arXiv subjects

Shyam Mohan Subbiah Pillai

Publications and source records attributed to Shyam Mohan Subbiah Pillai.

4 recordsLinked to original sources

arXiv2026

Multi-index importance sampling for McKean--Vlasov stochastic differential equations

This work addresses the estimation of rare-event quantities expressed as expectations of smooth observables of solutions to a broad class of McKean--Vlasov stochastic differential equations (MV-SDEs). Building on the double loop Monte Carlo (DLMC) method with stochastic optimal control-based importance sampling (IS) introduced by Ben Rached et al. (2024a), this work extends this framework to the multi-index Monte Carlo (MIMC) setting. The resulting multi-index DLMC estimator mitigates the explosion of the coefficient of variation for rare event quantities. Moreover, it exploits the sampling efficiency of MIMC by leveraging the propagation of chaos to ensure mixed-difference variances vanish in the mean-field limit. The complexity analysis relies on assumptions on mixed-difference bias and variance decay, similar to standard MIMC assumptions. Although not rigorously proved, this work presents strong numerical evidence in support of these assumptions. The primary contribution of this work is the novel numerical integration of the MIMC method with IS for MV-SDEs. This approach reduces the computational complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-2} (\log \mathrm{TOL}_{\mathrm{r}}^{-1})^2)$, enabling an accurate estimation of rare-event quantities within a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. Numerical experiments on the Kuramoto model from statistical physics demonstrate computational savings of several orders of magnitude for the multi-index DLMC estimator with IS, compared with the standard Monte Carlo (MC) method.

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arXiv2025

Optimal power procurement for green cellular wireless networks under uncertainty and chance constraints

Given the increasing global emphasis on sustainable energy usage and the rising energy demands of cellular wireless networks, this work seeks an optimal short-term, continuous-time power procurement schedule to minimize operating expenditure and the carbon footprint of cellular wireless networks equipped with energy storage capacity, and hybrid energy systems comprising uncertain renewable energy sources. Despite the stochastic nature of wireless fading channels, the network operator must ensure a certain quality-of-service (QoS) constraint with high probability. This probabilistic constraint prevents using the dynamic programming principle to solve the stochastic optimal control problem. This work introduces a novel time-continuous Lagrangian relaxation approach tailored for real-time, near-optimal energy procurement in cellular networks, overcoming tractability problems associated with the probabilistic QoS constraint. The numerical solution procedure includes an efficient upwind finite-difference solver for the Hamilton--Jacobi--Bellman equation corresponding to the relaxed problem, and an effective combination of the limited memory bundle method (LMBM) for handling nonsmooth optimization and the stochastic subgradient method (SSM) to navigate the stochasticity of the dual problem. Numerical results, based on the German power system and daily cellular traffic data, demonstrate the computational efficiency of the proposed numerical approach, providing a near-optimal policy in a practical timeframe.

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arXiv2024

Multilevel Importance Sampling for Rare Events Associated With the McKean--Vlasov Equation

This work combines multilevel Monte Carlo (MLMC) with importance sampling to estimate rare-event quantities that can be expressed as the expectation of a Lipschitz observable of the solution to a broad class of McKean--Vlasov stochastic differential equations. We extend the double loop Monte Carlo (DLMC) estimator introduced in this context in (Ben Rached et al., 2023) to the multilevel setting. We formulate a novel multilevel DLMC estimator and perform a comprehensive cost-error analysis yielding new and improved complexity results. Crucially, we devise an antithetic sampler to estimate level differences guaranteeing reduced computational complexity for the multilevel DLMC estimator compared with the single-level DLMC estimator. To address rare events, we apply the importance sampling scheme, obtained via stochastic optimal control in (Ben Rached et al., 2023), over all levels of the multilevel DLMC estimator. Combining importance sampling and multilevel DLMC reduces computational complexity by one order and drastically reduces the associated constant compared to the single-level DLMC estimator without importance sampling. We illustrate the effectiveness of the proposed multilevel DLMC estimator on the Kuramoto model from statistical physics with Lipschitz observables, confirming the reduced complexity from $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ for the single-level DLMC estimator to $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-3})$ while providing a feasible estimate of rare-event quantities up to prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$.

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arXiv2023

Double-Loop Importance Sampling for McKean--Vlasov Stochastic Differential Equation

This paper investigates Monte Carlo (MC) methods to estimate probabilities of rare events associated with solutions to the $d$-dimensional McKean-Vlasov stochastic differential equation (MV-SDE). MV-SDEs are usually approximated using a stochastic interacting $P$-particle system, which is a set of $P$ coupled $d$-dimensional stochastic differential equations (SDEs). Importance sampling (IS) is a common technique for reducing high relative variance of MC estimators of rare-event probabilities. We first derive a zero-variance IS change of measure for the quantity of interest by using stochastic optimal control theory. However, when this change of measure is applied to stochastic particle systems, it yields a $P \times d$-dimensional partial differential control equation (PDE), which is computationally expensive to solve. To address this issue, we use the decoupling approach introduced in [dos Reis et al., 2023], generating a $d$-dimensional control PDE for a zero-variance estimator of the decoupled SDE. Based on this approach, we develop a computationally efficient double loop MC (DLMC) estimator. We conduct a comprehensive numerical error and work analysis of the DLMC estimator. As a result, we show optimal complexity of $\mathcal{O}(\mathrm{TOL}_{\mathrm{r}}^{-4})$ with a significantly reduced constant to achieve a prescribed relative error tolerance $\mathrm{TOL}_{\mathrm{r}}$. Subsequently, we propose an adaptive DLMC method combined with IS to numerically estimate rare-event probabilities, substantially reducing relative variance and computational runtimes required to achieve a given $\mathrm{TOL}_{\mathrm{r}}$ compared with standard MC estimators in the absence of IS. Numerical experiments are performed on the Kuramoto model from statistical physics.

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