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| 2. |
- Adolfsson, Klas, 1972, et al.
(författare)
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Discretization of integro-differential equations modeling dynamic fractional order viscoelasticity
- 2006
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Ingår i: Lecture Notes in Computer Science, Springer, ''Proceedings of Large-Scale Scientific Computations, 2005, Sozopol, Bulgaria'', I. Lirkov, S. Margenov, and J. Wasniewski (Eds.). - 3540319948 ; 3743, s. 76-83
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Konferensbidrag (refereegranskat)abstract
- We study a dynamic model for viscoelastic materials based on a constitutive equation of fractional order. This results in an integro-differential equation with a weakly singular convolution kernel. We discretize in the spatial variable by a standard Galerkin finite element method. We prove stability and regularity estimates which show how the convolution term introduces dissipation into the equation of motion. These are then used to prove a priori error estimates. A numerical experiment is included.
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| 6. |
- Agapiou, Sergios, et al.
(författare)
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Posterior consistency of the Bayesian approach to linear ill-posed inverse problems
- 2012
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Ingår i: arXiv:1203.5753v2 [math.ST]. ; :1203.5753, s. 1-30
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting, with Gaussian prior and noise distribution. A method of identifying the posterior distribution using its precision operator is presented. Work- ing with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to study posterior con- sistency in a frequentist sense, and in particular to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. We show how these rates may be optimized by a choice of the scale parameter in the prior covariance operator. Our methods assume a relatively weak relation between the prior covariance operator, the forward operator and the noise covariance operator; more precisely, we assume that appropriate powers of these operators induce equivalent norms. We compare our results to known minimax rates of convergence in the case where the forward operator and the prior and noise covariances are all simultaneously diagonaliz- able, and confirm that the PDE method provides the same rates for a wide range of parameters. An elliptic PDE inverse problem is used to illustrate the power of the general theory.
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| 7. |
- Agapiou, Sergios, et al.
(författare)
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Posterior contraction rates for the Bayesian approach to linear ill-posed inverse problems
- 2013
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Ingår i: Stochastic Processes and their Applications. - : Elsevier BV. - 0304-4149. ; 123:10, s. 3828-3860
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Tidskriftsartikel (refereegranskat)abstract
- We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications.
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| 8. |
- Akrivis, Georgios, et al.
(författare)
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Linearly implicit finite element methods for the time-dependent Joule heating problem
- 2005
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 45:3, s. 429-442
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Tidskriftsartikel (refereegranskat)abstract
- Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions.
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| 9. |
- Andersson, Adam, 1979, et al.
(författare)
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Duality in refined Sobolev–Malliavin spaces and weak approximation of SPDE
- 2016
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Ingår i: Stochastic Partial Differential Equations: Analysis and Computations. - : Springer Science and Business Media LLC. - 2194-0401 .- 2194-041X. ; 4:1, s. 113-149
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Tidskriftsartikel (refereegranskat)abstract
- We introduce a new family of refined Sobolev–Malliavin spaces that capture the integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine a standard Galerkin finite element method with backward Euler timestepping. The method of proof does not rely on the use of the Kolmogorov equation or the Itō formula and is therefore non-Markovian in nature. Test functions satisfying polynomial growth and mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate.
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| 10. |
- Andersson, Adam, 1979, et al.
(författare)
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Duality in refined Watanabe-Sobolev spaces and weak approximations of SPDE
- 2013
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- In this paper we introduce a new family of refined Watanabe- Sobolev spaces that capture in a fine way integrability in time of the Malliavin derivative. We consider duality in these spaces and derive a Burkholder type inequality in a dual norm. The theory we develop allows us to prove weak convergence with essentially optimal rate for numerical approximations in space and time of semilinear parabolic stochastic evolution equations driven by Gaussian additive noise. In particular, we combine Galerkin finite element methods with a backward Euler scheme in time. The method of proof does not rely on the use of the Kolmogorov equation or the It¯o formula and is therefore in nature non-Markovian. With this method polynomial growth test functions with mild smoothness assumptions are allowed, meaning in particular that we prove convergence of arbitrary moments with essentially optimal rate. Our Gronwall argument also yields weak error estimates which are uniform in time without any additional effort.
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