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Blow-up for the 1D cubic NLS
No full textWe consider the 1D cubic NLS on ℝ and prove a blow-up result for functions that are of borderline regularity, i.e. Hs for any s<−12 for the Sobolev scale and L∞ for the Fourier-Lebesgue scale. This is done by identifying at this regularity a certain functional framework from which solutions exit in finite time. This functional framework allows, after using a pseudo-conformal transformation, to reduce the problem to a large-time study of a periodic Schrödinger equation with non-autonomous cubic nonlinearity. The blow-up result corresponds to an asymptotic completeness result for the new equation. We prove it using Bourgain's method and exploiting the oscillatory nature of the coefficients involved in the time-evolution of the Fourier modes. Finally, as an application we exhibit singular solutions of the binormal flow. More precisely, we give conditions on the curvature and the torsion of an initial smooth curve such that the constructed solutions generate several singularities in finite time
Contributions to the mathematical modeling of estimation of distribution algorithms and pseudo-boolean functions
Get PDFMaximise or minimise an objective function defined over a discrete space. Since most such problems cannot be solved through exhaustive search, their resolution is often approximated by heuristic algorithms. However, there is no algorithm that performs better than all other algorithms for solving every instance of any given problem. Therefore, the ideal goal is, given an instance of a problem, to know which algorithm's resolution is the most efficient. The two main lines of research to achieve this goal are studying the definitions of problems and the possible instances that each problem can generate, and studying the designs and characteristics of the algorithms. In this thesis, both lines have been addressed. On one hand, we have studied pseudo-Boolean functions and several specific binary problems. On the other hand, a mathematical modelling has been presented to study Estimation of Distribution Algorithms designed to solve permutation-based problems. The main motivation has been to continue progressing in this field to better understand the relationships between Combinatorial Optimisation Problems and optimisation algorithms
PERTURBED INTERPOLATION FORMULAE AND APPLICATIONS
Get PDFWe employ functional analysis techniques in order to deduce some versions of classical and recent interpolation results in Fourier analysis with perturbed nodes. As an application of our techniques, we obtain generalizations of Kadec's 14 -theorem for interpolation formulae in the Paley–Wiener space both in the real and complex cases, as well as versions of the recent interpolation result of Radchenko and Viazovska (Publ. Math. Inst. Hautes Etudes Sci. 129 (2019), 51–81) and the result of Cohn, Kumar, Miller, Radchenko and Viazovska (Ann. Math (2) 196:3 (2022), 983–1082) for Fourier interpolation with derivatives in dimensions 8 and 24 with suitable perturbations of the interpolation nodes. We also provide several applications of the main results and techniques, relating to recent contributions in interpolation formulae and uniqueness sets for the Fourier transform
GLOBAL AND LOCAL MAXIMIZERS FOR SOME FOURIER EXTENSION ESTIMATES ON THE SPHERE
Get PDFIn this note we improve, for the case of low dimensions, the known range of exponents for which constant functions are the unique maximizers for the L2(Sd−1) to LpradL2ang(Rd) mixed-norm Fourier extension estimate on the sphere. Moreover, we show that in the same range of exponents for which constant functions are the unique maximizers for the L2(Sd−1) to LpredL2ang(Rd) mixed-norm Fourier extension estimates they are also local maximizers for the Lp(Sd−1) to Lp(Rd) Fourier extension estimates. As a by-product, we obtain that for the cases of dimensions d = 2, 3 constant functions are local maximizers for all p ≥ pst(d), where pst denotes the Stein-Tomas endpoint, pst(d) := 2(d + 1)/(d − 1)
Discretization-Based Feature Selection as a Bilevel Optimization Problem
Get PDFDiscretization-based feature selection (DBFS) approaches have shown interesting results when using several metaheuristic algorithms, such as particle swarm optimization (PSO), genetic algorithm (GA), ant colony optimization (ACO), etc. However, these methods share the same shortcoming which consists in encoding the problem solution as a sequence of cut-points. From this cut-points vector, the decision of deleting or selecting any feature is induced. Indeed, the number of generated cut-points varies from one feature to another. Thus, the higher the number of cut-points, the higher the probability of selecting the considered feature; and vice versa. This fact leads to the deletion of possibly important features having a single or a low number of cut-points, such as the infection rate, the glycemia level, and the blood pressure. In order to solve the issue of the dependency relation between the feature selection (or removal) event and the number of its generated potential cut-points, we propose to model the DBFS task as a bilevel optimization problem and then solve it using an improved version of an existing co-evolutionary algorithm, named I-CEMBA. The latter ensures the variation of the number of features during the migration process in order to deal with the multimodality aspect. The resulting algorithm, termed bilevel discretization-based feature selection (Bi-DFS), performs selection at the upper level while discretization is done at the lower level. The experimental results on several high-dimensional datasets show that Bi-DFS outperforms relevant state-of-the-art methods in terms of classification accuracy, generalization ability, and feature selection bias
Revisiting Implicit and Explicit Averaging for Noisy Optimization
Get PDFExplicit and implicit averaging are two well-known strategies for noisy optimization. Both strategies can counteract the disruptive effect of noise; however, a critical question remains: which one is more efficient? This question has been raised in many studies, with conflicting preferences and, in some cases, findings. Nevertheless, theoretical findings on the noisy sphere problem with additive Gaussian noise supports the superiority of implicit averaging, which may have had a strong impact on the preference of implicit averaging in more recent evolutionary methods for noisy optimization. This study speculates that the analytically supported superiority of implicit averaging relies on specific features of the noisy sphere problem with additive noise, which cannot be generalized to other problems. It enumerates these features and designs controlled numerical experiments to investigate this potential reliance. Each experiment gradually suppresses one specific feature, and the progress rate is numerically calculated for different values of the sample size given a fixed evaluation budget. Our empirical results indicate that for a wide range of noise strength and evaluation budget per iteration, the more these specific features are suppressed, the more the optimal averaging strategy deviates from implicit toward explicit averaging, which confirms our speculations. Consequently, the optimal sample size, which is regarded as the tradeoff between implicit and explicit averaging, depends on the problem characteristics and should be learned during optimization for maximum efficiency
Sturm–Liouville systems for the survival probability in first-passage time problems
Get PDFWe derive a Sturm–Liouville system of equations for the exact calculation of the survival probability in first-passage time problems. This system is the one associated with the Wiener–Hopf integral equation obtained from the theory of random walks. The derived approach is an alternative to the existing literature and we tested it against direct calculations from both discrete- and continuous-time random walks in a manageable, but meaningful, example. Within this framework, the Sparre Andersen theorem results to be a boundary condition for the system.Predoc Severo Ochoa 2018 grant PRE2018-08442
Variable selection with LASSO regression for complex survey data
Get PDFVariable selection is an important step to end up with good prediction models. LASSO regression
models are one of the most commonly used methods for this purpose, for which
cross-validation is the most widely applied validation technique to choose the tuning parameter
(λ). Validation techniques in a complex survey framework are closely related to
“replicate weights”. However, to our knowledge, they have never been used in a LASSO
regression context. Applying LASSO regression models to complex survey data could be
challenging. The goal of this paper is two-fold. On the one hand, we analyze the performance
of replicate weights methods to select the tuning parameter for fitting LASSO regression
models to complex survey data. On the other hand, we propose new replicate weights methods
for the same purpose. In particular, we propose a new design-based cross-validation
method as a combination of the traditional cross-validation and replicate weights. The performance
of all these methods has been analyzed and compared by means of an extensive
simulation study to the traditional cross-validation technique to select the tuning parameter
for LASSO regression models. The results suggest a considerable improvement when
the new proposal design-based cross-validation is used instead of the traditional crossvalidation.IT1456-22
PIF18/21
Weighted BMO estimates for singular integrals and endpoint extrapolation in Banach function spaces
Get PDFIn this paper we prove sharp weighted BMO estimates for singular integrals, and we show how such estimates can be extrapolated to Banach function spaces
Optimal vaccination strategies for a heterogenous population using multiple objectives: The case of L1 and L2-formulations
Get PDFThe choice of the objective functional in optimization problems coming from biomedical and epidemiological applications plays a key role in optimal control outcomes. In this study, we investigate the role of the objective functional on the structure of the optimal control solution for an epidemic model for sexually transmitted infections that includes a core group with higher sexual activity levels than the rest of the population. An optimal control problem is formulated to find a targeted vaccination program able to control the spread of the infection with minimum vaccine deployment. Both and objectives are considered as an attempt to explore the trade-offs between control dynamics and the functional form characterizing optimality. The results show that the optimal vaccination policies for both the and the formulation share one important qualitative property, that is, immunization of the core group should be prioritized by policymakers to achieve a fast reduction of the epidemic. However, quantitative aspects of this result can be significantly affected depending on the choice of the control weights between formulations. Overall, the results suggest that with appropriate weight constants, the optimal control outcomes are reasonably robust with respect to the or formulation. This is particularly true when the monetary cost of the control policy is substantially lower than the cost associated with the disease burden. Under these conditions, even if the formulation is more realistic from a modeling perspective, the formulation can be used as an approximation and yield qualitatively comparable outcomes