Reduction, Approximation, Machine Learning, Surrogates, Emulators and Simulators
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Article
A Multifidelity Monte Carlo Method for Realistic Computational Budgets
A method for the multifidelity Monte Carlo (MFMC) estimation of statistical quantities is proposed which is applicable to computational budgets of any size. Based on a sequence of optimization problems each wi...
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Article
Stochastic Collocation Method for Stochastic Optimal Boundary Control of the Navier–Stokes Equations
We consider the optimal control of a system governed by the Navier–Stokes equations with stochastic Dirichlet boundary conditions. Control conditions imposed only on the boundary are associated with reduced re...
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Article
Approximation of Probability Density Functions for PDEs with Random Parameters Using Truncated Series Expansions
The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The ...
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Article
High-precision computation of the weak Galerkin methods for the fourth-order problem
The weak Galerkin form of the finite element method, requiring only C0 basis function, is applied to the biharmonic equation. The computational procedure is thoroughly considered. Local orthogonal bases on triang...
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Chapter
Piecewise Polynomial Approximation of Probability Density Functions with Application to Uncertainty Quantification for Stochastic PDEs
The probability density function (PDF) associated with a given set of samples is approximated by a piecewise-linear polynomial constructed with respect to a binning of the sample space. The kernel functions ar...
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Article
An Improved Discrete Least-Squares/Reduced-Basis Method for Parameterized Elliptic PDEs
It is shown that the computational efficiency of the discrete least-squares (DLS) approximation of solutions of stochastic elliptic PDEs is improved by incorporating a reduced-basis method into the DLS framewo...
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Article
Convergence of finite element solutions of stochastic partial integro-differential equations driven by white noise
Numerical approximation of a stochastic partial integro-differential equation driven by a space-time white noise is studied by truncating a series representation of the noise, with finite element method for sp...
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Chapter and Conference Paper
An Applied/Computational Mathematician’s View of Uncertainty Quantification for Complex Systems
Uncertainty quantification (UQ) is defined differently by different disciplines. Here, we first review an applied and computational mathematician’s definition of UQ for complex systems, especially in the conte...
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Reference Work Entry In depth
Recent Progress in Mathematical and Computational Aspects of Peridynamics
Recent developments in the mathematical and computational aspects of the nonlocal peridynamic model for material mechanics are provided. Based on a recently developed vector calculus for nonlocal operators, a ...
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Article
Convergence analysis of multifidelity Monte Carlo estimation
The multifidelity Monte Carlo method provides a general framework for combining cheap low-fidelity approximations of an expensive high-fidelity model to accelerate the Monte Carlo estimation of statistics of t...
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Living Reference Work Entry In depth
Recent Progress in Mathematical and Computational Aspects of Peridynamics
Recent developments in the mathematical and computational aspects of the nonlocal peridynamic model for material mechanics are provided. Based on a recently developed vector calculus for nonlocal operators, a ...
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Reference Work Entry In depth
Sparse Collocation Methods for Stochastic Interpolation and Quadrature
In this chapter, the authors survey the family of sparse stochastic collocation methods (SCMs) for partial differential equations with random input data. The SCMs under consideration can be viewed as a special...
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Article
An efficient and long-time accurate third-order algorithm for the Stokes–Darcy system
A third-order in time numerical IMEX-type algorithm for the Stokes–Darcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel third-order Adams–Moulton scheme is used for the di...
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Article
Asymptotically compatible schemes for the approximation of fractional Laplacian and related nonlocal diffusion problems on bounded domains
Approximations of solutions of fractional Laplacian equations on bounded domains are considered. Such equations allow global interactions between points separated by arbitrarily large distances. Two approximat...
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Article
A two phase field model for tracking vesicle–vesicle adhesion
A multi-phase-field model for simulating the adhesion between two vesicles is constructed. Two phase field functions are introduced to simulate each of the two vesicles. An energy model is defined which accoun...
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Article
A generalized nonlocal vector calculus
A nonlocal vector calculus was introduced in Du et al. (Math Model Meth Appl Sci 23:493–540, 2013) that has proved useful for the analysis of the peridynamics model of nonlocal mechanics and nonlocal diffusion mo...
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Article
Peridynamics and Material Interfaces
The convergence of a peridynamic model for solid mechanics inside heterogeneous media in the limit of vanishing nonlocality is analyzed. It is shown that the operator of linear peridynamics for an isotropic he...
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Article
Fractional Diffusion on Bounded Domains
The mathematically correct specification of a fractional differential equation on a bounded domain requires specification of appropriate boundary conditions, or their fractional analogue. This paper discusses ...
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Reference Work Entry In depth
Least Squares Finite Element Methods
