关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Stochastic equations in infinite dimensions

In this chapter we introduce some basic notation and definitions. We discuss a few general results on interpolation spaces. The most important one is the Aronszajn-Gagliardo theorem.

General Properties of Interpolation Spaces

On the low-rank approximation by the pivoted Cholesky decomposition

A local convergence theorem for calculating canonical low-rank tensor approximations (PARAFAC, CANDECOMP) by the alternating least squares algorithm is established. The main assumption is that the Hessian matrix of the problem is positive definite modulo the scaling indeterminacy. A discussion, whether this is realistic, and numerical illustrations are included. Also regularization is addressed.

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Local Convergence of the Alternating Least Squares Algorithm for Canonical Tensor Approximation

We study the spatial regularity of semilinear parabolic stochastic partial differential equations on bounded Lipschitz domains &#x1D4AA;⊆ ℝ d in the scale , 1/τ=α/ d +1/ p, p ≥2 fixed. The Besov smoothness in this scale determines the order of convergence that can be achieved by adaptive numerical algorithms and other nonlinear approximation schemes. The proofs are performed by establishing weighted Sobolev estimates and combining them with wavelet characterizations of Besov spaces.

Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains

We use the scale of Besov spaces B^\alpha_{\tau,\tau}(O), \alpha>0, 1/\tau=\alpha/d+1/p, p fixed, to study the spatial regularity of the solutions of linear parabolic stochastic partial differential equations on bounded Lipschitz domains O\subset R^d. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

We investigate the regularity of linear stochastic parabolic equations with zero Dirichlet boundary condition on bounded Lipschitz domains $\mathcal{O}\subset \mathbb{R}^d$ with both theoretical and numerical purpose. We use N.V. Krylov's framework of stochastic parabolic weighted Sobole spaces $\mathfrak{H}^{\gamma,q}_{p,\theta } (\mathcal{O},T)$ . The summability parameters $p$ and $q$ in space and time may differ. Existence and uniqueness of solutions in these spaces is established and the H o lder regularity in time is analysed. Moreover, we prove a general embedding of weighte $L_p(\mathcal{O})$ - Sobolev spaces into the scale o Besov spaces $B^\alpha_{\tau,\tau}(\mathcal{O})$ , $1/\tau=\alpha/d+1/p$ , $\alpha>0$ . This leads to a H o lder- Besov regularity result for the solution process. The regularity in this Besov scale determines the order of convergence that can be achieved by certain nonlinear approximation schemes.

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On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

We use the scale of Besov spacesB ; (O), 1= = =d +1=p, > 0,p xed, to study the spatial regularity of solutions of linear parabolic stochastic partial dierential equations on bounded Lipschitz domainsO R. The Besov smoothness determines the order of convergence that can be achieved by nonlinear approximation schemes. The proofs are based on a combination of weighted Sobolev estimates and characterizations of Besov spaces by wavelet expansions.

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Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains

We study the Besov regularity as well as linear and nonlinear approximation of random functions on bounded Lipschitz domains in ℝ
 d
 . The random functions are given either (i) explicitly in terms of a wavelet expansion or (ii) as the solution of a Poisson equation with a right-hand side in terms of a wavelet expansion. In the case (ii) we derive an adaptive wavelet algorithm that achieves the nonlinear approximation rate at a computational cost that is proportional to the degrees of freedom. These results are matched by computational experiments.

Petru A. Cioica

Papers

Spatial Besov regularity for semilinear stochastic partial differential equations on bounded Lipschitz domains

Spatial Besov Regularity for Stochastic Partial Differential Equations on Lipschitz Domains

On the $L_q(L_p)$-regularity and Besov smoothness of stochastic parabolic equations on bounded Lipschitz domains

Spatial Besov regularity for stochastic partial differential equations on Lipschitz domains

Adaptive wavelet methods for the stochastic Poisson equation