We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (montecarlo) algorithms. This quadrature problem is shown to be closely related to the problem of qua...
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We study numerical integration of Lipschitz functionals on a Banach space by means of deterministic and randomized (montecarlo) algorithms. This quadrature problem is shown to be closely related to the problem of quantization and to the average Kolmogorov widths of the underlying probability measure. In addition to the general setting, we analyze, in particular, integration with respect to Gaussian measures and distributions of diffusion processes. We derive lower bounds for the worst case error of every algorithm in terms of its cost, and we present matching upper bounds, up to logarithms, and corresponding almost optimal algorithms. As auxiliary results, we determine the asymptotic behavior of quantization numbers and Kolmogorov widths for diffusion processes.
We consider initial value problems for parameter dependent ordinary differential equations with values in a Banach space and study their complexity both in the deterministic and the randomized setting, for input data ...
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We consider initial value problems for parameter dependent ordinary differential equations with values in a Banach space and study their complexity both in the deterministic and the randomized setting, for input data from various smoothness classes. We develop multilevelalgorithms, investigate the convergence of their deterministic and stochastic versions, and prove lower bounds. (C) 2014 Elsevier Inc. All rights reserved.
In this paper, we consider the weakly singular stochastic Volterra integral equations with variable exponent. Firstly, the existence and uniqueness of the equations are studied by the Banach contraction mapping princi...
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In this paper, we consider the weakly singular stochastic Volterra integral equations with variable exponent. Firstly, the existence and uniqueness of the equations are studied by the Banach contraction mapping principle. Secondly, we develop an Euler-Maruyama (EM) method and obtain its strong convergence rate. Moreover, we propose a fast EM method via the exponential-sum-approximation technique to reduce the EM method's computational cost. More specifically, if one disregards the montecarlo sampling error, then the fast EM method reduces the computational cost from O(N-2) to O(N log(2) N) and the storage from O(N) to O(log(2) N), where N is the total number of time steps. Moreover, if the sampling error is taken into account, we employ the multilevelmontecarlo method based on the fast EM method to reduce computational costs further. Significantly, the computational costs of the EM method and the fast EM method to achieve an accuracy of O(epsilon) (epsilon < 1) are reduced from O(epsilon(-2-2/)((alpha) over tilde)) and O(epsilon(-2-2/)((alpha) over tilde) log(2)(epsilon)), respectively, to O(epsilon(-1/)((alpha) over tilde)(log(epsilon(-1)))(3)), where (alpha) over tilde = min {1 - alpha*, 1/2 - beta*} is related to the exponents of the singular kernel in the equations. Finally, numerical examples are provided to illustrate our theoretical results and demonstrate the superiority of the fast EM method.
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