| 1. |
- Babuska, I., et al.
(författare)
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Solving elliptic boundary value problems with uncertain coefficients by the finite element method : the stochastic formulation
- 2005
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - : Elsevier BV. - 0045-7825 .- 1879-2138. ; 194:16-dec, s. 1251-1294
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Tidskriftsartikel (refereegranskat)abstract
- This work studies a linear elliptic problem with uncertainty. The introduction gives a survey of different formulations of the uncertainty and resulting numerical approximations. The major emphasis of this work is the probabilistic treatment of uncertainty, addressing the problem of solving linear elliptic boundary value problems with stochastic coefficients. If the stochastic coefficients are known functions of a random vector, then the stochastic elliptic boundary value problem is turned into a parametric deterministic one with solution u(y, x), y is an element of Gamma, x is an element of D, where D subset of R-d, d = 1, 2, 3, and Gamma is a high-dimensional cube. In addition, the function u is specified as the solution of a deterministic variational problem over Gamma x D. A tensor product finite element method, of h-version in D and k-, or, p-version in Gamma, is proposed for the approximation of it. A priori error estimates are given and an adaptive algorithm is also proposed. Due to the high dimension of Gamma, the Monte Carlo finite element method is also studied here. This work compares the asymptotic complexity of the numerical methods, and shows results from numerical experiments. Comments on the uncertainty in the probabilistic characterization of the coefficients in the stochastic formulation are included.
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| 2. |
- Björk, T., et al.
(författare)
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Monte Carlo Euler approximations of HJM term structure financial models
- 2013
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Ingår i: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 53:2, s. 341-383
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Tidskriftsartikel (refereegranskat)abstract
- We present Monte Carlo-Euler methods for a weak approximation problem related to the Heath-Jarrow-Morton (HJM) term structure model, based on Itô stochastic differential equations in infinite dimensional spaces, and prove strong and weak error convergence estimates. The weak error estimates are based on stochastic flows and discrete dual backward problems, and they can be used to identify different error contributions arising from time and maturity discretization as well as the classical statistical error due to finite sampling. Explicit formulas for efficient computation of sharp error approximation are included. Due to the structure of the HJM models considered here, the computational effort devoted to the error estimates is low compared to the work to compute Monte Carlo solutions to the HJM model. Numerical examples with known exact solution are included in order to show the behavior of the estimates.
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| 3. |
- Moon, K. S., et al.
(författare)
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A variational principle for adaptive approximation of ordinary differential equations
- 2003
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 96:1, s. 131-152
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Tidskriftsartikel (refereegranskat)abstract
- A variational principle, inspired by optimal control, yields a simple derivation of an error representation, global error = Sigma local error . weight, for general approximation of functions of solutions to ordinary differential equations. This error representation is then approximated by a sum of computable error indicators, to obtain a useful global error indicator for adaptive mesh refinements. A uniqueness formulation is provided for desirable error representations of adaptive algorithms.
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| 4. |
- Moon, K. S., et al.
(författare)
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Convergence rates for adaptive approximation of ordinary differential equations
- 2003
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Ingår i: Numerische Mathematik. - : Springer Science and Business Media LLC. - 0029-599X .- 0945-3245. ; 96:1, s. 99-129
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Tidskriftsartikel (refereegranskat)abstract
- This paper constructs an adaptive algorithm for ordinary differential equations and analyzes its asymptotic behavior as the error tolerance parameter tends to zero. An adaptive algorithm, based on the error indicators and successive subdivision of time steps, is proven to stop with the optimal number, N, of steps up to a problem independent factor defined in the algorithm. A version of the algorithm with decreasing tolerance also stops with the total number of steps, including all refinement levels, bounded by O(N). The alternative version with constant tolerance stops with O(N log N) total steps. The global error is bounded by the tolerance parameter asymptotically as the tolerance tends to zero. For a p-th order accurate method the optimal number of adaptive steps is proportional to the p-th root of the L 1/p+1 quasi-norm of the error density, while the number of uniform steps, with the same error, is proportional to the p-th root of the larger L-1-norm of the error density.
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| 5. |
- Szepessy, Anders, et al.
(författare)
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Adaptive weak approximation of stochastic differential equations
- 2001
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Ingår i: Communications on Pure and Applied Mathematics. - : Wiley. - 0010-3640 .- 1097-0312. ; 54:10, s. 1169-1214
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Tidskriftsartikel (refereegranskat)abstract
- Adaptive time-stepping methods based on the Monte Carlo Euler method for weak approximation of Ito stochastic differential equations are developed. The main result is new expansions of the computational error, with computable leading-order term in a posteriori form, based on stochastic flows and discrete dual backward problems. The expansions lead to efficient and accurate computation of error estimates. Adaptive algorithms for either stochastic time steps or deterministic time steps are described. Numerical examples illustrate when stochastic and deterministic adaptive time steps are superior to constant time steps and when adaptive stochastic steps are superior to adaptive deterministic steps. Stochastic time steps use Brownian bridges and require more work for a given number of time steps. Deterministic time steps may yield more time steps but require less work; for example, in the limit of vanishing error tolerance, the ratio of the computational error and its computable estimate tends to 1 with negligible additional work to determine the adaptive deterministic time steps.
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