An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk

Zou, Zilong; Kouri, Drew P.; Aquino, Wilkins

doi:10.1016/j.cma.2018.10.028

Cited by 19 publications

(20 citation statements)

References 34 publications

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“…Again, the proposed approach provides approximately two orders of magnitude improvement over the structured adaptive sparse grid methods due to the flexibility of working with an unstructured set of data points. Furthermore, convergence is comparable to the reduced‐basis method proposed in the work of Zou et al…”

Section: Numerical Examplesmentioning

confidence: 59%

“…This problem has previously been solved by Zou et al in a forthcoming work, which proposes an adaptive reduced‐basis method for similar problems. The results presented here are compared with those presented in the work of Zou et al and have been developed through personal correspondence with the authors.…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Convergence of the proposed method is quantified by computing the average relative L 2 error defined as

e_{u} = \frac{1}{N} true \sum_{j = 1}^{N} \frac{false| false| ũ_{j} false(\cdot, bold-italicξ false) - u_{j} false(\cdot, bold-italicξ false) false| |_{L_{2} false(D false)}}{false| false| u_{j} false(\cdot, bold-italicξ false) false| |_{L_{2} false(D false)}},

where N = 4000 Monte Carlo samples drawn from the joint beta pdf. Results from the proposed approach are compared with those of the ASGC (ASG1 refers to the method of Ganapathysubmaranian and Zabaras and ASG2 refers to the method of Ma and Zabaras) and the local reduced‐basis method of Zou et al in Figure . Results of the ASGC and reduced‐basis method are plotted as reported in the aforementioned work .…”

Section: Numerical Examplesmentioning

confidence: 99%

“…Results from the proposed approach are compared with those of the ASGC (ASG1 refers to the method of Ganapathysubmaranian and Zabaras and ASG2 refers to the method of Ma and Zabaras) and the local reduced‐basis method of Zou et al in Figure . Results of the ASGC and reduced‐basis method are plotted as reported in the aforementioned work . Again, the proposed approach provides approximately two orders of magnitude improvement over the structured adaptive sparse grid methods due to the flexibility of working with an unstructured set of data points.…”

Section: Numerical Examplesmentioning

confidence: 99%

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Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems

Giovanis

Shields

2018

Numerical Meth Engineering

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In this work, an adaptive simplex stochastic collocation method is introduced in which sample refinement is informed by variability in the solution of the system. The proposed method is based on the concept of multi-element stochastic collocation methods and is capable of dealing with very high-dimensional models whose solutions are expressed as a vector, a matrix, or a tensor.The method leverages random samples to create a multi-element polynomial chaos surrogate model that incorporates local anisotropy in the refinement, informed by the variance of the estimated solution. This feature makes it beneficial for strongly nonlinear and/or discontinuous problems with correlated non-Gaussian uncertainties. To solve large systems, a reduced-order model (ROM) of the high-dimensional response is identified using singular value decomposition (higher-order SVD for matrix/tensor solutions) and polynomial chaos is used to interpolate the ROM. The method is applied to several stochastic systems of varying type of response (scalar/vector/matrix) and it shows considerable improvement in performance compared to existing simplex stochastic collocation methods and adaptive sparse grid collocation methods.

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Section: Numerical Examplesmentioning

confidence: 59%

Section: Numerical Examplesmentioning

confidence: 99%

“…Convergence of the proposed method is quantified by computing the average relative L 2 error defined as

e_{u} = \frac{1}{N} true \sum_{j = 1}^{N} \frac{false| false| ũ_{j} false(\cdot, bold-italicξ false) - u_{j} false(\cdot, bold-italicξ false) false| |_{L_{2} false(D false)}}{false| false| u_{j} false(\cdot, bold-italicξ false) false| |_{L_{2} false(D false)}},

Section: Numerical Examplesmentioning

confidence: 99%

Section: Numerical Examplesmentioning

confidence: 99%

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Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems

Giovanis

Shields

2018

Numerical Meth Engineering

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show abstract

“…The adaptive nature of the construction which, moreover, is informed by the error estimate makes it potentially more effective than the approach of Reference 12. There have been other techniques in the literature to alleviate the RB offline cost such as the parameter domain adaptivity, 15,16 greedy sampling acceleration through nonlinear optimization, 17,18 and local RBM, 19‐22 and so forth.…”

Section: Introductionmentioning

confidence: 99%

Adaptive greedy algorithms based on parameter‐domain decomposition and reconstruction for the reduced basis method

Jiang

Chen

2020

Numerical Meth Engineering

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The reduced basis method (RBM) empowers repeated and rapid evaluation of parametrized partial differential equations through an offline–online decomposition, a.k.a. a learning‐execution process. A key feature of the method is a greedy algorithm repeatedly scanning the training set, a fine discretization of the parameter domain, to identify the next dimension of the parameter‐induced solution manifold along which we expand the surrogate solution space. Although successfully applied to problems with fairly high parametric dimensions, the challenge is that this scanning cost dominates the offline cost due to it being proportional to the cardinality of the training set which is exponential with respect to the parameter dimension. In this work, we review three recent attempts in effectively delaying this curse of dimensionality, and propose two new hybrid strategies through successive refinement and multilevel maximization of the error estimate over the training set. All five offline‐enhanced methods and the original greedy algorithm are tested and compared on two types of problems: the thermal block problem and the geometrically parameterized Helmholtz problem.

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Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

Heinkenschloss,

Magruder

2022

Realization and Model Reduction of Dynamical Systems

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An adaptive local reduced basis method for solving PDEs with uncertain inputs and evaluating risk

Cited by 19 publications

References 34 publications

Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems

Variance‐based simplex stochastic collocation with model order reduction for high‐dimensional systems

Adaptive greedy algorithms based on parameter‐domain decomposition and reconstruction for the reduced basis method

Reduced Order Model Hessian Approximations in Newton Methods for Optimal Control

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