This article extends the framework of Bayesian inverse problems in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451--559, 2010) and others, to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined.
It is shown that analogues of the Karhunen--Loève expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces.
Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger metric upon perturbations of the misfit function and observed data.
In this work, we consider a FDE (fractional diffusion equation)
$$C_{D^\alpha_t} u(x,t)-a(t)\mathcal{L}u(x,t)=F(x,t)$$ with a time-dependent diffusion
coefficient $a(t)$. This is an extension of [13],
which deals with this FDE in one-dimensional space.
For the direct problem, given an $a(t),$
we establish the existence, uniqueness and some regularity properties
with a more general domain $\Omega$ and right-hand side $F(x,t)$.
For the inverse problem--recovering $a(t),$
we introduce an operator $K$ one of whose fixed points is $a(t)$
and show its monotonicity, uniqueness and
existence of its fixed points. With these properties, a reconstruction
algorithm for $a(t)$ is created and some numerical results are provided
to illustrate the theories.
We propose a direct imaging method based on the reverse time migration method for finding extended obstacles with phaseless total field data in the half space.
We prove that the imaging resolution of the method is essentially the same as the imaging results using the
scattering data with full phase information when the obstacle is far away from the surface of the half-space where the
measurement is taken. Numerical experiments are included to illustrate the
powerful imaging quality.
Surface denoising is a fundamental problem in geometry processing and computer graphics. In this paper, we propose a wavelet frame based variational model to restore surfaces which are corrupted by mixed Gaussian and impulse noise, under the assumption that the region corrupted by impulse noise is unknown. The model contains a universal $\ell_1 + \ell_2$ fidelity term and an $\ell_1$-regularized term which makes additional use of the wavelet frame transform on surfaces in order to preserve key features such as sharp edges and corners. We then apply the augmented Lagrangian and accelerated proximal gradient methods to solve this model. In the end, we demonstrate the efficacy of our approach with numerical experiments both on surfaces and functions defined on surfaces. The experimental results show that our method is competitive relative to some existing denoising methods.
This paper examines issues of data completion and location uncertainty, popular in many practical
PDE-based
inverse problems, in the context of option calibration via recovery of local volatility surfaces.
While real data is usually more accessible for this application than for many others, the data
is often given only at
a restricted set of locations. We show that attempts to “complete missing data”
by approximation or interpolation,
proposed and applied in the literature, may produce results that are inferior to treating the data as scarce.
Furthermore, model uncertainties may arise which translate to uncertainty in data locations,
and we show how a model-based
adjustment of the asset price may prove advantageous in such situations.
We further compare a carefully calibrated Tikhonov-type regularization approach
against a similarly adapted EnKF method,
in an attempt to fine-tune the data assimilation process.
The EnKF method offers reassurance as a different method
for assessing the solution in a problem where information about the true solution is
difficult to come by.
However, additional advantage in the latter
approach turns out to be limited in our context.
