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Patrick Guidotti

Publications and source records attributed to Patrick Guidotti.

At least 19 recordsLinked to original sources
arXiv2026

Geometric Kernel Interpolation and Regression

Exploiting the variational interpretation of kernel interpolation we exhibit a direct connection between interpolation and regression, where interpolation appears as a limiting case of regression. By applying this framework to point clouds or samples of smooth manifolds (hypersurfaces, in particular), we show how fundamental geometric quantities such as tangent plane and principal curvatures can be computed numerically using a kernel based (approximate) level set function (often a defining function) for smooth hypersurfaces. In the case of point clouds, the approach generates an interpolated hypersurface, which is an approximation of the underlying manifold when the cloud is a sample of it. It is shown how the geometric quantities obtained can be used in the numerical approximation/computation of geometric operators like the surface gradient or the Laplace-Beltrami operator in the spirit of kernel based meshfree methods. Kernel based interpolation can be extremely ill-posed, especially when using smooth kernels, and the regression approximation offers a natural regularization that proves also quite useful when dealing with geometric or functional data that are affected by errors or noise.

math.NA↗
arXiv2025

Analysis of a one-dimensional biofilm model

In this paper a reduced one-dimensional moving boundary model is studied that describes the evolution of a biofilm driven by the presence of a reaction limiting substrate. Global well-posedness is established for the resulting parabolic free boundary value problem in strong form in Sobolev spaces and for a quasi-stationary approximation in spaces of classical regularity. The general existence results are complemented by results about the qualitative properties of solutions including the existence, in general, and, additionally, the uniqueness and stability of non-trivial equilibria, in a special case.

math.AP↗
arXiv2024

Extracting Manifold Information from Point Clouds

A kernel based method is proposed for the construction of signature (defining) functions of subsets of $\mathbb{R}^d$. The subsets can range from full dimensional manifolds (open subsets) to point clouds (a finite number of points) and include bounded (closed) smooth manifolds of any codimension. The interpolation and analysis of point clouds are the main application. Two extreme cases in terms of regularity are considered, where the data set is interpolated by an analytic surface, at the one extreme, and by a H\"older continuous surface, at the other. The signature function can be computed as a combination of translated kernels, the coefficients of which are the solution of a Fredholm integral equation (matrix equation in the finite dimensional case). Once it is obtained, it can be used to estimate the dimension as well as the normal and the curvatures of the interpolated manifold. The method is global and does not require the data set to be organized or structured in any particular way. It admits a variational formulation with a natural regularized counterpart, that proves useful in dealing with data sets corrupted by numerical error or noise. The underlying analytical structure of the approach is presented in general before it is applied to the case of point clouds.

cs.CV↗
arXiv2023

The Curve Shortening Flow for Curves of Finite Total (Absolute) Curvature

We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is valid for all times and is insensitive to singularities. The construction is characterized by discretization in time and the approximant, while still exhibiting the possibile formation of finitely many singularities at a finite set of singular times, exists globally and is well behaved and simpler to analyze than a solution of the CSF. A solution of the latter is obtained in the limit. Estimates for a natural (geometric) norm involving length and total absolute curvature allow passage to the limit. Many classical qualitative results about the flow can be recovered by exploiting the simplicity of the approximant and new ones can be proved. The construction also suggests a numerical procedure for the computation of the flow which proves very effective as demonstrated by a series of numerical experiments scattered throughout the paper.

math.AP↗
arXiv2022

Dimension Independent Data Sets Approximation and Applications to Classification

We revisit the classical kernel method of approximation/interpolation theory in a very specific context motivated by the desire to obtain a robust procedure to approximate discrete data sets by (super)level sets of functions that are merely continuous at the data set arguments but are otherwise smooth. Special functions, called data signals, are defined for any given data set and are used to succesfully solve supervised classification problems in a robust way that depends continuously on the data set. The efficacy of the method is illustrated with a series of low dimensional examples and by its application to the standard benchmark high dimensional problem of MNIST digit classification.

cs.LG↗
arXiv2020

On Wiener's Violent Oscillations, Popov's curves and Hopf's Supercritical Bifurcation for a Scalar Heat Equation

A parameter dependent perturbation of the spectrum of the scalar Laplacian is studied for a class of nonlocal and non-self-adjoint rank one perturbations. A detailed description of the perturbed spectrum is obtained both for Dirichlet boundary conditions on a bounded interval as well as for the problem on the full real line. The perturbation results are applied to the study of a related parameter dependent nonlinear and nonlocal parabolic equation. The equation models a feedback system that e.g. can be interpreted as a thermostat device or in the context of an agent based price formation model for a market. The existence and the stability of periodic self-oscillations of the related nonlinear and nonlocal heat equation that arise from a Hopf bifurcation is proved. The bifurcation and stability results are obtained both for the nonlinear parabolic equation with Dirichlet boundary conditions and for a related problem with nonlinear Neumann boundary conditions that model feedback boundary control. The bifurcation and stability results follow from a Popov criterion for integral equations after reducing the stability analysis for the nonlinear parabolic equation to the study of a related nonlinear Volterra integral equation. While the problem is studied in the scalar case only it can be extended naturally to arbitrary euclidean dimension and to manifolds.

math.AP↗
arXiv2019

On the Maximal Parameter Range of Global Stability for a Nonlocal Thermostat Model

The global asymptotic stability of the unique steady state of a nonlinear scalar parabolic equation with a nonlocal boundary condition is studied. The equation describes the evolution of the temperature profile that is subject to a feedback control loop. It can be viewed as a model of a rudimentary thermostat, where a parameter controls the intensity of the heat flow in response to the magnitude of the deviation from the reference temperature at a boundary point. The system is known to undergo a Hopf bifurcation when the parameter exceeds a critical value. Results on the characterization of the maximal parameter range where the reference steady state is globally asymptotically stable are obtained by analyzing a closely related nonlinear Volterra integral equation. Its kernel is derived from the trace of a fundamental solution of a linear heat equation. A version of the Popov criterion is adapted and applied to the Volterra integral equation to obtain a sufficient condition for the asymptotic decay of its solutions.

math.AP↗
arXiv2019

The Smooth Selection Embedding Method with Chebyshev Polynomials

We propose an implementation of the Smooth Selection Embedding Method (SSEM) in the setting of Chebyshev polynomials. The SSEM is a hybrid fictitious domain / collocation method which solves boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, the SSEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non-periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve problems in complex geometries, with non-constant coefficients, and for general boundary conditions.

math.NA↗
arXiv2018

A Novel Optimization Approach to Fictitious Domain Methods

A new approach to the solution of boundary value problems within the so-called fictitious domain methods philosophy is proposed which avoids well known shortcomings of other fictitious domain methods, including the need to generate extensions of the data. The salient feature of the novel method, which we refer to as SSEM (Smooth Selection Embedding Method), is that it reduces the whole boundary value problem to a linear constraint for an appropriate optimization problem formulated in a larger simpler set containing the domain on which the boundary value problem is posed and which allows for the use of straightforward discretizations. The proposed method in essence computes a (discrete) extension of the solution to the boundary value problem by selecting it as a smooth element of the complete affine family of solutions of the extended, yet unmodified, under-determined problem. The actual regularity of this extension is determined by that of the analytic solution and the choice of obejctive functional. Numerical experiments will demonstrate that it can be stably used to efficiently deal with non-constant coefficients, general geometries, and different boundary conditions in dimensions d=1,2,3 and that it produces solutions of tunable (and high) accuracy.

math.NA↗
arXiv2018

Admission predictors for success in a mathematics graduate program

There are many factors that can influence the outcome for students in a mathematics PhD program: bachelor's GPA (BGPA), bachelor's major, GRE scores, gender, Under-Represented Minority (URM) status, institution tier, etc. Are these variables equally important predictors of a student's likelihood of succeeding in a math PhD program? In this paper, we present and analyze admission data of students from different groups entering a math PhD program at a southern California university. We observe that some factors correlate with success in the PhD program (defined as obtaining a PhD degree within a time-limit). According to our analysis, GRE scores correlate with success, but interestingly, the verbal part of the GRE score has a higher predictive power compared to the quantitative part. Further, we observe that undergraduate student GPA does not correlate with success (there is even a slight negative slope in the relationship between GPA and the probability of success). This counterintuitive observation is explained once undergraduate institutions are separated by tiers: students from "higher tiers" have undergone a more rigorous training program; they on average have a slightly lower GPA but run a slightly higher probability to succeed. Finally, a gender gap is observed in the probability to succeed with female students having a lower probability to finish with a PhD despite the same undergraduate performance, compared to males. This gap is reversed if we only consider foreign graduate students. It is our hope that this study will encourage other universities to perform similar analyses, in order to design better admission and retention strategies for Math PhD programs.

math.HO↗
arXiv2017

On a model for a sliding droplet:Well-posedness and stability of translating circular solutions

In this paper the model for a highly viscous droplet sliding down an inclined plane is analyzed. It is shown that, provided the slope is not too steep, the corresponding moving boundary problem possesses classical solutions. Well-posedness is lost when the relevant linearization ceases to be parabolic. This occurs above a critical incline which depends on the shape of the initial wetted region as well as on the liquid's mass. It is also shown that translating circular solutions are asymptotically stable if the kinematic boundary condition is given by an affine functionand provided the incline is small.

math.AP↗
arXiv2017

Improving LBP and its variants using anisotropic diffusion

The main purpose of this paper is to propose a new preprocessing step in order to improve local feature descriptors and texture classification. Preprocessing is implemented by using transformations which help highlight salient features that play a significant role in texture recognition. We evaluate and compare four different competing methods: three different anisotropic diffusion methods including the classical anisotropic Perona-Malik diffusion and two subsequent regularizations of it and the application of a Gaussian kernel, which is the classical multiscale approach in texture analysis. The combination of the transformed images and the original ones are analyzed. The results show that the use of the preprocessing step does lead to improved texture recognition.

cs.CV↗
arXiv2016

Wellposedness of a nonlocal nonlinear diffusion equation of image processing

Existence and uniqueness are established for a degenerate regularization of the well-known Perona-Malik equation proposed by the first author for non-smooth initial data. The results heavily rely on the choice of appropriate functional setting inspired by a recent approach to degenerate parabolic equations via so-called singular Riemannian manifolds introduced by Herbert Amann.

math.AP↗
arXiv2016

Domain Variations for Moving Boundary Problems

In the past few decades maximal regularity theory has successfully been applied to moving boundary problems. The basic idea is to reduce the system with varying domains to one in a fixed domain. This is done by a transformation (the so-called Hanzawa transformation) and yields a typically nonlocal and nonlinear coupled system of (evolution) equations. Well-posedness results can then often be established as soon as it is proved that the relevant linearization is the generator of an analytic semigroup or admits maximal regularity. To implement this program, it is necessary to somehow parametrize to space of boundaries/domains (typically the space of compact hypersurfaces $Γ$ in $\mathbb{R}^n$, in the Euclidean setting). This has traditionally been achieved by means of the already mentioned Hanzawa transformation. The approach, while successful, requires the introduction of a smooth manifold $Γ_\infty$ close to the manifold $Γ_0$ in which one cares to linearize. This prevents one to use coordinates in which $Γ_0$ lies at their "center". As a result formulæ tend to contain terms that would otherwise not be present were one able to linearize in a neighborhood emanating from $Γ_0$ instead of from $Γ_\infty$. In this paper it is made use of flows (curves of diffeomorphisms) to obtain a general form of the relevant linearization in combination with an alternative coordinatization of the manifold of hypersurfaces, which circumvents the need for the introduction of a "phantom" reference manifold $Γ_\infty$ by, in its place, making use of a "phantom geometry" on $Γ_0$. The upshot is a clear insight into the structure of the linearization, simplified calculations, and simpler formulæ for the resulting linear operators, which are useful in applications.

math.AP↗
arXiv2016

A Class Of Elliptic Equations with Interior Degeneration

A class of linear degenerate elliptic equations inspired by nonlinear diffusions of image processing is considered. It is characterized by an interior degeneration of the diffusion coefficient. It is shown that no particularly natural, unique interpretation of the equation is possible. This phenomenon is reflected in the behavior of numerical schemes for its resolution and points to similar issues that potentially affect its nonlinear counterpart.

math.AP↗
arXiv2016

Using an Encompassing Periodic Box to Perform Numerical Calculations on General Domains

This paper shows how numerical methods on a regular grid in a box can be used to generate numerical schemes for problems in general smooth domains contained in the box with no need for a domain specific discretization. The focus is mainly be on spectral discretizations due to their ability to accurately resolve the interaction of finite order distributions (generalized functions) and smooth functions. Mimicking the analytical structure of the relevant (pseudodifferential) operators leads to viable and accurate numerical representations and algorithms. An important byproduct of the structural insights gained in the process is the introduction of smooth kernels (at the discrete level) to replace classical singular kernels which are typically used in the (numerical) representations of the solution. The new kernel representations yield enhanced numerical resolution and, while they necessarily lead to significantly higher condition numbers, they also suggest natural and effective ways to precondition the systems.

math.NA↗
arXiv2013

Equilibria and Their Stability for a Viscous Droplet Model

A classical model of fluid dynamics is considered which describes the shape evolution of a viscous liquid droplet on a homogeneous substrate. All equilibria are characterized and their stability is analyzed by a geometric reduction argument.

math.AP↗