Patrick Guidotti

Daniel Agress, Patrick Guidotti, Dong Yan

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.

Daniel Agress, Patrick Guidotti

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.

Patrick Guidotti

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.

Patrick Guidotti

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.

Patrick Guidotti

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 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 Hölder continuous surface, at the other. The signature function can be computed as a linear combination of translated kernels, the coefficients of which are the solution of a finite dimensional linear problem. Once it is obtained, it can be used to estimate the dimension as well as the normal and the curvatures of the interpolated surface. The method is global and does not require explicit knowledge of local neighborhoods or any other structure present in the data set. It admits a variational formulation with a natural ``regularized'' counterpart, that proves to be 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.

Mariane B. Neiva, Patrick Guidotti, Odemir M. Bruno

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.