5,393 research outputs found
Continuation of Nesterov's Smoothing for Regression with Structured Sparsity in High-Dimensional Neuroimaging
Predictive models can be used on high-dimensional brain images for diagnosis
of a clinical condition. Spatial regularization through structured sparsity
offers new perspectives in this context and reduces the risk of overfitting the
model while providing interpretable neuroimaging signatures by forcing the
solution to adhere to domain-specific constraints. Total Variation (TV)
enforces spatial smoothness of the solution while segmenting predictive regions
from the background. We consider the problem of minimizing the sum of a smooth
convex loss, a non-smooth convex penalty (whose proximal operator is known) and
a wide range of possible complex, non-smooth convex structured penalties such
as TV or overlapping group Lasso. Existing solvers are either limited in the
functions they can minimize or in their practical capacity to scale to
high-dimensional imaging data. Nesterov's smoothing technique can be used to
minimize a large number of non-smooth convex structured penalties but
reasonable precision requires a small smoothing parameter, which slows down the
convergence speed. To benefit from the versatility of Nesterov's smoothing
technique, we propose a first order continuation algorithm, CONESTA, which
automatically generates a sequence of decreasing smoothing parameters. The
generated sequence maintains the optimal convergence speed towards any globally
desired precision. Our main contributions are: To propose an expression of the
duality gap to probe the current distance to the global optimum in order to
adapt the smoothing parameter and the convergence speed. We provide a
convergence rate, which is an improvement over classical proximal gradient
smoothing methods. We demonstrate on both simulated and high-dimensional
structural neuroimaging data that CONESTA significantly outperforms many
state-of-the-art solvers in regard to convergence speed and precision.Comment: 11 pages, 6 figures, accepted in IEEE TMI, IEEE Transactions on
Medical Imaging 201
A robust all-at-once multigrid method for the Stokes control problem
In this paper we present an all-at-once multigrid method for a distributed Stokes control problem (velocity tracking problem). For solving such a problem, we use the fact that the solution is characterized by the optimality system (Karush-Kuhn-Tucker-system). The discretized optimality system is a large-scale linear system whose condition number depends on the grid size and on the choice of the regularization parameter forming a part of the problem. Recently, block-diagonal preconditioners have been proposed, which allow to solve the problem using a Krylov space method with convergence rates that are robust in both, the grid size and the regularization parameter or cost parameter. In the present paper, we develop an all-at-once multigrid method for a Stokes control problem and show robust convergence, more precisely, we show that the method converges with rates which are bounded away from one by a constant which is independent of the grid size and the choice of the regularization or cost parameter
A cell-based smoothed finite element method for kinematic limit analysis
This paper presents a new numerical procedure for kinematic limit analysis problems, which incorporates the cell-based smoothed finite element method with second-order cone programming. The application of a strain smoothing technique to the standard displacement finite element both rules out volumetric locking and also results in an efficient method that can provide accurate solutions with minimal computational effort. The non-smooth optimization problem is formulated as a problem of minimizing a sum of Euclidean norms, ensuring that the resulting optimization problem can be solved by an efficient second-order cone programming algorithm. Plane stress and plane strain problems governed by the von Mises criterion are considered, but extensions to problems with other yield criteria having a similar conic quadratic form or 3D problems can be envisaged
Variational Downscaling, Fusion and Assimilation of Hydrometeorological States via Regularized Estimation
Improved estimation of hydrometeorological states from down-sampled
observations and background model forecasts in a noisy environment, has been a
subject of growing research in the past decades. Here, we introduce a unified
framework that ties together the problems of downscaling, data fusion and data
assimilation as ill-posed inverse problems. This framework seeks solutions
beyond the classic least squares estimation paradigms by imposing proper
regularization, which are constraints consistent with the degree of smoothness
and probabilistic structure of the underlying state. We review relevant
regularization methods in derivative space and extend classic formulations of
the aforementioned problems with particular emphasis on hydrologic and
atmospheric applications. Informed by the statistical characteristics of the
state variable of interest, the central results of the paper suggest that
proper regularization can lead to a more accurate and stable recovery of the
true state and hence more skillful forecasts. In particular, using the Tikhonov
and Huber regularization in the derivative space, the promise of the proposed
framework is demonstrated in static downscaling and fusion of synthetic
multi-sensor precipitation data, while a data assimilation numerical experiment
is presented using the heat equation in a variational setting
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