990 research outputs found
Convergence of Entropic Schemes for Optimal Transport and Gradient Flows
Replacing positivity constraints by an entropy barrier is popular to
approximate solutions of linear programs. In the special case of the optimal
transport problem, this technique dates back to the early work of
Schr\"odinger. This approach has recently been used successfully to solve
optimal transport related problems in several applied fields such as imaging
sciences, machine learning and social sciences. The main reason for this
success is that, in contrast to linear programming solvers, the resulting
algorithms are highly parallelizable and take advantage of the geometry of the
computational grid (e.g. an image or a triangulated mesh). The first
contribution of this article is the proof of the -convergence of the
entropic regularized optimal transport problem towards the Monge-Kantorovich
problem for the squared Euclidean norm cost function. This implies in
particular the convergence of the optimal entropic regularized transport plan
towards an optimal transport plan as the entropy vanishes. Optimal transport
distances are also useful to define gradient flows as a limit of implicit Euler
steps according to the transportation distance. Our second contribution is a
proof that implicit steps according to the entropic regularized distance
converge towards the original gradient flow when both the step size and the
entropic penalty vanish (in some controlled way)
Variational Data Assimilation via Sparse Regularization
This paper studies the role of sparse regularization in a properly chosen
basis for variational data assimilation (VDA) problems. Specifically, it
focuses on data assimilation of noisy and down-sampled observations while the
state variable of interest exhibits sparsity in the real or transformed domain.
We show that in the presence of sparsity, the -norm regularization
produces more accurate and stable solutions than the classic data assimilation
methods. To motivate further developments of the proposed methodology,
assimilation experiments are conducted in the wavelet and spectral domain using
the linear advection-diffusion equation
Second-order Shape Optimization for Geometric Inverse Problems in Vision
We develop a method for optimization in shape spaces, i.e., sets of surfaces
modulo re-parametrization. Unlike previously proposed gradient flows, we
achieve superlinear convergence rates through a subtle approximation of the
shape Hessian, which is generally hard to compute and suffers from a series of
degeneracies. Our analysis highlights the role of mean curvature motion in
comparison with first-order schemes: instead of surface area, our approach
penalizes deformation, either by its Dirichlet energy or total variation.
Latter regularizer sparks the development of an alternating direction method of
multipliers on triangular meshes. Therein, a conjugate-gradients solver enables
us to bypass formation of the Gaussian normal equations appearing in the course
of the overall optimization. We combine all of the aforementioned ideas in a
versatile geometric variation-regularized Levenberg-Marquardt-type method
applicable to a variety of shape functionals, depending on intrinsic properties
of the surface such as normal field and curvature as well as its embedding into
space. Promising experimental results are reported
Geotomography with solar and supernova neutrinos
We show how by studying the Earth matter effect on oscillations of solar and
supernova neutrinos inside the Earth one can in principle reconstruct the
electron number density profile of the Earth. A direct inversion of the
oscillation problem is possible due to the existence of a very simple analytic
formula for the Earth matter effect on oscillations of solar and supernova
neutrinos. From the point of view of the Earth tomography, these oscillations
have a number of advantages over the oscillations of the accelerator or
atmospheric neutrinos, which stem from the fact that solar and supernova
neutrinos are coming to the Earth as mass eigenstates rather than flavour
eigenstates. In particular, this allows reconstruction of density profiles even
over relatively short neutrino path lengths in the Earth, and also of
asymmetric profiles. We study the requirements that future experiments must
meet to achieve a given accuracy of the tomography of the Earth.Comment: 35 pages, 7 figures; minor textual changes in section
The Minimization of Piecewise Functions: Pseudo Stationarity
There are many significant applied contexts that require the solution of
discontinuous optimization problems in finite dimensions. Yet these problems
are very difficult, both computationally and analytically. With the functions
being discontinuous and a minimizer (local or global) of the problems, even if
it exists, being impossible to verifiably compute, a foremost question is what
kind of ''stationary solutions'' one can expect to obtain; these solutions
provide promising candidates for minimizers; i.e., their defining conditions
are necessary for optimality. Motivated by recent results on sparse
optimization, we introduce in this paper such a kind of solution, termed
''pseudo B- (for Bouligand) stationary solution'', for a broad class of
discontinuous piecewise continuous optimization problems with objective and
constraint defined by indicator functions of the positive real axis composite
with functions that are possibly nonsmooth. We present two approaches for
computing such a solution. One approach is based on lifting the problem to a
higher dimension via the epigraphical formulation of the indicator functions;
this requires the addition of some auxiliary variables. The other approach is
based on certain continuous (albeit not necessarily differentiable) piecewise
approximations of the indicator functions and the convergence to a pseudo
B-stationary solution of the original problem is established. The conditions
for convergence are discussed and illustrated by an example
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