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Register a new userSome approximation results for mild solutions of stochastic fractional order evolution equations driven by Gaussian noise
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Added by
Mihaly Kovacs Dr
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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We investigate the quality of space approximation of a class of stochastic integral equations of convolution type with Gaussian noise. Such equations arise, for example, when considering mild solutions of stochastic fractional order partial differential equations but also when considering mild solutions of classical stochastic partial differential equations. The key requirement for the equations is a smoothing property of the deterministic evolution operator which is typical in parabolic type problems. We show that if one has access to nonsmooth data estimates for the deterministic error operator together with its derivative of a space discretization procedure, then one obtains error estimates in pathwise Holder norms with rates that can be read off the deterministic error rates. We illustrate the main result by considering a class of stochastic fractional order partial differential equations and space approximations performed by spectral Galerkin methods and finite elements. We also improve an existing result on the stochastic heat equation.
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We establish a general theory of optimal strong error estimation for numerical approximations of a second-order parabolic stochastic partial differential equation with monotone drift driven by a multiplicative infinite-dimensional Wiener process. The equation is spatially discretized by Galerkin methods and temporally discretized by drift-implicit Euler and Milstein schemes. By the monotone and Lyapunov assumptions, we use both the variational and semigroup approaches to derive a spatial Sobolev regularity under the $L_omega^p L_t^infty dot H^{1+gamma}$-norm and a temporal Holder regularity under the $L_omega^p L_x^2$-norm for the solution of the proposed equation with an $dot H^{1+gamma}$-valued initial datum for $gammain [0,1]$. Then we make full use of the monotonicity of the equation and tools from stochastic calculus to derive the sharp strong convergence rates $O(h^{1+gamma}+tau^{1/2})$ and $O(h^{1+gamma}+tau^{(1+gamma)/2})$ for the Galerkin-based Euler and Milstein schemes, respectively.
In this paper, we consider the strong convergence of the time-space fractional diffusion equation driven by fractional Gaussion noise with Hurst index $Hin(frac{1}{2},1)$. A sharp regularity estimate of the mild solution and the numerical scheme constructed by finite element method for integral fractional Laplacian and backward Euler convolution quadrature for Riemann-Liouville time fractional derivative are proposed. With the help of inverse Laplace transform and fractional Ritz projection, we obtain the accurate error estimates in time and space. Finally, our theoretical results are accompanied by numerical experiments.
Fractional order controllers become increasingly popular due to their versatility and superiority in various performance. However, the bottleneck in deploying these tools in practice is related to their analog or numerical implementation. Numerical approximations are usually employed in which the approximation of fractional differintegrator is the foundation. Generally, the following three identical equations always hold, i.e., $frac{1}{s^alpha}frac{1}{s^{1-alpha}} = frac{1}{s}$, $s^alpha frac{1}{s^alpha} = 1$ and $s^alpha s^{1-alpha} = s$. However, for the approximate models of fractional differintegrator $s^alpha$, $alphain(-1,0)cup(0,1)$, there usually exist some conflicts on the mentioned equations, which might enlarge the approximation error or even cause fallacies in multiple orders occasion. To overcome the conflicts, this brief develops a piecewise approximate model and provides two procedures for designing the model parameters. The comparison with several existing methods shows that the proposed methods do not only satisfy the equalities but also achieve high approximation accuracy. From this, it is believed that this work can serve for simulation and realization of fractional order controllers more friendly.
For semilinear stochastic evolution equations whose coefficients are more general than the classical global Lipschitz, we present results on the strong convergence rates of numerical discretizations. The proof of them provides a new approach to strong convergence analysis of numerical discretizations for a large family of second order parabolic stochastic partial differential equations driven by space-time white noises. We apply these results to the stochastic advection-diffusion-reaction equation with a gradient term and multiplicative white noise, and show that the strong convergence rate of a fully discrete scheme constructed by spectral Galerkin approximation and explicit exponential integrator is exactly $frac12$ in space and $frac14$ in time. Compared with the optimal regularity of the mild solution, it indicates that the spetral Galerkin approximation is superconvergent and the convergence rate of the exponential integrator is optimal. Numerical experiments support our theoretical analysis.
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 spatial discretization and convolution quadrature for time discretization. Sharp-order convergence of the numerical solutions is proved up to a logarithmic factor. Numerical examples are provided to support the theoretical analysis.
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