1,685 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
Maximum-likelihood estimation of lithospheric flexural rigidity, initial-loading fraction, and load correlation, under isotropy
Topography and gravity are geophysical fields whose joint statistical
structure derives from interface-loading processes modulated by the underlying
mechanics of isostatic and flexural compensation in the shallow lithosphere.
Under this dual statistical-mechanistic viewpoint an estimation problem can be
formulated where the knowns are topography and gravity and the principal
unknown the elastic flexural rigidity of the lithosphere. In the guise of an
equivalent "effective elastic thickness", this important, geographically
varying, structural parameter has been the subject of many interpretative
studies, but precisely how well it is known or how best it can be found from
the data, abundant nonetheless, has remained contentious and unresolved
throughout the last few decades of dedicated study. The popular methods whereby
admittance or coherence, both spectral measures of the relation between gravity
and topography, are inverted for the flexural rigidity, have revealed
themselves to have insufficient power to independently constrain both it and
the additional unknown initial-loading fraction and load-correlation fac- tors,
respectively. Solving this extremely ill-posed inversion problem leads to
non-uniqueness and is further complicated by practical considerations such as
the choice of regularizing data tapers to render the analysis sufficiently
selective both in the spatial and spectral domains. Here, we rewrite the
problem in a form amenable to maximum-likelihood estimation theory, which we
show yields unbiased, minimum-variance estimates of flexural rigidity,
initial-loading frac- tion and load correlation, each of those separably
resolved with little a posteriori correlation between their estimates. We are
also able to separately characterize the isotropic spectral shape of the
initial loading processes.Comment: 41 pages, 13 figures, accepted for publication by Geophysical Journal
Internationa
Reduced order modeling with time-dependent bases for PDEs with stochastic boundary conditions
Low-rank approximation using time-dependent bases (TDBs) has proven effective
for reduced-order modeling of stochastic partial differential equations
(SPDEs). In these techniques, the random field is decomposed to a set of
deterministic TDBs and time-dependent stochastic coefficients. When applied to
SPDEs with non-homogeneous stochastic boundary conditions (BCs), appropriate BC
must be specified for each of the TDBs. However, determining BCs for TDB is not
trivial because: (i) the dimension of the random BCs is different than the rank
of the TDB subspace; (ii) TDB in most formulations must preserve orthonormality
or orthogonality constraints and specifying BCs for TDB should not violate
these constraints in the space-discretized form. In this work, we present a
methodology for determining the boundary conditions for TDBs at no additional
computational cost beyond that of solving the same SPDE with homogeneous BCs.
Our methodology is informed by the fact the TDB evolution equations are the
optimality conditions of a variational principle. We leverage the same
variational principle to derive an evolution equation for the value of TDB at
the boundaries. The presented methodology preserves the orthonormality or
orthogonality constraints of TDBs. We present the formulation for both the
dynamically bi-orthonormal (DBO) decomposition as well as the dynamically
orthogonal (DO) decomposition. We show that the presented methodology can be
applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We
assess the performance of the presented method for linear advection-diffusion
equation, Burgers' equation, and two-dimensional advection-diffusion equation
with constant and temperature-dependent conduction coefficient
A computational framework for infinite-dimensional Bayesian inverse problems: Part II. Stochastic Newton MCMC with application to ice sheet flow inverse problems
We address the numerical solution of infinite-dimensional inverse problems in
the framework of Bayesian inference. In the Part I companion to this paper
(arXiv.org:1308.1313), we considered the linearized infinite-dimensional
inverse problem. Here in Part II, we relax the linearization assumption and
consider the fully nonlinear infinite-dimensional inverse problem using a
Markov chain Monte Carlo (MCMC) sampling method. To address the challenges of
sampling high-dimensional pdfs arising from Bayesian inverse problems governed
by PDEs, we build on the stochastic Newton MCMC method. This method exploits
problem structure by taking as a proposal density a local Gaussian
approximation of the posterior pdf, whose construction is made tractable by
invoking a low-rank approximation of its data misfit component of the Hessian.
Here we introduce an approximation of the stochastic Newton proposal in which
we compute the low-rank-based Hessian at just the MAP point, and then reuse
this Hessian at each MCMC step. We compare the performance of the proposed
method to the original stochastic Newton MCMC method and to an independence
sampler. The comparison of the three methods is conducted on a synthetic ice
sheet inverse problem. For this problem, the stochastic Newton MCMC method with
a MAP-based Hessian converges at least as rapidly as the original stochastic
Newton MCMC method, but is far cheaper since it avoids recomputing the Hessian
at each step. On the other hand, it is more expensive per sample than the
independence sampler; however, its convergence is significantly more rapid, and
thus overall it is much cheaper. Finally, we present extensive analysis and
interpretation of the posterior distribution, and classify directions in
parameter space based on the extent to which they are informed by the prior or
the observations.Comment: 31 page
Computational Inverse Problems for Partial Differential Equations
The problem of determining unknown quantities in a PDE from measurements of (part of) the solution to this PDE arises in a wide range of applications in science, technology, medicine, and finance. The unknown quantity may e.g. be a coefficient, an initial or a boundary condition, a source term, or the shape of a boundary. The identification of such quantities is often computationally challenging and requires profound knowledge of the analytical properties of the underlying PDE as well as numerical techniques. The focus of this workshop was on applications in phase retrieval, imaging with waves in random media, and seismology of the Earth and the Sun, a further emphasis was put on stochastic aspects in the context of uncertainty quantification and parameter identification in stochastic differential equations. Many open problems and mathematical challenges in application fields were addressed, and intensive discussions provided an insight into the high potential of joining deep knowledge in numerical analysis, partial differential equations, and regularization, but also in mathematical statistics, homogenization, optimization, differential geometry, numerical linear algebra, and variational analysis to tackle these challenges
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