| 1. |
- Björk, T., et al.
(author)
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Monte Carlo Euler approximations of HJM term structure financial models
- 2013
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 53:2, s. 341-383
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Journal article (peer-reviewed)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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| 2. |
- Hall, E. J., et al.
(author)
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Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data
- 2016
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In: SIAM Journal on Scientific Computing. - : Society for Industrial and Applied Mathematics Publications. - 1064-8275 .- 1095-7197. ; 38:6, s. A3773-A3807
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Journal article (peer-reviewed)abstract
- We derive computable error estimates for finite element approximations of linear elliptic partial differential equations with rough stochastic coefficients. In this setting, the exact solutions contain high frequency content that standard a posteriori error estimates fail to capture. We propose goal-oriented estimates, based on local error indicators, for the pathwise Galerkin and expected quadrature errors committed in standard, continuous, piecewise linear finite element approximations. Derived using easily validated assumptions, these novel estimates can be computed at a relatively low cost and have applications to subsurface flow problems in geophysics where the conductivities are assumed to have lognormal distributions with low regularity. Our theory is supported by numerical experiments on test problems in one and two dimensions.
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| 3. |
- Hoel, H., et al.
(author)
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Implementation and analysis of an adaptive multilevel Monte Carlo algorithm
- 2014
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In: Monte Carlo Methods and Applications. - : Walter de Gruyter GmbH. - 0929-9629. ; 20:1, s. 1-41
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Journal article (peer-reviewed)abstract
- We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic dierential equations (SDE). The work [11] proposed and analyzed an MLMC method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a single level Euler-Maruyama Monte Carlo method from O(TOL-3) to O(TOL-2 log(TOL-1)2) for a mean square error of O(TOL2). Later, the work [17] presented an MLMC method using a hierarchy of adaptively re ned, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretizationMLMC method. This work improves the adaptiveMLMC algorithms presented in [17] and it also provides mathematical analysis of the improved algorithms. In particular, we show that under some assumptions our adaptive MLMC algorithms are asymptotically accurate and essentially have the correct complexity but with improved control of the complexity constant factor in the asymptotic analysis. Numerical tests include one case with singular drift and one with stopped diusion, where the complexity of a uniform single level method is O(TOL-4). For both these cases the results con rm the theory, exhibiting savings in the computational cost for achieving the accuracy O(TOL) from O(TOL-3) for the adaptive single level algorithm to essentially O(TOL-2 log(TOL-1)2) for the adaptive MLMC algorithm.
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| 4. |
- Karlsson, J., et al.
(author)
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An Error Estimate for Symplectic Euler Approximation of Optimal Control Problems
- 2015
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In: SIAM Journal on Scientific Computing. - : Society for Industrial & Applied Mathematics (SIAM). - 1064-8275 .- 1095-7197. ; 37:2
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Journal article (peer-reviewed)abstract
- This work focuses on numerical solutions of optimal control problems. A time discretization error representation is derived for the approximation of the associated value function. It concerns symplectic Euler solutions of the Hamiltonian system connected with the optimal control problem. The error representation has a leading-order term consisting of an error density that is computable from symplectic Euler solutions. Under an assumption of the pathwise convergence of the approximate dual function as the maximum time step goes to zero, we prove that the remainder is of higher order than the leading-error density part in the error representation. With the error representation, it is possible to perform adaptive time stepping. We apply an adaptive algorithm originally developed for ordinary differential equations. The performance is illustrated by numerical tests.
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| 5. |
- Litvinenko, A., et al.
(author)
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Computation of electromagnetic fields scattered from objects with uncertain shapes using multilevel monte carlo method
- 2019
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In: IEEE Journal on Multiscale and Multiphysics Computational Techniques. - : Institute of Electrical and Electronics Engineers Inc.. - 2379-8793. ; 4, s. 51-64
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Journal article (peer-reviewed)abstract
- Computational tools for characterizing electromagnetic scattering from objects with uncertain shapes are needed in various applications ranging from remote sensing at microwave frequencies to Raman spectroscopy at optical frequencies. Often, such computational tools use the Monte Carlo (MC) method to sample a parametric space describing geometric uncertainties. For each sample, which corresponds to a realization of the geometry, a deterministic electromagnetic solver computes the scattered fields. However, for an accurate statistical characterization, the number of MC samples has to be large. In this paper, to address this challenge, the continuation multilevel Monte Carlo (CMLMC) method is used together with a surface integral equation solver. The CMLMC method optimally balances statistical errors due to sampling of the parametric space, and numerical errors due to the discretization of the geometry using a hierarchy of discretizations, from coarse to fine. The number of realizations of finer discretizations can be kept low, with most samples computed on coarser discretizations to minimize computational cost. Consequently, the total execution time is significantly reduced, in comparison to the standard MC scheme.
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| 6. |
- Malenova, Gabriela, et al.
(author)
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A Sparse Stochastic Collocation Technique for High-Frequency Wave Propagation with Uncertainty
- 2016
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In: SIAM/ASA Journal on Uncertainty Quantification. - : Society for Industrial and Applied Mathematics Publications. - 2166-2525. ; 4:1, s. 1084-1110
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Journal article (peer-reviewed)abstract
- We consider the wave equation with highly oscillatory initial data, where there is uncertainty in the wave speed, initial phase, and/or initial amplitude. To estimate quantities of interest related to the solution and their statistics, we combine a high-frequency method based on Gaussian beams with sparse stochastic collocation. Although the wave solution, u(epsilon), is highly oscillatory in both physical and stochastic spaces, we provide theoretical arguments for simplified problems and numerical evidence that quantities of interest based on local averages of |u(epsilon)|(2) are smooth, with derivatives in the stochastic space uniformly bounded in epsilon, where epsilon denotes the short wavelength. This observable related regularity makes the sparse stochastic collocation approach more efficient than Monte Carlo methods. We present numerical tests that demonstrate this advantage.
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| 7. |
- Moon, Kyoung-Sook, et al.
(author)
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Convergence rates for an adaptive dual weighted residual finite element algorithm
- 2006
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In: BIT Numerical Mathematics. - : Springer Science and Business Media LLC. - 0006-3835 .- 1572-9125. ; 46:2, s. 367-407
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Journal article (peer-reviewed)abstract
- Basic convergence rates are established for an adaptive algorithm based on the dual weighted residual error representation, [GRAPHICS] applied to isoparametric d-linear quadrilateral finite element approximation of functionals of multi scale solutions to second order elliptic partial differential equations in bounded domains of R-d. In contrast to the usual aim to derive an a posteriori error estimate, this work derives, as the mesh size tends to zero, a uniformly convergent error expansion for the error density, with computable leading order term. It is shown that the optimal adaptive isotropic mesh uses a number of elements proportional to the d/2 power of the Ld/d+2 quasi-norm of the error density; the same error for approximation with a uniform mesh requires a number of elements proportional to the d/2 power of the larger L-1 norm of the same error density. A point is that this measure recognizes different convergence rates for multi scale problems, although the convergence order may be the same. The main result is a proof that the adaptive algorithm based on successive subdivisions of elements reduces the maximal error indicator with a factor or stops with the error asymptotically bounded by the tolerance using the optimal number of elements, up to a problem independent factor. An important step is to prove uniform convergence of the expansion for the error density, which is based on localized averages of second order difference quotients of the primal and dual finite element solutions. The averages are used since the difference quotients themselves do not converge pointwise for adapted meshes. The proof uses weak convergence techniques with a symmetrizer for the second order difference quotients and a splitting of the error into a dominating contribution, from elements with no hanging nodes or edges on the initial mesh, and a remaining asymptotically negligible part. Numerical experiments for an elasticity problem with a crack and different variants of the averages show that the algorithm is useful in practice also for relatively large tolerances, much larger than the small tolerances needed to theoretically guarantee that the algorithm works well.
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