2,030 research outputs found
A mixed regularization approach for sparse simultaneous approximation of parameterized PDEs
We present and analyze a novel sparse polynomial technique for the
simultaneous approximation of parameterized partial differential equations
(PDEs) with deterministic and stochastic inputs. Our approach treats the
numerical solution as a jointly sparse reconstruction problem through the
reformulation of the standard basis pursuit denoising, where the set of jointly
sparse vectors is infinite. To achieve global reconstruction of sparse
solutions to parameterized elliptic PDEs over both physical and parametric
domains, we combine the standard measurement scheme developed for compressed
sensing in the context of bounded orthonormal systems with a novel mixed-norm
based regularization method that exploits both energy and sparsity. In
addition, we are able to prove that, with minimal sample complexity, error
estimates comparable to the best -term and quasi-optimal approximations are
achievable, while requiring only a priori bounds on polynomial truncation error
with respect to the energy norm. Finally, we perform extensive numerical
experiments on several high-dimensional parameterized elliptic PDE models to
demonstrate the superior recovery properties of the proposed approach.Comment: 23 pages, 4 figure
Disparity and Optical Flow Partitioning Using Extended Potts Priors
This paper addresses the problems of disparity and optical flow partitioning
based on the brightness invariance assumption. We investigate new variational
approaches to these problems with Potts priors and possibly box constraints.
For the optical flow partitioning, our model includes vector-valued data and an
adapted Potts regularizer. Using the notation of asymptotically level stable
functions we prove the existence of global minimizers of our functionals. We
propose a modified alternating direction method of minimizers. This iterative
algorithm requires the computation of global minimizers of classical univariate
Potts problems which can be done efficiently by dynamic programming. We prove
that the algorithm converges both for the constrained and unconstrained
problems. Numerical examples demonstrate the very good performance of our
partitioning method
On the monotone and primal-dual active set schemes for -type problems,
Nonsmooth nonconvex optimization problems involving the quasi-norm,
, of a linear map are considered. A monotonically convergent
scheme for a regularized version of the original problem is developed and
necessary optimality conditions for the original problem in the form of a
complementary system amenable for computation are given. Then an algorithm for
solving the above mentioned necessary optimality conditions is proposed. It is
based on a combination of the monotone scheme and a primal-dual active set
strategy. The performance of the two algorithms is studied by means of a series
of numerical tests in different cases, including optimal control problems,
fracture mechanics and microscopy image reconstruction
Simultaneous reconstruction of evolutionary history and epidemiological dynamics from viral sequences with the birth-death SIR model
The evolution of RNA viruses such as HIV, Hepatitis C and Influenza virus
occurs so rapidly that the viruses' genomes contain information on past
ecological dynamics. Hence, we develop a phylodynamic method that enables the
joint estimation of epidemiological parameters and phylogenetic history. Based
on a compartmental susceptible-infected-removed (SIR) model, this method
provides separate information on incidence and prevalence of infections.
Detailed information on the interaction of host population dynamics and
evolutionary history can inform decisions on how to contain or entirely avoid
disease outbreaks.
We apply our Birth-Death SIR method (BDSIR) to two viral data sets. First,
five human immunodeficiency virus type 1 clusters sampled in the United Kingdom
between 1999 and 2003 are analyzed. The estimated basic reproduction ratios
range from 1.9 to 3.2 among the clusters. All clusters show a decline in the
growth rate of the local epidemic in the middle or end of the 90's.
The analysis of a hepatitis C virus (HCV) genotype 2c data set shows that the
local epidemic in the C\'ordoban city Cruz del Eje originated around 1906
(median), coinciding with an immigration wave from Europe to central Argentina
that dates from 1880--1920. The estimated time of epidemic peak is around 1970.Comment: Journal link:
http://rsif.royalsocietypublishing.org/content/11/94/20131106.ful
Efficient Reconstruction of Piecewise Constant Images Using Nonsmooth Nonconvex Minimization
We consider the restoration of piecewise constant images where the number of the regions and their
values are not fixed in advance, with a good difference of piecewise constant values between neighboring regions, from noisy data obtained at the output of a linear operator (e.g., a blurring kernel or
a Radon transform). Thus we also address the generic problem of unsupervised segmentation in the
context of linear inverse problems. The segmentation and the restoration tasks are solved jointly
by minimizing an objective function (an energy) composed of a quadratic data-fidelity term and a
nonsmooth nonconvex regularization term. The pertinence of such an energy is ensured by the analytical properties of its minimizers. However, its practical interest used to be limited by the difficulty
of the computational stage which requires a nonsmooth nonconvex minimization. Indeed, the existing
methods are unsatisfactory since they (implicitly or explicitly) involve a smooth approximation
of the regularization term and often get stuck in shallow local minima. The goal of this paper is to
design a method that efficiently handles the nonsmooth nonconvex minimization. More precisely,
we propose a continuation method where one tracks the minimizers along a sequence of approximate
nonsmooth energies {Jε}, the first of which being strictly convex and the last one the original energy
to minimize. Knowing the importance of the nonsmoothness of the regularization term for the segmentation task, each Jε is nonsmooth and is expressed as the sum of an l1 regularization term and
a smooth nonconvex function. Furthermore, the local minimization of each Jε is reformulated as the
minimization of a smooth function subject to a set of linear constraints. The latter problem is solved
by the modified primal-dual interior point method, which guarantees the descent direction at each
step. Experimental results are presented and show the effectiveness and the efficiency of the proposed
method. Comparison with simulated annealing methods further shows the advantage of our method.published_or_final_versio
Analog hardware for detecting discontinuities in early vision
The detection of discontinuities in motion, intensity, color, and depth is a well-studied but difficult problem in computer vision [6]. We discuss the first hardware circuit that explicitly implements either analog or binary line processes in a deterministic fashion. Specifically, we show that the processes of smoothing (using a first-order or membrane type of stabilizer) and of segmentation can be implemented by a single, two-terminal nonlinear voltage-controlled resistor, the “resistive fuse”; and we derive its current-voltage relationship from a number of deterministic approximations to the underlying stochastic Markov random fields algorthms. The concept that the quadratic variation functionals of early vision can be solved via linear resistive networks minimizing power dissipation [37] can be extended to non-convex variational functionals with analog or binary line processes being solved by nonlinear resistive networks minimizing the electrical co-content.
We have successfully designed, tested, and demonstrated an analog CMOS VLSI circuit that contains a 1D resistive network of fuses implementing piecewise smooth surface interpolation. We furthermore demonstrate the segmenting abilities of these analog and deterministic “line processes” by numerically simulating the nonlinear resistive network computing optical flow in the presence of motion discontinuities. Finally, we discuss various circuit implementations of the optical flow computation using these circuits
Continuous-time segmentation networks
Segmentation is a basic problem in computer vision. The tiny-tanh network, a continuous-time network that segments scenes based upon intensity, motion, or depth is introduced. The tiny- tanh algorithm maps naturally to analog circuitry since it was inspired by previous experiments with analog VLSI segmentation hardware. A convex Lyapunov energy is utilized so that the system does not get stuck in local minima. No annealing algorithms of any kind are necessary- -a sharp contrast to previous software/hardware solutions of this problem
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