2,864 research outputs found
A variational Bayesian method for inverse problems with impulsive noise
We propose a novel numerical method for solving inverse problems subject to
impulsive noises which possibly contain a large number of outliers. The
approach is of Bayesian type, and it exploits a heavy-tailed t distribution for
data noise to achieve robustness with respect to outliers. A hierarchical model
with all hyper-parameters automatically determined from the given data is
described. An algorithm of variational type by minimizing the Kullback-Leibler
divergence between the true posteriori distribution and a separable
approximation is developed. The numerical method is illustrated on several one-
and two-dimensional linear and nonlinear inverse problems arising from heat
conduction, including estimating boundary temperature, heat flux and heat
transfer coefficient. The results show its robustness to outliers and the fast
and steady convergence of the algorithm.Comment: 20 pages, to appear in J. Comput. Phy
Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors
This paper addresses the variational multiscale stabilization of standard
finite element methods for linear partial differential equations that exhibit
multiscale features. The stabilization is of Petrov-Galerkin type with a
standard finite element trial space and a problem-dependent test space based on
pre-computed fine-scale correctors. The exponential decay of these correctors
and their localisation to local cell problems is rigorously justified. The
stabilization eliminates scale-dependent pre-asymptotic effects as they appear
for standard finite element discretizations of highly oscillatory problems,
e.g., the poor approximation in homogenization problems or the pollution
effect in high-frequency acoustic scattering
A Partially Reflecting Random Walk on Spheres Algorithm for Electrical Impedance Tomography
In this work, we develop a probabilistic estimator for the voltage-to-current
map arising in electrical impedance tomography. This novel so-called partially
reflecting random walk on spheres estimator enables Monte Carlo methods to
compute the voltage-to-current map in an embarrassingly parallel manner, which
is an important issue with regard to the corresponding inverse problem. Our
method uses the well-known random walk on spheres algorithm inside subdomains
where the diffusion coefficient is constant and employs replacement techniques
motivated by finite difference discretization to deal with both mixed boundary
conditions and interface transmission conditions. We analyze the global bias
and the variance of the new estimator both theoretically and experimentally. In
a second step, the variance is considerably reduced via a novel control variate
conditional sampling technique
Quantitative photoacoustic tomography with piecewise constant material parameters
The goal of quantitative photoacoustic tomography is to determine optical and
acoustical material properties from initial pressure maps as obtained, for
instance, from photoacoustic imaging. The most relevant parameters are
absorption, diffusion and Grueneisen coefficients, all of which can be
heterogeneous. Recent work by Bal and Ren shows that in general, unique
reconstruction of all three parameters is impossible, even if multiple
measurements of the initial pressure (corresponding to different laser
excitation directions at a single wavelength) are available.
Here, we propose a restriction to piecewise constant material parameters. We
show that in the diffusion approximation of light transfer, piecewise constant
absorption, diffusion and Gr\"uneisen coefficients can be recovered uniquely
from photoacoustic measurements at a single wavelength. In addition, we
implemented our ideas numerically and tested them on simulated
three-dimensional data
A Computational Study of the Weak Galerkin Method for Second-Order Elliptic Equations
The weak Galerkin finite element method is a novel numerical method that was
first proposed and analyzed by Wang and Ye for general second order elliptic
problems on triangular meshes. The goal of this paper is to conduct a
computational investigation for the weak Galerkin method for various model
problems with more general finite element partitions. The numerical results
confirm the theory established by Wang and Ye. The results also indicate that
the weak Galerkin method is efficient, robust, and reliable in scientific
computing.Comment: 19 page
Sparse operator compression of higher-order elliptic operators with rough coefficients
We introduce the sparse operator compression to compress a self-adjoint
higher-order elliptic operator with rough coefficients and various boundary
conditions. The operator compression is achieved by using localized basis
functions, which are energy-minimizing functions on local patches. On a regular
mesh with mesh size , the localized basis functions have supports of
diameter and give optimal compression rate of the solution
operator. We show that by using localized basis functions with supports of
diameter , our method achieves the optimal compression rate of
the solution operator. From the perspective of the generalized finite element
method to solve elliptic equations, the localized basis functions have the
optimal convergence rate for a th-order elliptic problem in the
energy norm. From the perspective of the sparse PCA, our results show that a
large set of Mat\'{e}rn covariance functions can be approximated by a rank-
operator with a localized basis and with the optimal accuracy
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