2,220 research outputs found
Templates for Convex Cone Problems with Applications to Sparse Signal Recovery
This paper develops a general framework for solving a variety of convex cone
problems that frequently arise in signal processing, machine learning,
statistics, and other fields. The approach works as follows: first, determine a
conic formulation of the problem; second, determine its dual; third, apply
smoothing; and fourth, solve using an optimal first-order method. A merit of
this approach is its flexibility: for example, all compressed sensing problems
can be solved via this approach. These include models with objective
functionals such as the total-variation norm, ||Wx||_1 where W is arbitrary, or
a combination thereof. In addition, the paper also introduces a number of
technical contributions such as a novel continuation scheme, a novel approach
for controlling the step size, and some new results showing that the smooth and
unsmoothed problems are sometimes formally equivalent. Combined with our
framework, these lead to novel, stable and computationally efficient
algorithms. For instance, our general implementation is competitive with
state-of-the-art methods for solving intensively studied problems such as the
LASSO. Further, numerical experiments show that one can solve the Dantzig
selector problem, for which no efficient large-scale solvers exist, in a few
hundred iterations. Finally, the paper is accompanied with a software release.
This software is not a single, monolithic solver; rather, it is a suite of
programs and routines designed to serve as building blocks for constructing
complete algorithms.Comment: The TFOCS software is available at http://tfocs.stanford.edu This
version has updated reference
On the error propagation of semi-Lagrange and Fourier methods for advection problems
In this paper we study the error propagation of numerical schemes for the
advection equation in the case where high precision is desired. The numerical
methods considered are based on the fast Fourier transform, polynomial
interpolation (semi-Lagrangian methods using a Lagrange or spline
interpolation), and a discontinuous Galerkin semi-Lagrangian approach (which is
conservative and has to store more than a single value per cell).
We demonstrate, by carrying out numerical experiments, that the worst case
error estimates given in the literature provide a good explanation for the
error propagation of the interpolation-based semi-Lagrangian methods. For the
discontinuous Galerkin semi-Lagrangian method, however, we find that the
characteristic property of semi-Lagrangian error estimates (namely the fact
that the error increases proportionally to the number of time steps) is not
observed. We provide an explanation for this behavior and conduct numerical
simulations that corroborate the different qualitative features of the error in
the two respective types of semi-Lagrangian methods.
The method based on the fast Fourier transform is exact but, due to round-off
errors, susceptible to a linear increase of the error in the number of time
steps. We show how to modify the Cooley--Tukey algorithm in order to obtain an
error growth that is proportional to the square root of the number of time
steps.
Finally, we show, for a simple model, that our conclusions hold true if the
advection solver is used as part of a splitting scheme.Comment: submitted to Computers & Mathematics with Application
An inexact Newton-Krylov algorithm for constrained diffeomorphic image registration
We propose numerical algorithms for solving large deformation diffeomorphic
image registration problems. We formulate the nonrigid image registration
problem as a problem of optimal control. This leads to an infinite-dimensional
partial differential equation (PDE) constrained optimization problem.
The PDE constraint consists, in its simplest form, of a hyperbolic transport
equation for the evolution of the image intensity. The control variable is the
velocity field. Tikhonov regularization on the control ensures well-posedness.
We consider standard smoothness regularization based on - or
-seminorms. We augment this regularization scheme with a constraint on the
divergence of the velocity field rendering the deformation incompressible and
thus ensuring that the determinant of the deformation gradient is equal to one,
up to the numerical error.
We use a Fourier pseudospectral discretization in space and a Chebyshev
pseudospectral discretization in time. We use a preconditioned, globalized,
matrix-free, inexact Newton-Krylov method for numerical optimization. A
parameter continuation is designed to estimate an optimal regularization
parameter. Regularity is ensured by controlling the geometric properties of the
deformation field. Overall, we arrive at a black-box solver. We study spectral
properties of the Hessian, grid convergence, numerical accuracy, computational
efficiency, and deformation regularity of our scheme. We compare the designed
Newton-Krylov methods with a globalized preconditioned gradient descent. We
study the influence of a varying number of unknowns in time.
The reported results demonstrate excellent numerical accuracy, guaranteed
local deformation regularity, and computational efficiency with an optional
control on local mass conservation. The Newton-Krylov methods clearly
outperform the Picard method if high accuracy of the inversion is required.Comment: 32 pages; 10 figures; 9 table
Finite Mechanical Proxies for a Class of Reducible Continuum Systems
We present the exact finite reduction of a class of nonlinearly perturbed
wave equations, based on the Amann-Conley-Zehnder paradigm. By solving an
inverse eigenvalue problem, we establish an equivalence between the spectral
finite description derived from A-C-Z and a discrete mechanical model, a well
definite finite spring-mass system. By doing so, we decrypt the abstract
information encoded in the finite reduction and obtain a physically sound proxy
for the continuous problem.Comment: 15 pages, 3 figure
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