Håkon Hoel

Adaptive Multilevel Monte Carlo Simulation

This work generalizes a multilevel forward Euler Monte Carlo method introduced in Michael B. Giles. (Michael Giles. Oper. Res. 56(3):607–617, 2008.) for the approximation of expected values depending on the solution to an Ito stochastic differential equation. The work (Michael Giles. Oper. Res. 56(3):607– 617, 2008.) proposed and analyzed a forward Euler multilevelMonte Carlo method based on a hierarchy of uniform time discretizations and control variates to reduce the computational effort required by a standard, single level, Forward Euler Monte Carlo method. This work introduces an adaptive hierarchy of non uniform time discretizations, generated by an adaptive algorithmintroduced in (AnnaDzougoutov et al. Raul Tempone. Adaptive Monte Carlo algorithms for stopped diffusion. In Multiscale methods in science and engineering, volume 44 of Lect. Notes Comput. Sci. Eng., pages 59–88. Springer, Berlin, 2005; Kyoung-Sook Moon et al. Stoch. Anal. Appl. 23(3):511–558, 2005; Kyoung-Sook Moon et al. An adaptive algorithm for ordinary, stochastic and partial differential equations. In Recent advances in adaptive computation, volume 383 of Contemp. Math., pages 325–343. Amer. Math. Soc., Providence, RI, 2005.). This form of the adaptive algorithm generates stochastic, path dependent, time steps and is based on a posteriori error expansions first developed in (Anders Szepessy et al. Comm. Pure Appl. Math. 54(10):1169– 1214, 2001). Our numerical results for a stopped diffusion problem, exhibit savings in the computational cost to achieve an accuracy of \( \vartheta{\rm(TOL),\, from\,(TOL^{-3})}\), from using a single level version of the adaptive algorithm to \( \vartheta\left( \begin{array}{lll}\left({(TOL^{-1})\,log(TOL)}\right)^2\end{array}\right).\)

Implementation and analysis of an adaptive multilevel Monte Carlo algorithm

Abstract. We present an adaptive multilevel Monte Carlo (MLMC) method for weak approximations of solutions to Itô stochastic differential equations (SDE). The work [Oper. Res. 56 (2008), 607–617] 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 )

On Nonasymptotic Optimal Stopping Criteria In Monte Carlo Simulations

{{{\mathcal {O}}({\mathrm {TOL}}^{-3})}}

Multilevel ensemble Kalman filtering

to 𝒪( TOL -2 log( TOL -1 ) 2 )

Construction of a Mean Square Error Adaptive Euler–Maruyama Method With Applications in Multilevel Monte Carlo

{{{\mathcal {O}}({\mathrm {TOL}}^{-2}\log ({\mathrm {TOL}}^{-1})^{2})}}

An Extension of Clarke's Model With Stochastic Amplitude Flip Processes

for a mean square error of 𝒪( TOL 2 )

Computable error estimates for finite element approximations of elliptic partial differential equations with rough stochastic data

{{{\mathcal {O}}({\mathrm {TOL}}^2)}}

A numerical scheme using multi-shockpeakons to compute solutions of the Degasperis-Procesi equation

. Later, the work [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217–234] presented an MLMC method using a hierarchy of adaptively refined, non-uniform time discretizations, and, as such, it may be considered a generalization of the uniform time discretization MLMC method. This work improves the adaptive MLMC algorithms presented in [Lect. Notes Comput. Sci. Eng. 82, Springer-Verlag, Berlin (2012), 217–234] 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 diffusion, where the complexity of a uniform single level method is 𝒪( TOL -4 )

How accurate is molecular dynamics

{{{\mathcal {O}}({\mathrm {TOL}}^{-4})}}

MULTILEVEL ENSEMBLE KALMAN FILTERING FOR SPATIO-TEMPORAL PROCESSES

. For both these cases the results confirm the theory, exhibiting savings in the computational cost for achieving the accuracy 𝒪( TOL )