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Reduced chaos expansions with random coefficients in reduced-dimensional stochastic modeling of coupled problems Arnst, Maarten ; ; et al in International Journal for Numerical Methods in Engineering (2014), 97 We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed ... [more ▼] We address the curse of dimensionality in methods for solving stochastic coupled problems with an emphasis on stochastic expansion methods such as those involving polynomial chaos expansions. The proposed method entails a partitioned iterative solution algorithm that relies on a reduced-dimensional representation of information exchanged between subproblems to allow each subproblem to be solved within its own stochastic dimension while interacting with a reduced projection of the other subproblems. The proposed method extends previous work by the authors by introducing a reduced chaos expansion with random coefficients. The representation of the exchanged information by using this reduced chaos expansion with random coefficients enables an expeditious construction of doubly stochastic polynomial chaos expansions that separate the effect of uncertainty local to a subproblem from the effect of statistically independent uncertainty coming from other subproblems through the coupling. After laying out the theoretical framework, we apply the proposed method to a multiphysics problem from nuclear engineering. [less ▲] Detailed reference viewed: 51 (25 ULg)Stochastic Dimension Reduction of Multi Physics Systems through Measure Transformation ; ; et al Conference (2013, February 26) Uncertainty quantification of multiphysics systems represents numerous mathematical and computational challenges. Indeed, uncertainties that arise in each physics in a fully coupled system must be ... [more ▼] Uncertainty quantification of multiphysics systems represents numerous mathematical and computational challenges. Indeed, uncertainties that arise in each physics in a fully coupled system must be captured throughout the whole system, the so-called curse of dimensionality. We present techniques for mitigating the curse of dimensionality in network-coupled multiphysics systems by using the structure of the network to transform uncertainty representations as they pass between components. Examples from the simulation of nuclear power plants will be discussed. [less ▲] Detailed reference viewed: 90 (14 ULg)Hybrid Sampling/Spectral Method for Solving Stochastic Coupled Problems Arnst, Maarten ; ; in SIAM/ASA Journal on Uncertainty Quantification (2013), 1(1), 218-243 In this paper, we present a hybrid method that combines Monte Carlo sampling and spectral methods for solving stochastic coupled problems. After partitioning the stochastic coupled problem into subsidiary ... [more ▼] In this paper, we present a hybrid method that combines Monte Carlo sampling and spectral methods for solving stochastic coupled problems. After partitioning the stochastic coupled problem into subsidiary subproblems, the proposed hybrid method entails iterating between these subproblems in a way that enables the use of the Monte Carlo sampling method for subproblems that depend on a very large number of uncertain parameters and the use of spectral methods for subproblems that depend on only a small or moderate number of uncertain parameters. To facilitate communication between the subproblems, the proposed hybrid method shares between the subproblems a reference representation of all the solution random variables in the form of an ensemble of samples; for each subproblem solved by a spectral method, it uses a dimension-reduction technique to transform this reference representation into a subproblem-specific reduced-dimensional representation to facilitate a computationally efficient solution in a reduced-dimensional space. After laying out the theoretical framework, we provide an example relevant to microelectomechanical systems. [less ▲] Detailed reference viewed: 22 (8 ULg)Measure transformation and efficient quadrature in reduced-dimensional stochastic modeling of coupled problems Arnst, Maarten ; ; et al in International Journal for Numerical Methods in Engineering (2012), 92 Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of ... [more ▼] Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower-dimensional space than the sources themselves. In this work, we thus propose to use a dimension reduction technique for obtaining the representation of the exchanged information, and we propose a measure transformation technique that allows subproblem implementations to exploit this dimension reduction to achieve computational gains. The effectiveness of the proposed dimension reduction and measure transformation methodology is demonstrated through a multiphysics problem relevant to nuclear engineering. [less ▲] Detailed reference viewed: 24 (12 ULg)Dimension reduction in stochastic modeling of coupled problems Arnst, Maarten ; ; et al in International Journal for Numerical Methods in Engineering (2012), 92 Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of ... [more ▼] Coupled problems with various combinations of multiple physics, scales, and domains are found in numerous areas of science and engineering. A key challenge in the formulation and implementation of corresponding coupled numerical models is to facilitate the communication of information across physics, scale, and domain interfaces, as well as between the iterations of solvers used for response computations. In a probabilistic context, any information that is to be communicated between subproblems or iterations should be characterized by an appropriate probabilistic representation. Although the number of sources of uncertainty can be expected to be large in most coupled problems, our contention is that exchanged probabilistic information often resides in a considerably lower dimensional space than the sources themselves. This work thus presents an investigation into the characterization of the exchanged information by a reduced-dimensional representation and in particular by an adaptation of the Karhunen-Loève decomposition. The effectiveness of the proposed dimension–reduction methodology is analyzed and demonstrated through a multiphysics problem relevant to nuclear engineering. [less ▲] Detailed reference viewed: 45 (16 ULg)Stochastic Dimension Reduction Techniques for Uncertainty Quantification of Multiphysics Systems ; Arnst, Maarten ; et al Conference (2012, April 02) Uncertainty quantification of multiphysics systems represents numerous mathematical and computational challenges. Indeed, uncertainties that arise in each physics in a fully coupled system must be ... [more ▼] Uncertainty quantification of multiphysics systems represents numerous mathematical and computational challenges. Indeed, uncertainties that arise in each physics in a fully coupled system must be captured throughout the whole system, the so-called curse of dimensionality. We present techniques for mitigating the curse of dimensionality in network-coupled multiphysics systems by using the structure of the network to transform uncertainty representations as they pass between components. Examples from the simulation of nuclear power plants will be discussed. [less ▲] Detailed reference viewed: 77 (8 ULg)Dimension Reduction and Measure Transformation in Stochastic Multiphysics Modeling Arnst, Maarten ; ; et al Conference (2012, April 02) We present a computational framework based on stochastic expansion methods for the efficient propagation of uncertainties through multiphysics models. The framework leverages an adaptation of the Karhunen ... [more ▼] We present a computational framework based on stochastic expansion methods for the efficient propagation of uncertainties through multiphysics models. The framework leverages an adaptation of the Karhunen-Loeve decomposition to extract a low-dimensional representation of information passed from component to component in a stochastic coupled model. After a measure transformation, the reduced-dimensional interface thus created enables a more efficient solution in a reduced-dimensional space. We demonstrate the proposed approach on an illustration problem from nuclear engineering [less ▲] Detailed reference viewed: 24 (2 ULg)Dimension Reduction and Measure Transformation in Stochastic Analysis of Coupled Systems Arnst, Maarten ; ; et al Scientific conference (2011, September 29) Detailed reference viewed: 45 (9 ULg)Dimension Reduction and Measure Transformation in Stochastic Simulations of Coupled Systems Arnst, Maarten ; ; et al Conference (2011, July 25) Detailed reference viewed: 25 (1 ULg)Uncertain Handshaking for Coupled Physics ; Arnst, Maarten ; et al Conference (2011, July 18) Detailed reference viewed: 30 (4 ULg)Random Handshaking and Information Recovery Between Scales and Models ; Arnst, Maarten ; et al Conference (2011, July 05) Detailed reference viewed: 27 (4 ULg)Dimension reduction and measure transformation in stochastic multiphysics modeling Arnst, Maarten ; ; et al Scientific conference (2011, March 31) Detailed reference viewed: 37 (7 ULg)Coupling Algorithms for Stochastic Multiphysics Arnst, Maarten ; ; et al Conference (2011, March 02) Detailed reference viewed: 25 (4 ULg)A variational-inequality approach to stochastic boundary value problems with inequality constraints and its application to contact and elastoplasticity Arnst, Maarten ; in International Journal for Numerical Methods in Engineering (2011) This paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well ... [more ▼] This paper is concerned with stochastic boundary value problems (SBVPs) whose formulation involves inequality constraints. A class of stochastic variational inequalities (SVIs) is defined, which is well adapted to characterize the solution of specified inequality-constrained SBVPs. A methodology for solving such SVIs is proposed, which involves their discretization by projection onto polynomial chaos and collocation of the inequality constraints, followed by the solution of a finite-dimensional constrained optimization problem. Simulation studies in contact and elastoplasticity are provided to demonstrate the proposed framework. [less ▲] Detailed reference viewed: 37 (4 ULg)Maximum entropy approach to the identification of stochastic reduced-order models of nonlinear dynamical systems Arnst, Maarten ; ; in Aeronautical Journal (2010), 114(1160), 637-650 Data-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These ... [more ▼] Data-driven methodologies based on the restoring force method have been developed over the past few decades for building predictive reduced-order models (ROMs) of nonlinear dynamical systems. These methodologies involve fitting a polynomial expansion of the restoring force in the dominant state variables to observed states of the system. ROMs obtained in this way are usually prone to errors and uncertainties due to the approximate nature of the polynomial expansion and experimental limitations. We develop in this article a stochastic methodology that endows these errors and uncertainties with a probabilistic structure in order to obtain a quantitative description of the proximity between the ROM and the system that it purports to represent. Specifically, we propose an entropy maximization procedure for constructing a multi-variate probability distribution for the coefficients of power-series expansions of restoring forces. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed framework. [less ▲] Detailed reference viewed: 48 (2 ULg)Identification of Bayesian posteriors for coefficients of chaos expansions Arnst, Maarten ; ; in Journal of Computational Physics (2010), 229(9), 3134-3154 This article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data. The data used for the identification consist of measurements of ... [more ▼] This article is concerned with the identification of probabilistic characterizations of random variables and fields from experimental data. The data used for the identification consist of measurements of several realizations of the uncertain quantities that must be characterized. The random variables and fields are approximated by a polynomial chaos expansion, and the coefficients of this expansion are viewed as unknown parameters to be identified. It is shown how the Bayesian paradigm can be applied to formulate and solve the inverse problem. The estimated polynomial chaos coefficients are hereby themselves characterized as random variables whose probability density function is the Bayesian posterior. This allows to quantify the impact of missing experimental information on the accuracy of the identified coefficients, as well as on subsequent predictions. An illustration in stochastic aeroelastic stability analysis is provided to demonstrate the proposed methodology. [less ▲] Detailed reference viewed: 31 (7 ULg)Probabilistic Electromechanical Modeling of Nanostructures with Random Geometry Arnst, Maarten ; in Journal of Computational and Theoretical Nanoscience (2009), 6(10), 2256-2272 This article is concerned with the probabilistic modeling of the electromechanical behavior of nanostructures to assess the effect of variations in geometrical characteristics on the device performance ... [more ▼] This article is concerned with the probabilistic modeling of the electromechanical behavior of nanostructures to assess the effect of variations in geometrical characteristics on the device performance. The topological uncertainty that may be present in the position of the boundaries of nanostructures is accommodated by treating these boundaries as stochastic processes. It is shown how the probabilistic electromechanical models thus obtained can be discretized with the help of Galerkin projections on polynomial chaos expansions. An illustration is provided to demonstrate the proposed framework. [less ▲] Detailed reference viewed: 19 (3 ULg)Probabilistic equivalence and stochastic model reduction in multiscale analysis Arnst, Maarten ; in Computer Methods in Applied Mechanics & Engineering (2008), 197(43-44), 3584-3592 This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic ... [more ▼] This paper presents a probabilistic upscaling of mechanics models. A reduced-order probabilistic model is constructed as a coarse-scale representation of a specified fine-scale model whose probabilistic structure can be accurately determined. Equivalence of the fine- and coarse-scale representations is identified such that a reduction in the requisite degrees of freedom can be achieved while accuracy in certain quantities of interest is maintained. A significant stochastic model reduction can a priori be expected if a separation of spatial and temporal scales exists between the fine- and coarse-scale representations. The upscaling of probabilistic models is subsequently formulated as an optimization problem suitable for practical computations. An illustration in stochastic structural dynamics is provided to demonstrate the proposed framework. [less ▲] Detailed reference viewed: 32 (8 ULg) |
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