scholarly journals Kernel-independent adaptive construction of $$\mathcal {H}^2$$-matrix approximations

2021 ◽  
Author(s):  
M. Bauer ◽  
M. Bebendorf ◽  
B. Feist
Keyword(s):  
Error Estimates ◽  
Adaptive Cross Approximation ◽  
Local Operators ◽  
Non Local ◽  
Side Result ◽  
Matrix Approximations

AbstractA method for the kernel-independent construction of $$\mathcal {H}^2$$ H 2 -matrix approximations to non-local operators is proposed. Special attention is paid to the adaptive construction of nested bases. As a side result, new error estimates for adaptive cross approximation (ACA) are presented which have implications on the pivoting strategy of ACA.

scholarly journals Conformally invariant non-local operators

2001 ◽  
Vol 201 (1) ◽  
pp. 19-60 ◽  
Author(s):  
Thomas Branson ◽  
A. Rod Gover
Keyword(s):  
Conformally Invariant ◽  
Local Operators ◽  
Non Local

The bond-based peridynamic system with Dirichlet-type volume constraint

2014 ◽  
Vol 144 (1) ◽  
pp. 161-186 ◽  
Author(s):  
Tadele Mengesha ◽  
Qiang Du
Keyword(s):  
Poisson Ratio ◽  
Variational Problems ◽  
Compact Embedding ◽  
Volume Constraint ◽  
Positive Definite Kernels ◽  
Local Boundary ◽  
Local Operators ◽  
Non Local ◽  
Poincaré Type Inequalities ◽  
Non Locality

In this paper, the bond-based peridynamic system is analysed as a non-local boundary-value problem with volume constraint. The study extends earlier works in the literature on non-local diffusion and non-local peridynamic models, to include non-positive definite kernels. We prove the well-posedness of both linear and nonlinear variational problems with volume constraints. The analysis is based on some non-local Poincaré-type inequalities and the compactness of the associated non-local operators. It also offers careful characterizations of the associated solution spaces, such as compact embedding, separability and completeness. In the limit of vanishing non-locality, the convergence of the peridynamic system to the classical Navier equations of elasticity with Poisson ratio ¼ is demonstrated.

scholarly journals Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

2021 ◽  
Vol 7 (1) ◽  
pp. 260-275
Author(s):  
Zihan Cai ◽  
◽  
Yan Liu ◽  
Baiping Ouyang ◽  
Keyword(s):  
Cauchy Problem ◽  
Phase Space ◽  
Coupled Systems ◽  
Decay Properties ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Thermoelastic Plate ◽  
Local Operators ◽  
Non Local

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

scholarly journals Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

2021 ◽  
Author(s):  
Yan Liu ◽  
Zihan Cai ◽  
Shuanghu Zhang
Keyword(s):  
Cauchy Problem ◽  
Phase Space ◽  
Coupled Systems ◽  
Decay Properties ◽  
Representation Of Solutions ◽  
The Cauchy Problem ◽  
Plate Equations ◽  
Thermoelastic Plate ◽  
Local Operators ◽  
Non Local

In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $L^p-L^q$ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.

scholarly journals Chernoff approximation for semigroups generated by killed Feller processes and Feynman formulae for time-fractional Fokker–Planck–Kolmogorov equations

2018 ◽  
Vol 21 (5) ◽  
pp. 1203-1237 ◽  
Author(s):  
Yana A. Butko
Keyword(s):  
Evolution Equations ◽  
Approximate Solutions ◽  
Dynamics Simulation ◽  
Fractional Evolution Equations ◽  
Fractional Equations ◽  
Dirichlet Type ◽  
Chernoff Theorem ◽  
Local Operators ◽  
Non Local ◽  
Distributed Order

Abstract We consider operator semigroups generated by Feller processes killed upon leaving a given domain. These semigroups correspond to Cauchy–Dirichlet type initial-exterior value problems in this domain for a class of evolution equations with (possibly non-local) operators. The considered semigroups are approximated by means of the Chernoff theorem. For a class of killed Feller processes, the constructed Chernoff approximation leads to a representation of the solution of the corresponding Cauchy–Dirichlet type problem by a Feynman formula, i.e. by a limit of n-fold iterated integrals of certain functions as n → ∞. Feynman formulae can be used for direct calculations, modelling of underlying dynamics, simulation of underlying stochastic processes. Further, a method to approximate solutions of time-fractional evolution equations is suggested. The method is based on connections between time-fractional and time-non-fractional evolution equations as well as on Chernoff approximations for the latter ones. This method leads to Feynman formulae for solutions of time-fractional evolution equations. A class of distributed order time-fractional equations is considered; Feynman formulae for solutions of the corresponding Cauchy and Cauchy–Dirichlet type problems are obtained.

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