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The probability density function (PDF) of a random variable associated with the solution of a partial differential equation (PDE) with random parameters is approximated using a truncated series expansion. The random PDE is solved using two…
In this work, we revisit the use of the virtual density method to model uniform geometrical perturbations. We propose a general algorithm in order to estimate explicitly the effect of geometrical perturbations in continuous-energy Monte…
Over the last couple of decades, several copula based methods have been proposed in the literature to test for the independence among several random variables. But these existing tests are not invariant under monotone transformations of the…
Current reserve procurement approaches ignore the stochastic nature of reserve asset availability itself and thus limit the type and volume of reserve offers. This paper develops a reliability-aware probabilistic approach that allows…
With the fast growth of wind power penetration, power systems need additional flexibility to cope with wind power ramping. Several electricity markets have established requirements for flexible ramping capacity (FRC) reserves. This paper…
In this work we consider the problem of approximating the statistics of a given Quantity of Interest (QoI) that depends on the solution of a linear elliptic PDE defined over a random domain parameterized by $N$ random variables. The random…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
In the study of natural and artificial complex systems, responses that are not completely determined by the considered decision variables are commonly modelled probabilistically, resulting in response distributions varying across decision…
Many random processes can be simulated as the output of a deterministic model accepting random inputs. Such a model usually describes a complex mathematical or physical stochastic system and the randomness is introduced in the input…
A probabilistic method for solving time-dependent load-transfer models of fracture is developed. It is applicable to any rule of load redistribution, i.e, local, hierarchical, etc. In the new method, the fluctuations are generated during…
Non-random sample selection is a commonplace amongst many empirical studies and it appears when an output variable of interest is available only for a restricted non-random sub-sample of data. We introduce an extension of the generalized…
We investigate Monte Carlo based algorithms for solving stochastic control problems with probabilistic constraints. Our motivation comes from microgrid management, where the controller tries to optimally dispatch a diesel generator while…
Many problems in the geophysical sciences demand the ability to calibrate the parameters and predict the time evolution of complex dynamical models using sequentially-collected data. Here we introduce a general methodology for the joint…
We overview a series of recent works addressing numerical simulations of partial differential equations in the presence of some elements of randomness. The specific equations manipulated are linear elliptic, and arise in the context of…
We present a class of flexible and tractable static factor models for the term structure of joint default probabilities, the factor copula models. These high-dimensional models remain parsimonious with pair-copula constructions, and nest…
Solving decision problems in complex, stochastic environments is often achieved by estimating the expected outcome of decisions via Monte Carlo sampling. However, sampling may overlook rare, but important events, which can severely impact…
In this paper we present a basis selection method that can be used with $\ell_1$-minimization to adaptively determine the large coefficients of polynomial chaos expansions (PCE). The adaptive construction produces anisotropic basis sets…
Being the limits of copulas of componentwise maxima in independent random samples, extreme-value copulas can be considered to provide appropriate models for the dependence structure between rare events. Extreme-value copulas not only arise…
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…
Principal component analysis (PCA) is a well-established tool in machine learning and data processing. The principal axes in PCA were shown to be equivalent to the maximum marginal likelihood estimator of the factor loading matrix in a…