Begell House Inc.
International Journal for Uncertainty Quantification
IJUQ
2152-5080
8
3
2018
AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES
193-210
10.1615/Int.J.UncertaintyQuantification.2018024436
Heyrim
Cho
Department of Mathematics, University of Maryland, College Park, MD 20742
Howard C.
Elman
Department of Computer Science and Institute for Advanced Computer Studies, University of Maryland, USA
reduced basis method
ANOVA decomposition
probabilistic collocation method
anisotropic stochasticity
high-dimensionality
The combination of reduced basis and collocation methods enables efficient and accurate evaluation of the solutions
to parametrized partial differential equations (PDEs). In this paper, we study the stochastic collocation methods that
can be combined with reduced basis methods to solve high-dimensional parametrized stochastic PDEs. We also propose
an adaptive algorithm using a probabilistic collocation method (PCM) and ANOVA decomposition. This procedure
involves two stages. First, the method employs an ANOVA decomposition to identify the effective dimensions, i.e.,
subspaces of the parameter space in which the contributions to the solution are larger, and sort the reduced basis
solution in a descending order of error. Then, the adaptive search refines the parametric space by increasing the order
of polynomials until the algorithm is terminated by a saturation constraint. We demonstrate the effectiveness of the proposed algorithm for solving a stationary stochastic convection-diffusion equation, a benchmark problem chosen
because solutions contain steep boundary layers and anisotropic features. We show that two stages of adaptivity are critical in a benchmark problem with anisotropic stochasticity.
INTERVAL-VALUED INTUITIONISTIC FUZZY POWER MACLAURIN SYMMETRIC MEAN AGGREGATION OPERATORS AND THEIR APPLICATION TO MULTIPLE ATTRIBUTE GROUP DECISION-MAKING
211-232
10.1615/Int.J.UncertaintyQuantification.2018020702
Zhengmin
Liu
School of Management Science and Engineering, Shandong University of Finance and
Economics, Jinan Shandong 250014, China
Fei
Teng
School of Management Science and Engineering, Shandong University of Finance and
Economics, Jinan Shandong 250014, China
Peide
Liu
School of Management Science and Engineering, Shandong University of Finance and
Economics, Jinan Shandong 250014, China; School of Economics and Management, Civil Aviation University of China, Tianjin 300300,
China
Qian
Ge
School of Science, Shandong Jianzhu University, Jinan Shandong 250014, China
multiple attribute group decision-making
interval-valued intuitionistic fuzzy set
power average operator
Maclaurin symmetric mean
interval-valued intuitionistic fuzzy power Maclaurin symmetric mean operator
The power average operator (PA), originally introduced by Yager (IEEE Trans. Syst. Man Cybern. Part A, 31(6):724–731, 2001), can reduce the negative impact of unreasonable evaluation values on the decision result. The Maclaurin
symmetric mean (MSM), originally introduced by Maclaurin (Phil. Trans., 36:59–96, 1729), can reflect the interrelationship among the multi-input arguments. However, in some complex decision-making situations, we need to
reduce the influence of unreasonable evaluation values and reflect the interrelationship among the multi-input arguments at the same time. In this paper, in order to solve such situations, we combine the ordinary PA operator with the traditional MSM in interval-valued intuitionistic context and propose two novel interval-valued intuitionistic fuzzy aggregation operators, i.e., the interval-valued intuitionistic fuzzy power Maclaurin symmetric mean operator and the weighted interval-valued intuitionistic fuzzy power Maclaurin symmetric mean operator. Then, some desirable
properties of these new proposed operators are investigated and some special cases are discussed. Furthermore, based
on these proposed operators, we develop a new approach to multiple attribute group decision-making under interval-valued intuitionistic fuzzy environment. Finally, two examples are provided to illustrate the feasibility and validity of the proposed approach by comparing to other existing representative methods.
STOCHASTIC MULTIOBJECTIVE OPTIMIZATION ON A BUDGET: APPLICATION TO MULTIPASS WIRE DRAWING WITH QUANTIFIED UNCERTAINTIES
233-249
10.1615/Int.J.UncertaintyQuantification.2018021315
Piyush
Pandita
Probabilistic Design, GE Research, Niskayuna, New York, 12309, USA
Ilias
Bilionis
School of Mechanical Engineering, Purdue University, Mechanical Engineering Room 1069, 585 Purdue Mall, West Lafayette, IN 47907-2088, USA
Jitesh
Panchal
School of Mechanical Engineering, Purdue University, West Lafayette, Indiana, 47907
B. P.
Gautham
TATA Research Development and Design Centre, TATA Consultancy Services, Pune, India
Amol
Joshi
TATA Research Development and Design Centre, TATA Consultancy Services, Pune, India
Pramod
Zagade
TATA Research Development and Design Centre, TATA Consultancy Services, Pune, India
Bayesian optimization
Gaussian processes
expected improvement over dominated hypervolume
information acquisition
Vorob'ev expectation
uncertainty quantification
stochastic optimization
Design optimization of engineering systems with multiple competing objectives is a painstakingly tedious process
especially when the objective functions are expensive-to-evaluate computer codes with parametric uncertainties. The
effectiveness of the state-of-the-art techniques is greatly diminished because they require a large number of objective
evaluations, which makes them impractical for problems of the above kind. Bayesian global optimization (BGO) has
managed to deal with these challenges in solving single-objective optimization problems and has recently been extended to multiobjective optimization (MOO). BGO models the objectives via probabilistic surrogates and uses the epistemic uncertainty to define an information acquisition function (IAF) that quantifies the merit of evaluating the objective at new designs. This iterative data acquisition process continues until a stopping criterion is met. The most commonly used IAF for MOO is the expected improvement over the dominated hypervolume (EIHV) which in its original form is unable to deal with parametric uncertainties or measurement noise. In this work, we provide a systematic reformulation of EIHV to deal with stochastic MOO problems. The primary contribution of this paper lies in being able to filter out the noise and reformulate the EIHV without having to observe or estimate the stochastic parameters. An addendum of the probabilistic nature of our methodology is that it enables us to characterize our confidence about the predicted Pareto front. We verify and validate the proposed methodology by applying it to synthetic test problems with known solutions. We demonstrate our approach on an industrial problem of die pass design for a steel-wire drawing process.
A METRIC ON UNCERTAIN VARIABLES
251-266
10.1615/Int.J.UncertaintyQuantification.2018020455
Tingqing
Ye
School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu,
China
Yuanguo
Zhu
School of Science, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu,
China
metric
uncertain variable
uncertain sequence
convergence
Metric is a vital concept on a space. In this paper, a new metric is proposed to study the space consisting of uncertain
variables. Then, some formulas of calculating the metric in some cases are presented. Finally, the convergence of
uncertain sequences in metric is defined and the relationships among convergence in metric, convergence in measure,
convergence in mean, convergence in distribution and uniform convergence almost surely are studied.
HESITANT PYTHAGOREAN FUZZY SETS AND THEIR AGGREGATION OPERATORS IN MULTIPLE ATTRIBUTE DECISION-MAKING
267-289
10.1615/Int.J.UncertaintyQuantification.2018020979
Harish
Garg
School of Mathematics, Thapar Institute of Engineering and Technology (Deemed University)
Patiala 147004, Punjab, India
hesitant Pythagorean fuzzy set
hesitant fuzzy set
Pythagorean fuzzy set
aggregation operators
multi-attribute decision-making
In this article, a new concept of the hesitant Pythagorean fuzzy sets has been presented by combining the concept
of the Pythagorean as well as the Hesitant fuzzy sets. Some of the basic operations laws and their properties have
been investigated. Further, we have developed some new weighted averaging and geometric aggregation operators
named as hesitant Pythagorean fuzzy weighted average and geometric, ordered weighted average and geometric, hybrid
average and geometric with hesitant Pythagorean fuzzy information. The properties of these aggregation operators
are investigated. The proposed set is the generalization of the sets of fuzzy, intuitionistic fuzzy, hesitant fuzzy, and
Pythagorean fuzzy. Additionally, a multiple-attribute decision-making approach is established based on these operators under hesitant Pythagorean fuzzy environment and an example is given to illustrate the application of it. Finally, we compare the results with the existing methods to show the effectiveness of it.