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We investigate the problem of computing a nested expectation of the form $\mathbb{P}[\mathbb{E}[X|Y] \!\geq\!0]\!=\!\mathbb{E}[\textrm{H}(\mathbb{E}[X|Y])]$ where $\textrm{H}$ is the Heaviside function. This nested expectation appears, for…
This paper considers the challenging computational task of estimating nested expectations. Existing algorithms, such as nested Monte Carlo or multilevel Monte Carlo, are known to be consistent but require a large number of samples at both…
One of the most challenging issues in applied mathematics is to develop and analyze algorithms which are able to approximately compute solutions of high-dimensional nonlinear partial differential equations (PDEs). In particular, it is very…
The estimation of repeatedly nested expectations is a challenging task that arises in many real-world systems. However, existing methods generally suffer from high computational costs when the number of nestings becomes large. Fix any…
In this article we consider the approximation of expectations w.r.t. probability distributions associated to the solution of partial differential equations (PDEs); this scenario appears routinely in Bayesian inverse problems. In practice,…
We consider ordinary differential equations (ODEs) which involve expectations of a random variable. These ODEs are special cases of McKean-Vlasov stochastic differential equations (SDEs). A plain vanilla Monte Carlo approximation method for…
Full-history recursive multilevel Picard (MLP) approximation schemes have been shown to overcome the curse of dimensionality in the numerical approximation of high-dimensional semilinear partial differential equations (PDEs) with general…
We consider the problem of estimating a nested structure of two expectations taking the form $U_0 = E[\max\{U_1(Y), \pi(Y)\}]$, where $U_1(Y) = E[X\ |\ Y]$. Terms of this form arise in financial risk estimation and option pricing. When…
The recently introduced full-history recursive multilevel Picard (MLP) approximation methods have turned out to be quite successful in the numerical approximation of solutions of high-dimensional nonlinear PDEs. In particular, there are…
In the literatur there exist approximation methods for McKean-Vlasov stochastic differential equations which have a computational effort of order $3$. In this article we introduce full-history recursive multilevel Picard (MLP)…
Many problems in machine learning and statistics involve nested expectations and thus do not permit conventional Monte Carlo (MC) estimation. For such problems, one must nest estimators, such that terms in an outer estimator themselves…
It is one of the most challenging problems in applied mathematics to approximatively solve high-dimensional partial differential equations (PDEs). In particular, most of the numerical approximation schemes studied in the scientific…
This article reviews the application of advanced Monte Carlo techniques in the context of Multilevel Monte Carlo (MLMC). MLMC is a strategy employed to compute expectations which can be biased in some sense, for instance, by using the…
We study the estimation of repeatedly nested expectations (RNEs) with a constant horizon (number of nestings) using quantum computing. We propose a quantum algorithm that achieves $\varepsilon$-error with cost $\tilde O(\varepsilon^{-1})$,…
The multilevel Monte Carlo (MLMC) method has been used for a wide variety of stochastic applications. In this paper we consider its use in situations in which input random variables can be replaced by similar approximate random variables…
Estimating nested expectations is an important task in computational mathematics and statistics. In this paper we propose a new Monte Carlo method using post-stratification to estimate nested expectations efficiently without taking samples…
Nested Monte Carlo is widely used for risk estimation, but its efficiency is limited by the discontinuity of the indicator function and high computational cost. This paper proposes a nested Multilevel Monte Carlo (MLMC) method combined with…
We introduce multilevel Picard (MLP) approximations for McKean--Vlasov stochastic differential equations (SDEs) with nonconstant diffusion coefficient. Under standard Lipschitz assumptions on the coefficients, we show that the MLP algorithm…
We leverage multilevel Monte Carlo (MLMC) to improve the performance of multi-step look-ahead Bayesian optimization (BO) methods that involve nested expectations and maximizations. Often these expectations must be computed by Monte Carlo…
Motivated by various computational applications, we investigate the problem of estimating nested expectations. Building upon recent work by the authors, we propose a novel Monte Carlo estimator for nested expectations, inspired by sparse…