-subsequence exactly \(\lambda\) times. The central question is: for given \(n\) and \(k\), what is the smallest value of \(\lambda\) (denoted \(g(n,k)\)) for which such a configuration exists? We interpret the sequences of a perfect sequence covering array as elements of the symmetric group \(S_n\), and constrain its structure to be a union of cosets of a prescribed subgroup of \(S_n\). By adapting a search algorithm due to Mathon and van Trung for finding spreads, we obtain highly structured examples of perfect sequence covering arrays. In particular, we determine that \(g(6,3) = g(7,3) = g(7,4) = 2\) and that \(g(8,3)\) is either 2 or 3.
This is joint work with Jingzhou Na.
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speaker: Jonathan Jedwab (Simon Fraser University, Canada)
- titulo: The status of the \((r, l)\) well-covered graph sandwich problem
abstract: |
An \((r, l)\)-partition of a graph \(G\) is a partition of its vertex set into \(r\) independent sets and \(l\) cliques. A graph is \((r,l)\) if it admits an \((r, l)\)-partition. A graph is well-covered if every maximal independent set is also maximum. A graph is \((r,l)\)-well-covered if it is both \((r,l)\) and well-covered. We have proved the full classification of the decision problem \((r,l)\)-Well-Covered Graph problem, where we are given a graph \(G\), and the question is whether \(G\) is an \((r, l)\)-well-covered graph. We have shown that this problem is polynomial only in the cases \((0, 1)\), \((0, 2)\), \((1, 0)\), \((1, 1)\), \((1, 2)\), and \((2, 0)\) and hard otherwise. The Sandwich Problem for the Property \(\pi\) has as an instance a pair of graphs \(G1 = (V, E_1)\) and \(G2 = (V, E_2)\) with \(E_1 \subset E_2\), plus the question whether there is a graph \(G=(V,E)\) with \(E_1 \subset E\subset E_2\), such that \(G\) has the property \(\pi\). When a recognition problem for a property \(\pi\) is hard, we can consider the sets \(E_1 = E_2\) to obtain that the sandwich problem for the property \(\pi\) reduces to the recognition and so also hard. Hence, the only cases where the \((r, l)\)-well-covered sandwich problem can be no longer hard are in the \(6\) polynomial cases. In this talk we prove that \((r, l)\)-well-covered sandwich problem is polynomial in the cases that \((r, l) = (0, 1), (1, 0), (1, 1)\) or \((0, 2)\), and NP-complete if we consider the property of being \((1, 2)\)-well-covered.
This work was done with the collaboration of Sancrey Rodrigues Alves (FAETEC), Fernanda Couto (UFRRJ), Luerbio Faria (UERJ), Sylvain Gravier (CNRS), and Uéverton S. Souza (UFF).
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speaker: Sulamita Klein (Universidade Federal do Rio de Janeiro, Brasil)
- titulo: Counting Segment, Rays, Lines in Quasimetric spaces
abstract: In the Euclidean plane, it is a well-known result that a set of n points defines exactly \(n(n-1)/2\) distinct segments, at least \(2(n-1)-1\) different rays and, when the points are not collinear, at least \(n\) different lines. Segments, rays and lines can be define in any quasimetric space by using its quasimetric. In this talk we survey recent results concerning the generalization of the above mentioned properties of the Euclidean plane, specially those of lines, to finite quasimetric spaces, mainly those defined by connected graphs and strongly connected digraphs.
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speaker: Martín Matamala (Universidad de Chile, Chile)
- titulo: Uniformly Optimally-Reliable Graphs
abstract: |
The reliability polynomial of a simple graph G represents the probability that G will remain connected given a fixed probability \(1-p\) of each edge failing. A graph \(G\) is uniformly most reliable if its reliability polynomial is greater than or equal to the reliability polynomial of all other graphs with the same number of nodes and edges for all \(p \in [0, 1]\).
Which is the most-reliable graph with n nodes and m edges?
This problem has several variants, according to the notion of optimality (in a local or uniform sense), failure type (either nodes or edges), or reliability model (all-terminal connectedness, source-terminal or general multi-terminal setting).
This presentation shows a summary of the multiple proposals to attack the problem, together with recent trends and important conjectures proposed decades ago which require further research.
Particularly, we will present recently proved conjectures regarding UMGR (Uniformly Most-Reliable Graphs) as well as new evaluation techniques for this property.
Keywords: uniformly most-reliable graph, uniformly least-reliable graph, all-terminal reliability, source-terminal reliability, failure-type, graph theory.
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speaker: Franco Robledo (Universidad de la Republica, Uruguay)
- titulo: Circularly compatible ones, \(D\)-circularity, and proper circular-arc bigraphs
abstract: In 1969, Tucker characterized proper circular-arc graphs as those graphs whose augmented adjacency matrices have the circularly compatible ones property. Moreover, he also found a polynomial-time algorithm for deciding whether any given augmented adjacency matrix has the circularly compatible ones property. These results led to the first polynomial-time recognition algorithm for proper circular-arc graphs. However, as remarked there, this work did not solve the problems of finding a structure theorem and an efficient recognition algorithm for the circularly compatible ones property in arbitrary matrices (i.e., not restricted to augmented adjacency matrices only). We solve these problems. More precisely, we give a minimal forbidden submatrix characterization for the circularly compatible ones property in arbitrary matrices and a linear-time recognition algorithm for the same property. We derive these results from analogous ones for the related \(D\)-circular property. Interestingly, these results lead to a minimal forbidden induced subgraph characterization and a linear-time recognition algorithm for proper circular-arc bigraphs, solving a problem first posed by Basu et al. [J. Graph Theory, 73 (2013), pp. 361--376]. Our findings generalize some known results about \(D\)-interval hypergraphs and proper interval bigraphs.
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speaker: Martín Safe, (Universidad Nacional del Sur, Argentina)
- titulo: Active clustering
abstract: |
Given a set (known to us), and a partition of this set, unknown to us, consider the following task. We have to determine the partition by making successive queries about pairs of elements, each time querying whether the pair belong to the same class or not. Later queries are allowed to depend on the outcome of earlier queries. There is a natural way to assign a graph to each step of the querying process, by starting with an edgeless graph where each vertex represents an element of the set, and after queries identifying vertices for a yes-answer and adding edges for no-answers. We prove the at first sight surprising result that the partitioning algorithms reaching the minimal average number of queries are exactly those that keep these graphs chordal. (The average is taken over all possible partitions of the base set.) The optimal algorithms even share the same distribution on the number of queries, which we characterize. We also analyse a related model where the number of partition classes is fixed, and each element of the base set chooses its partition class randomly.
This is joint work with Élie de Panafieu, Quentin Lutz and Alex Scott.
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speaker: Maya Stein (Universidad de Chile, Chile)
- sesion: Análisis Numérico
organizadores:
- nombre: Gabriel Acosta
@ -444,6 +589,60 @@
mail: pmorin@fiq.unl.edu.ar
- nombre: Rodolfo Rodríguez
mail: rodolfo@ing-mat.udec.cl
charlas:
- titulo: Aproximación unificada del problema acoplado de Stokes-Darcy
abstract: |
En esta charla nos abocaremos al análisis y la resolución numérica, por elementos finitos mixtos, del problema acoplado de Stokes-Darcy en el plano. Introduciremos una formulación modificada del problema con el propósito de permitir el uso de la misma familia de elementos, en el sector del dominio gobernado por la ecuación de Stokes y en la porción del dominio gobernada por la ecuación de Darcy.
En primer lugar consideraremos el caso de que el dominio sea poligonal, donde presentaremos resultados tanto teóricos como numéricos de la resolución del problema utilizando MINI-elements, los cuales son uno de los más sencillos de implementar. También, aprovechando las ventajas de nuestra formulación, mostraremos la buena performance de la aplicación de otros elementos como los P2-P1 o P1-P1 estabilizados.
Luego trataremos el problema en dominios curvos, incluso con la posibilidad de tener interfase curva, haciendo uso de triángulos curvos. Con este enfoque obtendremos también estimaciones de error de orden óptimo, extendiendo así los resultados obtenidos para el caso poligonal. Por último presentaremos experimentos numéricos que confirman la buena performance del método propuesto.
Trabajo en colaboración con María Lorena Stockdale. Departamento de Matematica, FCEyN, Universidad de Buenos Aires.
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speaker: Gabriela Armentano (Universidad de Buenos Aires, Argentina)
- titulo: 'Transversally-Enriched Pipe Element Method: an efficient approach for computational hemodynamics'
abstract: The development of novel numerical and computational strategies for the simulation of blood flow in complex patient-specific vasculatures has gained strong attention in the scientific community aiming both theoretical and applied research. The need for fast, yet accurate methods is at the core of any translational research enterprise. In this talk, we will introduce a middle fidelity numerical approach to tackle the demanding blood-flow simulation in deformable arteries. This strategy can be placed in-between oversimplified (low-fidelity) reduced 1D models and expensive (high-fidelity) full 3D models. The main feature of the method is the discretization of the pipe-like domain in which the fluid flow problem is addressed. This pipe-oriented discretization is exploited by placing a differentiated polynomial approximation for cross-sectional and longitudinal directions. We will show the basic idea of the method, a numerical study of its convergence properties and applications for patient-specific blood flow modeling in large branching arterial geometries.
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speaker: Pablo Blanco (Laboratório Nacional de Computação Científica, Brasil) joint with Alonso M. Alvarez, Raúl A. Feijóo
- titulo: Local error estimates for nonlocal problems
abstract: |
The integral fractional Laplacian of order \(s \in (0,1)\) is a nonlocal operator. It is known that solutions to the Dirichlet problem involving such an operator exhibit an algebraic boundary singularity regardless of the domain regularity. This, in turn, deteriorates the global regularity of solutions and as a result the global convergence rate of the numerical solutions.
In this talk we shall discuss regularity of solutions on bounded Lipschitz domains. For finite element discretizations, we derive local error estimates in the \(H^s\)-seminorm and show optimal convergence rates in the interior of the domain by only assuming meshes to be shape-regular. These estimates quantify the fact that the reduced approximation error is concentrated near the boundary of the domain. We illustrate our theoretical results with several numerical examples.
The talk is based on joint work with Dmitriy Leykekhman (University of Connecticut) and Ricardo Nochetto (University of Maryland).
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speaker: Juan Pablo Borthagaray (Universidad de la República, Uruguay)
- titulo: A priori and a posteriori error analysis of a momentum conservative mixed-FEM for the stationary Navier-Stokes problem
abstract: |
In this talk we present a new conforming mixed finite element method for the Navier-Stokes problem posed on non-standard Banach spaces, where a pseudostress tensor and the velocity are the main unknowns of the system. The associated Galerkin scheme can be defined by employing Raviart-Thomas elements of degree k for the pseudostress and discontinuous piecewise polynomials of degree k for the velocity. Next, by extending standard techniques commonly used on Hilbert spaces to the case of Banach spaces we derive a reliable and efficient residual-based a posteriori error estimator for the corresponding mixed scheme.
Joint work with S. Caucao (Universidad Católica de la Santísima Concepción, Chile), C. Garcia (Pontificia Universidad Católica de Chile), R. Oyarzúa (Universidad del Bío Bío, Chile) and S. Villa (Universidad del Bío Bío, Chile).
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speaker: Jessika Camaño (Universidad Católica de la Santísima Concepción, Concepción, Chile)
- titulo: A fully discrete adaptive scheme for parabolic equations
abstract: |
We present an adaptive algorithm for solving linear parabolic equations using hierarchical B-splines and the implicit Euler method for the spatial and time discretizations, respectively.
Our development improves upon one from 2018 by Gaspoz and collaborators, where fully discrete adaptive schemes have been analyzed within the framework of classical finite elements. Our approach is based on an a posteriori error estimation that essentially consists of four indicators: a time and a consistency error indicator that dictate the time-step size adaptation, and coarsening and a space error indicator that are used to obtain suitably adapted hierarchical meshes (at different time steps). Even though we use hierarchical B-splines for the space discretization, a straightforward generalization to other methods, such as FEM, is possible. The algorithm is guaranteed to reach the final time within a finite number of operations, and keep the space-time error below a prescribed tolerance. Some numerical tests document the practical performance of the proposed adaptive algorithm.
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speaker: Eduardo Garau (Universidad Nacional del Litoral, Santa Fe, Argentina)
- titulo: LSD for multiscale problems
abstract: |
The Localized Spectral Decomposition finite element method is based on a hybrid formulation of elliptic partial differential equations, that is then transformed via a FETI-like space decomposition. Such decomposition make the formulation embarrassingly parallel and efficient, in particular in the presence of multiscale coefficients. It differs from most of the methods out there since it requires solution's minimum regularity. The major novelty is that we base our decomposition in local spectral problems, resulting in methods that are robust with respect to high contrast coefficients.
This is a joint work with Marcus Sarkis, of WPI (Worcester Polytechnic Institute).
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speaker: Alexandre Madureira (Laboratório Nacional de Computação Científica, Brasil)
- titulo: Virtual Element Spectral Analysis for the Transmission Eigenvalue Problem.
abstract: The aim of this talk is to analyze a C1 Virtual Element Method (VEM) on polygonal meshes for solving a quadratic and non-selfadjoint fourth-order eigenvalue problem derived from the transmission eigenvalue problem. Optimal order error estimates for the eigenfunctions and a double order for the eigenvalues are obtained. Numerical experiments will be provided to verify the theoretical error estimates.
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speaker: David Mora (Universidad del Bio-Bio, Concepción, Chile)
- titulo: Analysis and approximation of fluids under singular forcing
abstract: We present a well-posedness theory for Newtonian and some non-Newtonian fluids under singular forcing in Lipschitz domains and in convex polytopes. The main idea, that allows us to deal with such forces, is that we study the fluid problems in suitably weighted Sobolev spaces. We develop an a priori approximation theory, which requires the development of the stability of the Stokes projection over weighted spaces. We conclude by presenting existence results for a singular Bousinessq system and, in the case that the forcing is a linear combination of Dirac deltas, we also present an a posteriori error estimator.
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speaker: Enrique Otárola (Universidad Técnica Federico Santamaría, Valparaíso, Chile)
- sesion: Análisis Funcional y Geometría
organizadores:
- nombre: Esteban Andruchow
@ -452,6 +651,86 @@
mail: massey@mate.unlp.edu.ar
- nombre: Lázaro Recht
mail: recht@usb.ve
charlas:
- titulo: Mínimos locales de problemas tipo Procusto en la variedad de matrices positivas
abstract: |
Sea\(\mathcal{M}_d(\mathbb{C})\)el espacio de matrices (cuadradas) de dimensión\(d\)y\(\mathcal{X}\subset \mathcal{M}_d(\mathbb{C})\) Consideremos una matriz\(A\in\mathcal{M}_d(\mathbb{C})\)(fija) y una métrica en\(\mathcal{M}_d(\mathbb{C})\)dada por una distancia\(\rm {\textbf d}\)
Un típico problema de aproximación de matrices (o de tipo Procusto) es estudiar la distancia mínima
$$\rm {\textbf d}(A,\mathcal{X}):= \inf\{ \rm {\textbf d}(A,C):\,C \in \mathcal{X}\}\,,$$
y en caso de que se alcance, estudiar el conjunto de mejores aproximantes de\(A\)en\(\mathcal{X}\)
$$\mathcal{A}^{\rm op}(A,\mathcal{X}) =\{C\in\mathcal{X}:\,\rm {\textbf d}(A,C)= \rm {\textbf d}(A,\mathcal{X})\}\,.$$
Algunas de las elecciones clásicas de \(\mathcal{X}\subset \mathcal{M}_d(\mathbb{C})\)son las matrices autoadjuntas, las semidefinidas positivas, los proyectores ortogonales, etc, y la métrica suele ser la inducida por la norma Frobenius, pero también podría provenir de cualquier otra norma, por ejemplo, de alguna que sea unitariamente invariante (nui).
El problema del que nos ocuparemos en esta charla es el siguiente: dada\(N\)una nui (estrictamente convexa) en\(\mathcal{M}_d(\mathbb{C})\) definimos en el cono de matrices positivas\(\mathcal{P}_d(\mathbb{C})\) la distancia
$$ {\bf{d}}_N(A,B):=N(\log(A^{-1/2} B A^{-1/2})) \quad \text{para } A,B\in\mathcal{P}_d(\mathbb{C}). $$
Entonces, si fijamos\(A,B\in \mathcal{P}_d(\mathbb{C})\)podemos considerar
$$\mathcal{X}=\mathcal{O}_B= \{ UBU^* : \, U\quad\text{es unitaria}\}\,.$$
Luego, el problema de Procusto asociado es el de estudiar la distancia
$$ \displaystyle{\bf{d}}_N(A,\mathcal{O}_B) =\inf_{C\in\mathcal{O}_B} {\bf{d}}_N(A,C)$$
y (en caso de ser posible) los mejores aproximantes de\(A\)en\(\mathcal{O}_B\) En 2019, Bhatia y Congedo probaron que esa distancia se alcanza en matrices de\(\mathcal{O}_B\)que conmutan con\(A\) Como\(\mathcal{O}_B\)es un espacio métrico con la métrica inducida por la norma usual de operadores, lo que proponemos en esta charla es estudiar los minimizadores globales la función\(F_{(N,A,B)}= F_N:\mathcal{O}_B \to \mathbb{R}_{>0}\)dada por
$$F_N(C)=N (\log (A^{-1/2}CA^{-1/2}))$$
para \(C\in \mathcal{O}_B\) En particular, vamos a dar una caracterización espectral de los minimizadores locales de\(F_N\)en \(\mathcal{O}_B\)(cuando\(N\)es una nui estrictamente convexa) utilizando técnicas geométricas aplicadas al caso de igualdad en la desigualdad de Lidskii (multiplicativa) y probaremos que los minimizadores locales son globales, independientemente de la nui estrictamente convexa elegida. La charla está basada en un trabajo en co-autoría con Pablo Calderón y Mariano Ruiz.
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speaker: Noelia Belén Rios (Universidad Nacional de La Plata, Argentina)
- titulo: On \(\lambda\)-Rings of Pseudo-differential Operators
abstract: |
The theory of\(\lambda\)Rings goes back to the work of Grothendieck on Chern classes in algebraic topology, it is a suitable axiomatization of the algebraic properties of exterior powers operations on vector bundles;\(\lambda\)rings were also used by Atiyah and coworkers in the study of representations of groups and\(K\)Theory. During this talk we will present recent results on the\(\lambda\)ring structure in algebras of pseudo-differential operators and their use in index theory.
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speaker: Alexander Cardona (Universidad de los Andes, Colombia)
- titulo: El cuarto problema de Hilbert para analistas
abstract: |
El cuarto problema de Hilbert pide construir y estudiar las distancias continuas (aunque no necesariamente simétricas) en abiertos convexos de espacios proyectivos para las cuales las rectas son lineas geodésicas. En dimensión dos, y para distancias simétricas, la solución de Busemann y Pogorelov es de una maravillosa simplicidad. Sin embargo, en dimensiones mayores a dos y para distancias no simétricas podemos seguir considerando el problema como abierto. Después de una breve reseña de la solución bidimensional de Busemann y Pogorelov y de varios resultados parciales en dimensión superior, la charla se centrará en la formulación de problemas relacionados a la transformada de Fourier y a otras transformadas integrales que llevarían a una solución o a un mejor entendimiento del problema de Hilbert.
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speaker: Juan Carlos Alvarez-Paiva (Université Lille 1, Francia)
- titulo: Productos de operadores positivos
abstract: |
La charla se basa en un trabajo en colaboración con Maximiliano Contino, Michael Dritschel y Stefania Marcantognini sobre el conjunto de operadores en un espacio de Hilbert separable que pueden escribirse como el producto de dos operadores lineales, acotados y positivos. En espacios de dimensión finita, es fácil ver que un operador es el producto de dos operadores positivos si y sólo si es similar a un operador positivo, es decir si y sólo si es un operador escalar con espectro positivo. La estructura de este conjunto en espacios de dimensión infinita es mucho más rica y compleja. La factorization de un elemento del conjunto no es única pero existen factorizaciones distinguidas. La pertenencia a este conjunto se vincula con la cuasi-similaridad y cuasi afinidad a un operador positivo aunque no es equivalente. Estudiamos también las propiedades espectrales de los elementos del conjunto y describimos varios ejemplos.
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speaker: Alejandra Maestripieri (Universidad de Buenos Aires, Argentina)
- titulo: Diseño óptimo de multicompletaciones con restricciones de norma
abstract: |
Consideremos una sucesión finita de números reales positivos\(\alpha=(\alpha_i)_{i=1}^n\) y una sucesión de números enteros positivos\(\mathbf d=(d_j)_{j=1}^m\), ambas ordenadas en forma no-creciente.
Un\((\alpha,\mathbf d)-\)diseño es una familia\(\Phi=(\mathcal F_j)_{j=1}^m\) tal que:\(\mathcal F_j=\{f_{ij}\}_{i=1}^n \in (\mathbb C^{d_j})^n\) de forma que se verifican las restricciones $$\sum_{j=1}^m\|f_{ij}\|^2=\alpha_i\,,\ i=1,\ldots,n.$$ Denotaremos con\(\mathcal D(\alpha,\mathbf d)\) al conjunto de todos los\((\alpha,\mathbf d)-\)diseños.
Sea\(\Phi^0 =(\mathcal F^0_j)_{j=1}^m\) tal que\(\mathcal F^0_j=\{f^0_{ij}\}_{i=1}^k\in (\mathbb C^{d_j})^k\) con\(j=1,\ldots,m\). Una\((\alpha,\mathbf d)-\){ \it multicompletación} de\(\Phi^0\) es
$$(\Phi^0,\Phi)=(\mathcal F^0_j,\mathcal F_j)_{j=1}^m \,\text{ con }\, \Phi\in \mathcal D(\alpha,\mathbf d)\,,$$
donde\((\mathcal F^0_j, \mathcal F_j)\in (\mathbb C^{d_j})^{k+n}\), para\(j=1, \ldots, m\). Dadas\((\Phi^0,\Phi)\) una\((\alpha,\mathbf d)-\)multicompletación y una función\(\varphi:\mathbb R_{\geq 0}\to \mathbb R_{\geq 0}\) estrictamente convexa, consideramos el potencial conjunto inducido por\(\varphi\), dado por:
$$\Psi_{\varphi}(\Phi)= \rm P_{\varphi}(\Phi^0,\Phi)=\sum_{j=1}^m \text{tr}(\varphi[S_{(\mathcal F^0_j , \mathcal F_j)}]),$$
donde\(S_{(\mathcal F^0_j, \mathcal F_j)}=S_{\mathcal F^0_j}+S_{ \mathcal F_j}\) denota el operador de marco de\((\mathcal F^0_j, \mathcal F_j)\in (\mathbb C^{d_j})^{k+n}\), para\(j=1, \ldots, m\). Es bien sabido que los mínimos de potenciales convexos (bajo restricciones en las normas de los vectores) dan lugar a sistemas de reconstrucción más estables: cuanto menor es el potencial, más estable es el sistema.
En esta charla consideraremos el problema de la existencia de\((\alpha,\mathbf d)-\)multicompletaciones\((\Phi^0,\Phi^{\text{op}})\) óptimas dentro de la clase de todas las\((\alpha,\mathbf d)-\)multicompletaciones, es decir, tales que
$$\rm P_{\varphi}(\Phi^0,\Phi^{\text{op}})\leq \rm P_{\varphi}(\Phi^0,\Phi),$$
para toda\((\alpha,\mathbf d)-\)multicompletación\((\Phi^0,\Phi)\) y para toda\(\varphi\).
Si\(m=1\) y\(\mathcal F_1^0\) es una sucesión inicial fija, entonces el problema anterior se reduce a hallar las completaciones óptimas de\(\mathcal F_1^0\) con normas predeterminadas por\(\alpha\) (este caso fue probado por P. Massey, N. Ríos y D. Stojanoff en 2018), que a su vez contiene el problema de diseño óptimo con normas predeterminadas i.e.\(\mathcal F_1^0=\{0\}\) (probado por M.B; P. Massey, M. Ruiz y D. Stojanoff en 2020). La charla está basada en un trabajo en co-autoría con P. Massey, M. Ruiz y D. Stojanoff.
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speaker: María José Benac (Universidad Nacional de Santiago del Estero, Argentina)
- titulo: From closures of unitary group actions to groupoid actions
abstract: |
The set of all normal operators on a Hilbert space is naturally acted on both by the group of unitary operators and by the groupoid of partial isometries. We show that the norm closures of these group orbits are related to the above groupoid orbits, and this relation allows one to study the differentiable structures of these orbit closures. Some of the earlier results on closed unitary orbits are recovered in the natural degree of generality in our framework. This presentation is based on joint work with Gabriel Larotonda.
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speaker: Daniel Beltita (Institute of Mathematics of the Romanian Academy, Rumania)
- titulo: A nonlocal Jacobian equation?
abstract: |
We study an operator that assigns to each function\(u:\mathbb{R}^d\to\mathbb{R}\) a mapping\(G_u:\mathbb{R}^d \to C_*(\mathbb{R}^d)\),
$$G_u(x)(h) := u(x+h)-u(x)\;\forall h \in \mathbb{R}^d.$$
This map\(G_u(x)\) has some similarities with the gradient map\(\nabla u(x)\), which is a central object of study in the theory of the Monge-Ampère equation and Jacobian equations in general. The image of the map\(G_u\) will be, in general, a\(d\)-dimensional submanifold inside the Banach space\(C_*(\mathbb{R}^d)\) (the space of continuous, bounded functions which vanish at the origin). Our goal is to find a relation, at least for some broad class of functions\(u\), between the oscillation of the function\(u\) in a compact domain\(D\) and the\(d\)-dimensional measure of the set\(G_u(D)\). Such a relation would be analogous to Aleksandrov's estimate for convex functions, a fundamental estimate in the theory of elliptic equations which can be traced back to the reverse Blaschke-Santaló inequality. The validity of an integro-differential version of this estimate would have significant implications for the study of nonlinear integro-differential equations. In this talk I will review this background and discuss some preliminary results about the map\(G_u\). This is work in progress.
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speaker: Nestor Guillen (Texas State University, Estados Unidos)
- titulo: Constant scalar curvature, scalar flat, and Einstein metrics
abstract: |
Let\((M^n,g)\) be a closed Riemannian manifold of dimension\(n\). Then:
<ul>
<li> If\(n=1\),\(g\) is flat.</li>
<li> If\(n=2\),\(g\) is Einstein if, and only if,\(g\) has constant scalar curvature. Generically, the scalar curvature is negative.</li>
<li> If\(n\geq 3\), then\(g\) has constant scalar curvature if, and only if,\(g\) is a critical point of the scale invariant normalized total scalar curvature (or Yamabe functional) in its conformal class. A metric of constant nonpositive scalar curvature is a Yamabe metric in its conformal class, while if\(g\) has positive scalar curvature and\((M,g)\) admits an isometric minimal embedding into the standard sphere, then\(g\) is Yamabe in its conformal class. Over the space of metrics of fixed volume, a metric\(g\) is Einstein if, and only if, it is critical point of the total scalar curvature, while a metric is scalar flat or Einstein if, and only if, it is a critical point of the squared\(L^2\) norm of the scalar curvature functional.</li>
</ul>
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speaker: Santiago Simanca (the results of this talk were obtained during his stay at Courant Institute of Mathematical Sciences until 2020)
- sesion: Geometría, Mecánica, Control y sus interconexiones
organizadores:
- nombre: Viviana Alejandra Díaz