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- Kovacs, Mihaly, 1977, et al.
(författare)
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Finite element approximation of the linear stochastic wave equation with additive noise
- 2010
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Ingår i: SIAM Journal on Numerical Analysis. - : Society for Industrial & Applied Mathematics (SIAM). - 0036-1429 .- 1095-7170. ; 48:2, s. 408-427
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Tidskriftsartikel (refereegranskat)abstract
- Semidiscrete finite element approximation of the linear stochastic wave equation (LSWE) with additive noise is studied in a semigroup framework. Optimal error estimates for the deterministic problem are obtained under minimal regularity assumptions. These are used to prove strong convergence estimates for the stochastic problem. The theory presented here applies to multidimensional domains and spatially correlated noise. Numerical examples illustrate the theory.
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- Kovacs, Mihaly, 1977, et al.
(författare)
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Strong convergence of the finite element method with truncated noise for semilinear parabolic stochastic equations with additive noise
- 2010
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Ingår i: Numerical Algorithms. - : Springer Science and Business Media LLC. - 1017-1398 .- 1572-9265. ; 53:2-3, s. 309-320
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Tidskriftsartikel (refereegranskat)abstract
- We consider a semilinear parabolic PDE driven by additive noise. The equation is discretized in space by a standard piecewise linear finite element method. We show that the orthogonal expansion of the finite-dimensional Wiener process, that appears in the discretized problem, can be truncated severely without losing the asymptotic order of the method, provided that the kernel of the covariance operator of the Wiener process is smooth enough. For example, if the covariance operator is given by the Gauss kernel, then the number of terms to be kept is the quasi-logarithm of the number of terms in the original expansion. Then one can reduce the size of the corresponding linear algebra problem enormously and hence reduce the computational complexity, which is a key issue when stochastic problems are simulated.
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- Kraft, Karin, 1977, et al.
(författare)
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The dual weighted residuals approach to optimal control of ordinary differential equations
- 2010
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 50:3, s. 587-607
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Tidskriftsartikel (refereegranskat)abstract
- The methodology of dual weighted residuals is applied to an optimal control problem for ordinary differential equations. The differential equations are discretized by finite element methods. An a posteriori error estimate is derived and an adaptive algorithm is formulated. The algorithm is implemented in Matlab and tested on a simple model problem from vehicle dynamics.
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- Larsson, Stig, 1952, et al.
(författare)
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Optimal closing of a pair trade with a model containing jumps
- 2010
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- A pair trade is a portfolio consisting of a long position in one asset and a short position in another, and it is a widely applied investment strategy in the financial industry. Recently, Ekström, Lindberg and Tysk studied the problem of optimally closing a pair trading strategy when the difference of the two assets is modelled by an Ornstein-Uhlenbeck process. In this paper we study the same problem, but the model is generalized to also include jumps. More precisely we assume that the above difference is an Ornstein-Uhlenbeck type process, driven by a Lévy process of finite activity. We prove a verification theorem and analyze a numerical method for the associated free boundary problem. We prove rigorous error estimates, which are used to draw some conclusions from numerical simulations.
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- Larsson, Stig, 1952, et al.
(författare)
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The continuous Galerkin method for an integro-differential equation modeling dynamic fractional order viscoelasticity
- 2010
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Ingår i: IMA Journal of Numerical Analysis. - : Oxford University Press (OUP). - 1464-3642 .- 0272-4979. ; 30:4, s. 964-986
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Tidskriftsartikel (refereegranskat)abstract
- We consider a fractional order integro-differential equation with a weakly singular convolution kernel. The equation with homogeneous mixed Dirichlet and Neumann boundary conditions is reformulated as an abstract Cauchy problem, and well-posedness is verified in the context of linear semigroup theory. Then we formulate a continuous Galerkin method for the problem, and we prove stability estimates. These are then used to prove a priori error estimates. The theory is illustrated by a numerical example.
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