IGA-based Multi-Index Stochastic Collocation for random PDEs on arbitrary domains

This paper proposes an extension of the Multi-Index Stochastic Collocation (MISC) method for forward uncertainty quantification (UQ) problems in computational domains of shape other than a square or cube, by exploiting isogeometric analysis (IGA) techniques. Introducing IGA solvers to the MISC algorithm is very natural since they are tensor-based PDE solvers, which are precisely what is required by the MISC machinery. Moreover, the combination-technique formulation of MISC allows the straight-forward reuse of existing implementations of IGA solvers. We present numerical results to showcase the effectiveness of the proposed approach.Comment: version 3, version after revisio

Pricing American Options by Exercise Rate Optimization

We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model

Approximation and interpolation of divergence free flows.

In many applications like meteorology, atmospheric pollution studies, eolic energy prospection, estimation of instantaneous velocity fields etc., one faces the problem of estimating a velocity field that is assumed to be incompressible. Very often the available data contains just a few and sparse velocity measurements and may be some boundary conditions imposed by solid boundaries. This inverse problem is studied here, and a new method to provide a numerical solution is presented. It is based on the Fourier transform, and allows to include the incompressibility constraint in a simple way, leading to an unconstrained least squares formulation, usually ill-posed. The Tikhonov regularization is applied to stabilize the solution, as well as to provide some smoothness in the estimated fow. As a consequence, the numerical solution will generally approximate the measurements up to a threshold given by the size of the regularization parameter. Moreover, if the available velocity measurements come from a smooth velocity field then the numerical solution can be usually constructed using just a small number of Fourier terms. The choice of the regularization parameter is done using the L curve method, balancing the perturbation and regularization contributions to the error. Perturbation bounds (i.e.), bounds for the condition number of the matrix from the Least Squares formulation are included. Numerical experiments with test problems and real data from the southern part of Uruguay are carried out. In addition, the results are compared with related work and the results are satisfactory

A note on tools for prediction under uncertainty and identifiability of SIR-like dynamical systems for epidemiology

We provide an overview of the methods that can be used for prediction under uncertainty and data fitting of dynamical systems, and of the fundamental challenges that arise in this context. The focus is on SIR-like models, that are being commonly used when attempting to predict the trend of the COVID-19 pandemic. In particular, we raise a warning flag about identifiability of the parameters of SIR-like models; often, it might be hard to infer the correct values of the parameters from data, even for very simple models, making it non-trivial to use these models for meaningful predictions. Most of the points that we touch upon are actually generally valid for inverse problems in more general setups