Begell House Inc. International Journal for Uncertainty Quantification IJUQ 2152-5080 8 1 2018
A STUDY ON Z-SOFT ROUGH FUZZY SEMIGROUPS AND ITS DECISION-MAKING 1-22 Qiumei Wang Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, Peoples Republic of China Jianming Zhan Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, P.R. China Muhammad Irfan Ali Department of Mathematics, Islamabad Model College for Girls, F-6/2, Islamabad, Pakistan Nayyar Mehmood Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan The hybrid soft set model is an important topic dealing with uncertain information. In this article, we firstly introduce the concept of Z-soft rough fuzzy sets of a semigroup, and obtain some basic operations about the upper (lower) approximations. Next, we give the definition of Z-soft rough fuzzy semigroups and study related properties. Particularly, we form an approach which is built on the basis of Z-soft rough fuzzy sets of a semigroup for decision-making, and we also give an example to test the validity of the approach. Finally, we compare the performance of three algorithms for decision-making, stated in terms of hybrid soft set models, inclusive of our Z-soft rough fuzzy sets. We use computer simulations in a medical environment with Matlab programs that are provided in an Appendix.
COMPARISON OF RISK ANALYSIS METHODOLOGIES IN A GEOSTATISTICAL CONTEXT: MONTE CARLO WITH JOINT PROXY MODELS AND DISCRETIZED LATIN HYPERCUBE 23-41 Susana Santos University of Campinas Ana Teresa F. S. Gaspar Department of Energy, School of Mechanical Engineering, University of Campinas, Brazil Denis J. Schiozer Department of Energy, School of Mechanical Engineering, University of Campinas, Brazil During the development of petroleum fields, uncertainty quantification is essential to base decisions. Several methods are presented in the literature, but its choice must agree with the complexity of the case study to ensure reliable results at minimum computational costs. In this study, we compared two risk analysis methodologies applied to a complex reservoir model comprising a large set of geostatistical realizations: (1) a generation of scenarios using the discretized Latin hypercube sampling technique combined with geostatistical realizations (DLHG) and (2) a generation of scenarios using the Monte Carlo sampling technique combined with joint proxy models, entitled the joint modeling method (JMM). For a reference response, we assessed risk using the pure Monte Carlo sampling combined with flow simulation using a very high sampling number. We compared the methodologies, looking at the (1) accuracy of the results, (2) computational cost, (3) difficulty in the application, and (4) limitations of the methods. Our results showed that both methods are reliable but revealed limitations in the JMM. Due to the way the JMM captures the effect of a geostatistical uncertainty, the number of required flow simulation runs increased exponentially and became unfeasible to consider more than 10 realizations. The DLHG method showed advantages in such a context, namely, because it generated precise results from less than half of the flow simulation runs, the risk curves were computed directly from the flow simulation results (i.e., a proxy model was not needed), and incorporated hundreds of geostatistical realizations. In addition, this method is fast, straightforward, and easy to implement.
CLUSTERING-BASED COLLOCATION FOR UNCERTAINTY PROPAGATION WITH MULTIVARIATE DEPENDENT INPUTS 43-59 Anne W. Eggels Centrum Wiskunde & Informatica, Amsterdam, the Netherlands D. T. Crommelin Centrum Wiskunde & Informatica, Amsterdam, the Netherlands; Korteweg−de Vries Institute for Mathematics, University of Amsterdam, the Netherlands J. A. S. Witteveen Centrum Wiskunde & Informatica, Amsterdam, the Netherlands In this paper, we propose the use of partitioning and clustering methods as an alternative to Gaussian quadrature for stochastic collocation. The key idea is to use cluster centers as the nodes for collocation. In this way, we can extend the use of collocation methods to uncertainty propagation with multivariate, dependent input, in which the output approximation is piecewise constant on the clusters. The approach is particularly useful in situations where the probability distribution of the input is unknown and only a sample from the input distribution is available. We examine several clustering methods and assess the convergence of collocation based on these methods both theoretically and numerically. We demonstrate good performance of the proposed methods, most notably for the challenging case of nonlinearly dependent inputs in higher dimensions. Numerical tests with input dimension up to 16 are included, using as benchmarks the Genz test functions and a test case from computational fluid dynamics (lid-driven cavity flow).
A MULTI-INDEX MARKOV CHAIN MONTE CARLO METHOD 61-73 Ajay Jasra Department of Statistics & Applied Probability National University of Singapore, Singapore Kengo Kamatani Graduate School of Engineering Science, Osaka University, Osaka, 565–0871, Japan Kody J. H. Law School of Mathematics, University of Manchester, Manchester, UK, M13 9PL Yan Zhou Department of Statistics & Applied Probability National University of Singapore, Singapore In this paper, we consider computing expectations with respect to probability laws associated with a certain class of stochastic systems. In order to achieve such a task, one must not only resort to numerical approximation of the expectation but also to a biased discretization of the associated probability. We are concerned with the situation for which the discretization is required in multiple dimensions, for instance in space-time. In such contexts, it is known that the multi-index Monte Carlo (MIMC) method of Haji-Ali, Nobile, and Tempone, (Numer. Math., 132, pp. 767– 806, 2016) can improve on independent identically distributed (i.i.d.) sampling from the most accurate approximation of the probability law. Through a nontrivial modification of the multilevel Monte Carlo (MLMC) method, this method can reduce the work to obtain a given level of error, relative to i.i.d. sampling and even to MLMC. In this paper, we consider the case when such probability laws are too complex to be sampled independently, for example a Bayesian inverse problem where evaluation of the likelihood requires solution of a partial differential equation model, which needs to be approximated at finite resolution. We develop a modification of the MIMC method, which allows one to use standard Markov chain Monte Carlo (MCMC) algorithms to replace independent and coupled sampling, in certain contexts. We prove a variance theorem for a simplified estimator that shows that using our MIMCMC method is preferable, in the sense above, to i.i.d. sampling from the most accurate approximation, under appropriate assumptions. The method is numerically illustrated on a Bayesian inverse problem associated to a stochastic partial differential equation, where the path measure is conditioned on some observations.
P-SOFT ROUGH FUZZY GROUPS AND P-SOFT FUZZY ROUGH GROUPS AND CORRESPONDING DECISION-MAKING 75-99 Wenjun Pan Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, P.R. China Jianming Zhan Department of Mathematics, Hubei University for Nationalities, Enshi, 445000, P.R. China The study of applying hybrid soft models to uncertain problems has become a hot topic. In this research study, we introduce the notions of P-soft rough fuzzy groups and P-soft fuzzy rough groups and investigate some of their basic properties. Further, we present certain notions, including P-soft rough fuzzy subgroups, P-soft rough fuzzy normal subgroups, P-soft fuzzy rough-subgroups and P-soft fuzzy rough normal, subgroups of a group. In particular, we propose two kinds of decision-making methods based on P-soft rough fuzzy sets and P-soft fuzzy rough sets in groups to illustrate the effectiveness and rationality. Finally, we present numerical experimentations of six algorithms, in which the comparison among six types of hybrid soft models are analyzed.