arXiv subjects

Raul Tempone

Publications and source records attributed to Raul Tempone.

At least 19 recordsLinked to original sources

arXiv2026

Score-Based Martingale Posteriors for Deep Neural Networks

In this paper we investigate the efficacy of the score-based martingale posteriors (SMP) (Cui & Walker, 2025; Fong et al., 2023) in the context of modern and large-scale machine learning problems and its potential for meaningful uncertainty quantification. SMPs work with a stochastic gradient ascent-type recursion on the parameter space of stochastic models and construct a martingale on the parameter space. Under simple mathematical assumptions, the recursion can be built so that the parameters form a martingale sequence which possesses a limiting, in time, random variable, the latter of which can be simulated very quickly, in contrast to Monte Carlo-based methods such as Markov chain Monte Carlo. In this expository paper we explore the SMP for inferring the parameters of deep neural networks (DNNs) and, where feasible, compare our results to the state-of-the-art Monte Carlo methods aimed at inferring conventional Bayesian posteriors.

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arXiv2026

Data-Driven Stochastic Optimal Control for Intraday Electricity Trading by Renewable Producers

The rapid growth of weather-dependent renewable generation increases price volatility and imbalance penalty risk in power markets, creating the need for advanced quantitative trading strategies. We develop a data-driven continuous-time stochastic optimal control framework for intraday electricity trading using stochastic differential equations with drift terms ensuring mean reversion to deterministic forecast trajectories. Production follows a Jacobi diffusion, while prices follow an asymmetric jump-diffusion to reflect the heavy-tailed behavior observed in intraday markets. The framework accounts for realistic market features by incorporating gate closure and energy-based imbalance settlement over the delivery window, where the path-dependent imbalance cost is handled by state augmentation to preserve the Markovian structure. The value function is characterized via the dynamic programming principle by a three-stage sequence of two linear Kolmogorov backward equations and a nonlinear Hamilton-Jacobi-Bellman partial integro-differential equation. To solve this problem efficiently, we propose a monotone IMEX finite-difference scheme with operator splitting, semi-implicit linearization, and a differential formulation for the jump operator. Numerical experiments based on German market data indicate that, under the provided forecasts, the computed strategy outperforms the TWAP benchmark and approaches the perfect-foresight benchmark. Sensitivity experiments further show how jump intensity, delivery-window length, and trading horizon affect the trading policy and the resulting profit-and-loss distribution.

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arXiv2026

Hierarchical Importance Sampling for Estimating Occupation Time for SDE Solutions

This study considers the estimation of the complementary cumulative distribution function of the occupation time (i.e., the time spent below a threshold) for a process governed by a stochastic differential equation. The focus is on the right tail, where the underlying event becomes rare, and using variance reduction techniques is essential to obtain computationally efficient estimates. Building on recent developments that relate importance sampling (IS) to stochastic optimal control, this work develops an optimal single level IS (SLIS) estimator based on the solution of an auxiliary Hamilton Jacobi Bellman (HJB) partial differential equation (PDE). The cost of solving the HJB-PDE is incorporated into the total computational work, and an optimized trade off between preprocessing and sampling is proposed to minimize the overall cost. The SLIS approach is extended to the multilevel setting to enhance efficiency, yielding a multilevel IS (MLIS) estimator. A necessary and sufficient condition under which the MLIS method outperforms the SLIS method is established, and a common likelihood MLIS formulation is introduced that satisfies this condition under appropriate regularity assumptions. The classical multilevel Monte Carlo complexity theory can be extended to accommodate settings where the single-level variance varies with the discretization level. As a special case, the variance-decay behavior observed in the IS framework stems from the zero variance property of the optimal control. Notably, the total work complexity of MLIS can be better than an order of two. Numerical experiments in the context of fade duration estimation demonstrate the benefits of the proposed approach and validate these theoretical results.

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arXiv2025

A Pontryagin Maximum Principle on the Belief Space for Continuous-Time Optimal Control with Discrete Observations

We study a continuous time stochastic optimal control problem under partial observations that are available only at discrete time instants. This hybrid setting, with continuous dynamics and intermittent noisy measurements, arises in applications ranging from robotic exploration and target tracking to epidemic control. We formulate the problem on the space of beliefs (information states), treating the controller's posterior distribution of the state as the state variable for decision making. On this belief space we derive a Pontryagin maximum principle that provides necessary conditions for optimality. The analysis carefully tracks both the continuous evolution of the state between observation times and the Bayesian jump updates of the belief at observation instants. A key insight is a relationship between the adjoint process in our maximum principle and the gradient of the value functional on the belief space, which links the optimality conditions to the dynamic programming approach on the space of probability measures. The resulting optimality system has a prediction and update structure that is closely related to the unnormalised Zakai equation and the normalised Kushner-Stratonovich equation in nonlinear filtering. Building on this analysis, we design a particle based numerical scheme to approximate the coupled forward (filter) and backward (adjoint) system. The scheme uses particle filtering to represent the evolving belief and regression techniques to approximate the adjoint, which yields a practical algorithm for computing near optimal controls under partial information. The effectiveness of the approach is illustrated on both linear and nonlinear examples and highlights in particular the benefits of actively controlling the observation process.

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arXiv2025

A function approximation algorithm using multilevel active subspaces

The Active Subspace (AS) method is a widely used technique for identifying the most influential directions in high-dimensional input spaces that affect the output of a computational model. The standard AS algorithm requires a sufficient number of gradient evaluations (samples) of the input output map to achieve quasi-optimal reconstruction of the active subspace, which can lead to a significant computational cost if the samples include numerical discretization errors which have to be kept sufficiently small. To address this issue, we propose a multilevel version of the Active Subspace method (MLAS) that utilizes samples computed with different accuracies and yields different active subspaces across accuracy levels, which can match the accuracy of single-level AS with reduced computational cost, making it suitable for downstream tasks such as function approximation. In particular, we propose to perform the latter via optimally-weighted least-squares polynomial approximation in the different active subspaces, and we present an adaptive algorithm to choose dynamically the dimensions of the active subspaces and polynomial spaces. We demonstrate the practical viability of the MLAS method with polynomial approximation through numerical experiments based on random partial differential equations (PDEs).

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arXiv2025

Unbiased Parameter Estimation for Bayesian Inverse Problems

In this paper we consider the estimation of unknown parameters in Bayesian inverse problems. In most cases of practical interest, there are several barriers to performing such estimation, This includes a numerical approximation of a solution of a differential equation and, even if exact solutions are available, an analytical intractability of the marginal likelihood and its associated gradient, which is used for parameter estimation. The focus of this article is to deliver unbiased estimates of the unknown parameters, that is, stochastic estimators that, in expectation, are equal to the maximize of the marginal likelihood, and possess no numerical approximation error. Based upon the ideas of [4] we develop a new approach for unbiased parameter estimation for Bayesian inverse problems. We prove unbiasedness and establish numerically that the associated estimation procedure is faster than the current state-of-the-art methodology for this problem. We demonstrate the performance of our methodology on a range of problems which include a PDE and ODE.

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arXiv2024

Residual Multi-Fidelity Neural Network Computing

In this work, we consider the general problem of constructing a neural network surrogate model using multi-fidelity information. Motivated by error-complexity estimates for ReLU neural networks, we formulate the correlation between an inexpensive low-fidelity model and an expensive high-fidelity model as a possibly non-linear residual function. This function defines a mapping between 1) the shared input space of the models along with the low-fidelity model output, and 2) the discrepancy between the outputs of the two models. The computational framework proceeds by training two neural networks to work in concert. The first network learns the residual function on a small set of high- and low-fidelity data. Once trained, this network is used to generate additional synthetic high-fidelity data, which is used in the training of the second network. The trained second network then acts as our surrogate for the high-fidelity quantity of interest. We present four numerical examples to demonstrate the power of the proposed framework, showing that significant savings in computational cost may be achieved when the output predictions are desired to be accurate within small tolerances.

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arXiv2024

Parameter Estimation for Partially Observed McKean-Vlasov Diffusions

In this article we consider likelihood-based estimation of static parameters for a class of partially observed McKean-Vlasov (POMV) diffusion process with discrete-time observations over a fixed time interval. In particular, using the framework of [5] we develop a new randomized multilevel Monte Carlo method for estimating the parameters, based upon Markovian stochastic approximation methodology. New Markov chain Monte Carlo algorithms for the POMV model are introduced facilitating the application of [5]. We prove, under assumptions, that the expectation of our estimator is biased, but with expected small and controllable bias. Our approach is implemented on several examples.

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arXiv2024

Estimation of uncertainties in the density driven flow in fractured porous media using MLMC

We use the Multi Level Monte Carlo method to estimate uncertainties in a Henry-like salt water intrusion problem with a fracture. The flow is induced by the variation of the density of the fluid phase, which depends on the mass fraction of salt. We assume that the fracture has a known fixed location but an uncertain aperture. Other input uncertainties are the porosity and permeability fields and the recharge. In our setting, porosity and permeability vary spatially and recharge is time-dependent. For each realisation of these uncertain parameters, the evolution of the mass fraction and pressure fields is modelled by a system of non-linear and time-dependent PDEs with a jump of the solution at the fracture. The uncertainties propagate into the distribution of the salt concentration, which is an important characteristic of the quality of water resources. We show that the multilevel Monte Carlo (MLMC) method is able to reduce the overall computational cost compared to classical Monte Carlo methods. This is achieved by balancing discretisation and statistical errors. Multiple scenarios are evaluated at different spatial and temporal mesh levels. The deterministic solver ug4 is run in parallel to calculate all stochastic scenarios.

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arXiv2024

Importance sampling for rare event tracking within the ensemble Kalman filtering framework

In this work we employ importance sampling (IS) techniques to track a small over-threshold probability of a running maximum associated with the solution of a stochastic differential equation (SDE) within the framework of ensemble Kalman filtering (EnKF). Between two observation times of the EnKF, we propose to use IS with respect to the initial condition of the SDE, IS with respect to the Wiener process via a stochastic optimal control formulation, and combined IS with respect to both initial condition and Wiener process. Both IS strategies require the approximation of the solution of Kolmogorov Backward equation (KBE) with boundary conditions. In multidimensional settings, we employ a Markovian projection dimension reduction technique to obtain an approximation of the solution of the KBE by just solving a one dimensional PDE. The proposed ideas are tested on three illustrative examples: Double Well SDE, Langevin dynamics and noisy Charney-deVore model, and showcase a significant variance reduction compared to the standard Monte Carlo method and another sampling-based IS technique, namely, multilevel cross entropy.

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arXiv2024

Approximating Hessian matrices using Bayesian inference: a new approach for quasi-Newton methods in stochastic optimization

Using quasi-Newton methods in stochastic optimization is not a trivial task given the difficulty of extracting curvature information from the noisy gradients. Moreover, pre-conditioning noisy gradient observations tend to amplify the noise. We propose a Bayesian approach to obtain a Hessian matrix approximation for stochastic optimization that minimizes the secant equations residue while retaining the extreme eigenvalues between a specified range. Thus, the proposed approach assists stochastic gradient descent to converge to local minima without augmenting gradient noise. We propose maximizing the log posterior using the Newton-CG method. Numerical results on a stochastic quadratic function and an $\ell_2$-regularized logistic regression problem are presented. In all the cases tested, our approach improves the convergence of stochastic gradient descent, compensating for the overhead of solving the log posterior maximization. In particular, pre-conditioning the stochastic gradient with the inverse of our Hessian approximation becomes more advantageous the larger the condition number of the problem is.

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arXiv2024

Quasi-Monte Carlo with Domain Transformation for Efficient Fourier Pricing of Multi-Asset Options

Efficiently pricing multi-asset options poses a significant challenge in quantitative finance. Fourier methods leverage the regularity properties of the integrand in the Fourier domain to accurately and rapidly value options that typically lack regularity in the physical domain. However, most of the existing Fourier approaches face hurdles in high-dimensional settings due to the tensor product (TP) structure of the commonly employed numerical quadrature techniques. To overcome this difficulty, this work advocates using the randomized quasi-MC (RQMC) quadrature to improve the scalability of Fourier methods with high dimensions. The RQMC technique benefits from the smoothness of the integrand and alleviates the curse of dimensionality while providing practical error estimates. Nonetheless, the applicability of RQMC on the unbounded domain, $\mathbb{R}^d$, requires a domain transformation to $[0,1]^d$, which may result in singularities of the transformed integrand at the corners of the hypercube, and hence deteriorate the performance of RQMC. To circumvent this difficulty, we design an efficient domain transformation procedure based on boundary growth conditions on the transformed integrand. The proposed transformation preserves sufficient regularity of the original integrand for fast convergence of the RQMC method. To validate our analysis, we demonstrate the efficiency of employing RQMC with an appropriate transformation to evaluate options in the Fourier space for various pricing models, payoffs, and dimensions. Finally, we highlight the computational advantage of applying RQMC over MC or TP in the Fourier domain, and over MC in the physical domain for options with up to 15 assets.

q-fin.CP↗

arXiv2024

Uncertainty quantification in the Henry problem using the multilevel Monte Carlo method

We investigate the applicability of the well-known multilevel Monte Carlo (MLMC) method to the class of density-driven flow problems, in particular the problem of salinisation of coastal aquifers. As a test case, we solve the uncertain Henry saltwater intrusion problem. Unknown porosity, permeability and recharge parameters are modelled by using random fields. The classical deterministic Henry problem is non-linear and time-dependent, and can easily take several hours of computing time. Uncertain settings require the solution of multiple realisations of the deterministic problem, and the total computational cost increases drastically. Instead of computing of hundreds random realisations, typically the mean value and the variance are computed. The standard methods such as the Monte Carlo or surrogate-based methods is a good choice, but they compute all stochastic realisations on the same, often, very fine mesh. They also do not balance the stochastic and discretisation errors. These facts motivated us to apply the MLMC method. We demonstrate that by solving the Henry problem on multi-level spatial and temporal meshes, the MLMC method reduces the overall computational and storage costs. To reduce the computing cost further, parallelization is performed in both physical and stochastic spaces. To solve each deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion.

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arXiv2024

Stochastic differential equations for performance analysis of wireless communication systems

This paper addresses the difficulty of characterizing the time-varying nature of fading channels. The current time-invariant models often fall short of capturing and tracking these dynamic characteristics. To overcome this limitation, we explore using of stochastic differential equations (SDEs) and Markovian projection to model signal envelope variations, considering scenarios involving Rayleigh, Rice, and Hoyt distributions. Furthermore, it is of practical interest to study the performance of channels modeled by SDEs. In this work, we investigate the fade duration metric, representing the time during which the signal remains below a specified threshold within a fixed time interval. We estimate the complementary cumulative distribution function (CCDF) of the fade duration using Monte Carlo simulations, and analyze the influence of system parameters on its behavior. Finally, we leverage importance sampling, a known variance-reduction technique, to estimate the tail of the CCDF efficiently.

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arXiv2023

Multilevel Monte Carlo with Numerical Smoothing for Robust and Efficient Computation of Probabilities and Densities

The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in (Quantitative Finance, 23(2), 209-227, 2023), in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence, and consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.

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arXiv2023

Modeling metallic fatigue data using the Birnbaum--Saunders distribution

This work employs the Birnbaum--Saunders distribution to model the fatigue life of metallic materials under cyclic loading and compares it with the normal distribution. Fatigue-limit models are fitted to three datasets of unnotched specimens of 75S-T6 aluminum alloys and carbon laminate with different loading types. A new equivalent stress definition that accounts for the effect of the experiment type is proposed. The results show that the Birnbaum--Saunders distribution consistently outperforms the normal distribution in fitting the fatigue data and provides more accurate predictions of fatigue life and survival probability.

stat.CO↗

arXiv2023

Physics-informed Spectral Learning: the Discrete Helmholtz--Hodge Decomposition

In this work, we further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. \cite{Esp21} based on a discrete $L^2$ projection to solve the discrete Hodge--Helmholtz decomposition from sparse data. Within this physics-informed statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. Moreover, our PiSL computational framework enjoys spectral (exponential) convergence. We regularize the minimization problems with the seminorm of the fractional Sobolev space in a Tikhonov fashion. In the Fourier setting, the divergence- and curl-free constraints become a finite set of linear algebraic equations. The proposed computational framework combines supervised and unsupervised learning techniques in that we use data concomitantly with the projection onto divergence- and curl-free spaces. We assess the capabilities of our method in various numerical examples including the `Storm of the Century' with satellite data from 1993.

cs.LG↗

arXiv2023

Uncertainty quantification in coastal aquifers using the multilevel Monte Carlo method

We consider a class of density-driven flow problems. We are particularly interested in the problem of the salinization of coastal aquifers. We consider the Henry saltwater intrusion problem with uncertain porosity, permeability, and recharge parameters as a test case. The reason for the presence of uncertainties is the lack of knowledge, inaccurate measurements, and inability to measure parameters at each spatial or time location. This problem is nonlinear and time-dependent. The solution is the salt mass fraction, which is uncertain and changes in time. Uncertainties in porosity, permeability, recharge, and mass fraction are modeled using random fields. This work investigates the applicability of the well-known multilevel Monte Carlo (MLMC) method for such problems. The MLMC method can reduce the total computational and storage costs. Moreover, the MLMC method runs multiple scenarios on different spatial and time meshes and then estimates the mean value of the mass fraction. The parallelization is performed in both the physical space and stochastic space. To solve every deterministic scenario, we run the parallel multigrid solver ug4 in a black-box fashion. We use the solution obtained from the quasi-Monte Carlo method as a reference solution.

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