Multilevel Monte Carlo Methods

By Stefan Heinrich. 2001

956 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Numerical Analysis

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Citations in the corpus

Control functionals for Monte Carlo integration (2014, methodology): “Our starting point is to establish a trade-off between random sampling and deterministic approximation, as suggested on several separate occasions by researchers including Bakhvalov (1959), Heinrich (1995), Speight (2009) and Giles (2013).”
Improved Multilevel Monte Carlo Convergence using the Milstein Scheme (2008, background): “There are also strong similarities to Heinrich’s multilevel approach for parametric integration [7].”
Machine Learning for High-entropy Alloys: Progress, Challenges and Opportunities (2022, methodology): “For instance, for a 250-atom CuZn supercell, the first-principles Monte-Carlo simulation requires the self-consistent (SCF) calculations of 600,000 DFT energies [46], which can easily consume a million CPU core-hours.”
Convergence of tamed Euler schemes for a class of stochastic evolution equations (2014, methodology): “Moreover Hutzenthaler, Jentzen and Kloeden [11] have demonstrated that to apply multilevel Monte Carlo methods (see Heinrich [7,8] and Giles [5]) to stochastic differential equations with super-linearly growing coefficients one must “tame” the explicit Euler scheme.”
Multilevel Monte Carlo Path Simulation (2008, methodology): “A similar multilevel Monte Carlo idea has been used by Heinrich [9] for parametric integration, in which one is interested in evaluating a quantity I(λ) which is defined as a multi-dimensional integral of a function which has a parametric dependence on λ.”
Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients (2011, methodology): “This family of methods was introduced, to the authors’ knowledge, by Giles [17,18] for Itô stochastic ordinary differential equations after earlier work by Heinrich on numerical quadrature (see [21]).”
Optimal Model Management for Multifidelity Monte Carlo Estimation (2016, methodology): “The multilevel Monte Carlo method [27, 22, 15, 46] uses coarse-grid approximations as surrogate models and distributes the work among the models using known error and cost rates.”
Survey of multifidelity methods in uncertainty propagation, inference, and optimization (2018, methodology): “evious iterations of the optimization problem are used as control variate. This means that data from previous iterations serve as a kind of low-delity \model". The multilevel Monte Carlo method [98, 76] uses the control variate framework 16 PEHERSTORFER, WILLCOX, AND GUNZBURGER to combine multi”
Adaptive Euler–Maruyama method for SDEs with nonglobally Lipschitz drift (2016, positive): “N` can be optimized following the same approach in the MLMC theorem in [6].”
Tractability of Multivariate Problems (2008, methodology): “Faster algorithms for computing L2 discrepancy for relatively small d have been found by Heinrich [99] and Frank and Heinrich [69].”
On Nesting Monte Carlo Estimators (2017, methodology): “Also related is the study of techniques for variance reduction, such as multilevel MC (Heinrich, 2001; Giles, 2008), and bias reduction, such as the multi-step Richardson-Romberg method (Pages, 2007; Lemaire et al., 2017) and Russian roulette sampling (Lyne et al., 2015), many of which are…”
Multilevel Richardson-Romberg Extrapolation (2014, methodology): “The standard Multilevel Monte Carlo allocation matrix used by [Hei01, Gil08] is derived from the telescopic summation This corresponds in our general framework to the allocation matrix T deﬁned by T j = e j − e j 1 for j ∈ 2 , . . . , R i.e.”