Speeding up SOR Solvers for Constraint-based GUIs with a Warm-Start Strategy

Many computer programs have graphical user interfaces (GUIs), which need good layout to make efficient use of the available screen real estate. Most GUIs do not have a fixed layout, but are resizable and able to adapt themselves. Constraints are a powerful tool for specifying adaptable GUI layouts: they are used to specify a layout in a general form, and a constraint solver is used to find a satisfying concrete layout, e.g.\ for a specific GUI size. The constraint solver has to calculate a new layout every time a GUI is resized or changed, so it needs to be efficient to ensure a good user experience. One approach for constraint solvers is based on the Gauss-Seidel algorithm and successive over-relaxation (SOR). Our observation is that a solution after resizing or changing is similar in structure to a previous solution. Thus, our hypothesis is that we can increase the computational performance of an SOR-based constraint solver if we reuse the solution of a previous layout to warm-start the solving of a new layout. In this paper we report on experiments to test this hypothesis experimentally for three common use cases: big-step resizing, small-step resizing and constraint change. In our experiments, we measured the solving time for randomly generated GUI layout specifications of various sizes. For all three cases we found that the performance is improved if an existing solution is used as a starting solution for a new layout

The Graphical Lasso: New Insights and Alternatives

The graphical lasso \citep{FHT2007a} is an algorithm for learning the structure in an undirected Gaussian graphical model, using $\ell_1$ regularization to control the number of zeros in the precision matrix {\B\Theta}={\B\Sigma}^{-1} \citep{BGA2008,yuan_lin_07}. The {\texttt R} package \GL\ \citep{FHT2007a} is popular, fast, and allows one to efficiently build a path of models for different values of the tuning parameter. Convergence of \GL\ can be tricky; the converged precision matrix might not be the inverse of the estimated covariance, and occasionally it fails to converge with warm starts. In this paper we explain this behavior, and propose new algorithms that appear to outperform \GL. By studying the "normal equations" we see that, \GL\ is solving the {\em dual} of the graphical lasso penalized likelihood, by block coordinate ascent; a result which can also be found in \cite{BGA2008}. In this dual, the target of estimation is \B\Sigma, the covariance matrix, rather than the precision matrix \B\Theta. We propose similar primal algorithms \PGL\ and \DPGL, that also operate by block-coordinate descent, where \B\Theta is the optimization target. We study all of these algorithms, and in particular different approaches to solving their coordinate sub-problems. We conclude that \DPGL\ is superior from several points of view.Comment: This is a revised version of our previous manuscript with the same name ArXiv id: http://arxiv.org/abs/1111.547

New Formation Models for the Kepler-36 System

Formation of the planets in the Kepler-36 system is modeled by detailed numerical simulations according to the core-nucleated accretion scenario. The standard model is updated to include the dissolution of accreting rocky planetesimals in the gaseous envelope of the planet, leading to substantial enrichment of the envelope mass in heavy elements and a non-uniform composition with depth. For Kepler-36 c, models involving in situ formation and models involving orbital migration are considered. The results are compared with standard formation models. The calculations include the formation (accretion) phase, as well as the subsequent cooling phase, up to the age of Kepler-36 (7 Gyr). During the latter phase, mass loss induced by stellar XUV radiation is included. In all cases, the results fit the measured mass, 7.84 M$_\oplus$, and radius, 3.68 R$_\oplus$, of Kepler-36 c. Two parameters are varied to obtain these fits: the disk solid surface density at the formation location, and the "efficiency" factor in the XUV mass loss rate. The updated models are hotter and therefore less dense in the silicate portion of the planet and in the overlying layers of H/He, as compared with standard models. The lower densities mean that only about half as much H/He is needed to be accreted to fit the present-day mass and radius constraints. For Kepler-36 b, an updated in situ calculation shows that the entire H/He envelope is lost, early in the cooling phase, in agreement with observation.Comment: 21 pages, 18 figures, 1 table. Accepted for publication in The Astrophysical Journa