Fuzzy-stochastic FEM-based homogenization framework for materials with polymorphic uncertainties in the microstructure

Uncertainties in the macroscopic response of heterogeneous materials result from two sources: the natural variability in the microstructure's geometry and the lack of sufficient knowledge regarding the microstructure. The first type of uncertainty is denoted aleatoric uncertainty and may be characterized by a known probability density function. The second type of uncertainty is denoted epistemic uncertainty. This kind of uncertainty cannot be described using probabilistic methods. Models considering both sources of uncertainties are called polymorphic. In the case of polymorphic uncertainties, some combination of stochastic methods and fuzzy arithmetic should be used. Thus, in the current work, we examine a fuzzy‐stochastic finite element method–based homogenization framework for materials with random inclusion sizes. We analyze an experimental radii distribution of inclusions and develop a stochastic representative volume element. The stochastic finite element method is used to obtain the material response in the case of random inclusion radii. Due to unavoidable noise in experimental data, an insufficient number of samples, and limited accuracy of the fitting procedure, the radii distribution density cannot be obtained exactly; thus, it is described in terms of fuzzy location and scale parameters. The influence of fuzzy input on the homogenized stress measures is analyzed

Laplacian Mixture Modeling for Network Analysis and Unsupervised Learning on Graphs

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionality reductions or domain decompositions for a variety of input data types, including mixture distributions, feature vectors, and graphs or networks. Provable optimal recovery using the algorithm is analytically shown for a nontrivial class of cluster graphs. Heuristic approximations for scalable high-performance implementations are described and empirically tested. Connections to PageRank and community detection in network analysis demonstrate the wide applicability of this approach. The origins of fuzzy spectral methods, beginning with generalized heat or diffusion equations in physics, are reviewed and summarized. Comparisons to other dimensionality reduction and clustering methods for challenging unsupervised machine learning problems are also discussed.Comment: 13 figures, 35 reference

Estructura Combinatoria de Politopos asociados a Medidas Difusas

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 23-11-2020This PhD thesis is devoted to the study of geometric and combinatorial aspects of polytopes associated to fuzzy measures. Fuzzy measures are an essential tool, since they generalize the concept of probability. This greater generality allows applications to be developed in various elds, from the Decision Theory to the Game Theory. The set formed by all fuzzy measures on a referential set is a polytope. In the same way, many of the most relevant subfamilies of fuzzy measures are also polytopes. Studying the combinatorial structure of these polytopes arises as a natural problem that allows us to better understand the properties of the associated fuzzy measures. Knowing the combinatorial structure of these polytopes helps us to develop algorithms to generate points uniformly at random inside these polytopes. Generating points uniformly inside a polytope is a complex problem from both a theoretical and a computational point of view. Having algorithms that allow us to sample uniformly in polytopes associated to fuzzy measures allows us to solve many problems, among them the identi cation problem, i.e. estimate the fuzzy measure that underlies an observed data set...La presente tesis doctoral esta dedicada al estudio de distintas propiedades geometricas y combinatorias de politopos de medidas difusas. Las medidas difusas son una herramienta esencial puesto que generalizan el concepto de probabilidad. Esta mayor generalidad permite desarrollar aplicaciones en diversos campos, desde la Teoría de la Decision a laTeoría de Juegos. El conjunto formado por todas las medidas difusas sobre un referencial tiene estructura de politopo. De la misma forma, la mayora de las subfamilias mas relevantes de medidas difusas son tambien politopos. Estudiar la estructura combinatoria de estos politopos surge como un problema natural que nos permite comprender mejor las propiedades delas medidas difusas asociadas. Conocer la estructura combinatoria de estos politopos tambien nos ayuda a desarrollar algoritmos para generar aleatoria y uniformemente puntos dentro de estos politopos. Generar puntos de forma uniforme dentro de un politopo es un problema complejo desde el punto de vista tanto teorico como computacional. Disponer de algoritmos que nos permitan generar uniformemente en politopos asociados a medidas difusas nos permite resolver muchos problemas, entre ellos el problema de identificacion que trata de estimarla medida difusa que subyace a un conjunto de datos observado...Fac. de Ciencias MatemáticasTRUEunpu

Confidence limits of evolutionary synthesis models. IV Moving forward to a probabilistic formulation

Synthesis models predict the integrated properties of stellar populations. Several problems exist in this field, mostly related to the fact that integrated properties are distributed. To date, this aspect has been either ignored (as in standard synthesis models, which are inherently deterministic) or interpreted phenomenologically (as in Monte Carlo simulations, which describe distributed properties rather than explain them). We approach population synthesis as a problem in probability theory, in which stellar luminosities are random variables extracted from the stellar luminosity distribution function (sLDF). We derive the population LDF (pLDF) for clusters of any size from the sLDF, obtaining the scale relations that link the sLDF to the pLDF. We recover the predictions of standard synthesis models, which are shown to compute the mean of the sLDF. We provide diagnostic diagrams and a simplified recipe for testing the statistical richness of observed clusters, thereby assessing whether standard synthesis models can be safely used or a statistical treatment is mandatory. We also recover the predictions of Monte Carlo simulations, with the additional bonus of being able to interpret them in mathematical and physical terms. We give examples of problems that can be addressed through our probabilistic formalism. Though still under development, ours is a powerful approach to population synthesis. In an era of resolved observations and pipelined analyses of large surveys, this paper is offered as a signpost in the field of stellar populations.Comment: Accepted by A&A. Substantially modified with respect to the 1st draft. 26 pages, 14 fig