99 research outputs found
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
Nonasymptotic Convergence Rate of Quasi-Monte Carlo: Applications to Linear Elliptic PDEs with Lognormal Coefficients and Importance Samplings
This study analyzes the nonasymptotic convergence behavior of the quasi-Monte
Carlo (QMC) method with applications to linear elliptic partial differential
equations (PDEs) with lognormal coefficients. Building upon the error analysis
presented in (Owen, 2006), we derive a nonasymptotic convergence estimate
depending on the specific integrands, the input dimensionality, and the finite
number of samples used in the QMC quadrature. We discuss the effects of the
variance and dimensionality of the input random variable. Then, we apply the
QMC method with importance sampling (IS) to approximate deterministic,
real-valued, bounded linear functionals that depend on the solution of a linear
elliptic PDE with a lognormal diffusivity coefficient in bounded domains of
, where the random coefficient is modeled as a stationary
Gaussian random field parameterized by the trigonometric and wavelet-type
basis. We propose two types of IS distributions, analyze their effects on the
QMC convergence rate, and observe the improvements
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