1,573 research outputs found
Poisson inverse problems
In this paper we focus on nonparametric estimators in inverse problems for
Poisson processes involving the use of wavelet decompositions. Adopting an
adaptive wavelet Galerkin discretization, we find that our method combines the
well-known theoretical advantages of wavelet--vaguelette decompositions for
inverse problems in terms of optimally adapting to the unknown smoothness of
the solution, together with the remarkably simple closed-form expressions of
Galerkin inversion methods. Adapting the results of Barron and Sheu [Ann.
Statist. 19 (1991) 1347--1369] to the context of log-intensity functions
approximated by wavelet series with the use of the Kullback--Leibler distance
between two point processes, we also present an asymptotic analysis of
convergence rates that justifies our approach. In order to shed some light on
the theoretical results obtained and to examine the accuracy of our estimates
in finite samples, we illustrate our method by the analysis of some simulated
examples.Comment: Published at http://dx.doi.org/10.1214/009053606000000687 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Nonlinear estimation for linear inverse problems with error in the operator
We study two nonlinear methods for statistical linear inverse problems when
the operator is not known. The two constructions combine Galerkin
regularization and wavelet thresholding. Their performances depend on the
underlying structure of the operator, quantified by an index of sparsity. We
prove their rate-optimality and adaptivity properties over Besov classes.Comment: Published in at http://dx.doi.org/10.1214/009053607000000721 the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Bayesian linear inverse problems in regularity scales
We obtain rates of contraction of posterior distributions in inverse problems
defined by scales of smoothness classes. We derive abstract results for general
priors, with contraction rates determined by Galerkin approximation. The rate
depends on the amount of prior concentration near the true function and the
prior mass of functions with inferior Galerkin approximation. We apply the
general result to non-conjugate series priors, showing that these priors give
near optimal and adaptive recovery in some generality, Gaussian priors, and
mixtures of Gaussian priors, where the latter are also shown to be near optimal
and adaptive. The proofs are based on general testing and approximation
arguments, without explicit calculations on the posterior distribution. We are
thus not restricted to priors based on the singular value decomposition of the
operator. We illustrate the results with examples of inverse problems resulting
from differential equations.Comment: 34 page
Review of modern numerical methods for a simple vanilla option pricing problem
Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better
Adaptive boundary element methods with convergence rates
This paper presents adaptive boundary element methods for positive, negative,
as well as zero order operator equations, together with proofs that they
converge at certain rates. The convergence rates are quasi-optimal in a certain
sense under mild assumptions that are analogous to what is typically assumed in
the theory of adaptive finite element methods. In particular, no
saturation-type assumption is used. The main ingredients of the proof that
constitute new findings are some results on a posteriori error estimates for
boundary element methods, and an inverse-type inequality involving boundary
integral operators on locally refined finite element spaces.Comment: 48 pages. A journal version. The previous version (v3) is a bit
lengthie
Deconvolution in white noise with a random blurring function
We consider the problem of denoising a function observed after a convolution
with a random filter independent of the noise and satisfying some mean
smoothness condition depending on an ill posedness coefficient. We establish
the minimax rates for the Lp risk over balls of periodic Besov spaces with
respect to the level of noise, and we provide an adaptive estimator achieving
these rates up to log factors. Simulations were performed to highlight the
effects of the ill posedness and of the distribution of the filter on the
efficiency of the estimator
An optimal adaptive Fictitious Domain Method
We consider a Fictitious Domain formulation of an elliptic partial
differential equation and approximate the resulting saddle-point system using
an inexact preconditioned Uzawa iterative algorithm. Each iteration entails the
approximation of an elliptic problems performed using adaptive finite element
methods. We prove that the overall method converges with the best possible rate
and illustrate numerically our theoretical findings
Adaptive Spectral Galerkin Methods with Dynamic Marking
The convergence and optimality theory of adaptive Galerkin methods is almost
exclusively based on the D\"orfler marking. This entails a fixed parameter and
leads to a contraction constant bounded below away from zero. For spectral
Galerkin methods this is a severe limitation which affects performance. We
present a dynamic marking strategy that allows for a super-linear relation
between consecutive discretization errors, and show exponential convergence
with linear computational complexity whenever the solution belongs to a Gevrey
approximation class.Comment: 20 page
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