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103
GALERKIN FINITE ELEMENT APPROXIMATIONS OF STOCHASTIC ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS
, 2004
"... We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the ..."
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Cited by 193 (11 self)
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We describe and analyze two numerical methods for a linear elliptic problem with stochastic coefficients and homogeneous Dirichlet boundary conditions. Here the aim of the computations is to approximate statistical moments of the solution, and, in particular, we give a priori error estimates for the computation of the expected value of the solution. The first method generates independent identically distributed approximations of the solution by sampling the coefficients of the equation and using a standard Galerkin finite element variational formulation. The Monte Carlo method then uses these approximations to compute corresponding sample averages. The second method is based on a finite dimensional approximation of the stochastic coefficients, turning the original stochastic problem into a deterministic parametric elliptic problem. A Galerkin finite element method, of either the h or pversion, then approximates the corresponding deterministic solution, yielding approximations of the desired statistics. We present a priori error estimates and include a comparison of the computational work required by each numerical approximation to achieve a given accuracy. This comparison suggests intuitive conditions for an optimal selection of the numerical approximation.
A generalized spectral decomposition technique to solve a class of linear stochastic . . .
, 2007
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Extensible Lattice Sequences For QuasiMonte Carlo Quadrature
 SIAM Journal on Scientific Computing
, 1999
"... Integration lattices are one of the main types of low discrepancy sets used in quasiMonte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first b m of which form a lattice for any nonnegative ..."
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Cited by 35 (11 self)
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Integration lattices are one of the main types of low discrepancy sets used in quasiMonte Carlo methods. However, they have the disadvantage of being of fixed size. This article describes the construction of an infinite sequence of points, the first b m of which form a lattice for any nonnegative integer m. Thus, if the quadrature error using an initial lattice is too large, the lattice can be extended without discarding the original points. Generating vectors for extensible lattices are found by minimizing a loss function based on some measure of discrepancy or nonuniformity of the lattice. The spectral test used for finding pseudorandom number generators is one important example of such a discrepancy. The performance of the extensible lattices proposed here is compared to that of other methods for some practical quadrature problems.
Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations
, 2009
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Learning variable impedance control
 International Journal of Robotics Research
, 2011
"... One of the hallmarks of the performance, versatility, and robustness of biological motor control is the ability to adapt the impedance of the overall biomechanical system to different task requirements and stochastic disturbances. A transfer of this principle to robotics is desirable, for instance t ..."
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Cited by 25 (9 self)
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One of the hallmarks of the performance, versatility, and robustness of biological motor control is the ability to adapt the impedance of the overall biomechanical system to different task requirements and stochastic disturbances. A transfer of this principle to robotics is desirable, for instance to enable robots to work robustly and safely in everyday human environments. It is, however, not trivial to derive variable impedance controllers for practical high degreeoffreedom (DOF) robotic tasks. In this contribution, we accomplish such variable impedance control with the reinforcement learning (RL) algorithm PI2 (Policy Improvement with Path Integrals). PI2 is a modelfree, sampling based learning method derived from first principles of stochastic optimal control. The PI2 algorithm requires no tuning of algorithmic parameters besides the exploration noise. The designer can thus fully focus on cost function design to specify the task. From the
A study on algorithms for optimization of Latin hypercubes
 Journal of Statistical Planning and Inference
, 2006
"... A crucial component in the statistical simulation of a computationally expensive model is a good design of experiments. In this paper we compare the efficiency of the columnwisepairwise (CP) and genetic algorithms for the optimization of Latin hypercubes (LH) for the purpose of sampling in statisti ..."
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Cited by 22 (4 self)
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A crucial component in the statistical simulation of a computationally expensive model is a good design of experiments. In this paper we compare the efficiency of the columnwisepairwise (CP) and genetic algorithms for the optimization of Latin hypercubes (LH) for the purpose of sampling in statistical investigations. The performed experiments indicate, among other results, that CP methods are most efficient for small and medium size LH while an adopted genetic algorithm performs better for large LH. Two optimality criteria suggested in the literature are evaluated with respect to statistical properties and efficiency. The obtained results lead us to favor a criterion based on the physical analogy of minimization of forces between charged particles suggested in [1] over a ‘maximin distance ’ criterion from [9].
Numerical methods and Smolyak quadrature for nonlinear stochastic partial differential equations
, 2003
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eXtended Stochastic Finite Element Method for the numerical simulation of heterogenous materials with random material interfaces
, 2010
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Rotor walks and Markov chains
 IN ALGORITHMIC PROBABILITY AND COMBINATORICS, AMERICAN MATHEMATICAL SOCIETY
, 2010
"... The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to countable Markov ..."
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Cited by 14 (6 self)
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The rotor walk is a derandomized version of the random walk on a graph. On successive visits to any given vertex, the walker is routed to each of the neighboring vertices in some fixed cyclic order, rather than to a random sequence of neighbors. The concept generalizes naturally to countable Markov chains. Subject to general conditions, we prove that many natural quantities associated with the rotor walk (including normalized hitting frequencies, hitting times and occupation frequencies) concentrate around their expected values for the random walk. Furthermore, the concentration is stronger than that associated with repeated runs of the random walk; the discrepancy is at most C/n after n runs (for an explicit constant C), rather than c / √ n. 1
Adaptive sampling using support vector machines
 in Proceeding of American Nuclear Society
, 2012
"... Reliability/safety analysis of stochastic dynamic systems (e.g., nuclear power plants, airplanes, chemical plants) is currently performed through a combination of EventTrees and FaultTrees [1]. However, these methods are characterized by ..."
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Cited by 13 (7 self)
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Reliability/safety analysis of stochastic dynamic systems (e.g., nuclear power plants, airplanes, chemical plants) is currently performed through a combination of EventTrees and FaultTrees [1]. However, these methods are characterized by