A three-level finite difference scheme for solving a dual-phase-lagging heat transport equation in spherical coordinates

Heat transport through thin films or micro-objects is of vital importance in microtechnology applications. For instance, metal thin films are important components of microelectronic devices. The reduction of the device size to microscale has the advantage of enhancing the switching speed of the device. On the other hand, size reduction increases the rate of heat generation, which leads to a high thermal load on the microelectronic devices. Heat transfer at the microscale is also important for the thermal possessing of materials with a pulsed-laser. Examples in metal processing are laser micromachining, laser patterning, laser processing of diamond films from carbon ion implanted copper substrates, and laser surface hardening. In thermal processing of materials, microvoids may be found owing to thermal expansion. When such defects begin in the workpiece, their thermal energy in the neighborhood of the defects may be amplified, resulting in severe material damage and, consequently, total failure of the thermal processing. A detailed understanding of the way in which the local defects dissipate the thermal energy is then necessary not only to avoid the damage but also to improve the efficiency of the thermal processing. The heat transport equation at the microscale is different from the traditional heat diffusion equation because a second-order derivative of temperature with respect to time and a third-order mixed derivative of temperature with respect to space and time are introduced. In this study, we consider the heat transport equation in three-dimensional spherical coordinates and develop a three-level finite difference scheme for solving the heat transport equation in a microsphere. Stability of the scheme is proved in this dissertation. It is shown that the scheme is unconditionally stable. The scheme is then employed to investigate the temperature rise in a gold sphere subjected to a short-pulse laser. Numerical results are obtained for the cases that the laser irradiation is symmetric on the surface of the sphere, and the laser irradiation is from the top to a portion of the surface of the sphere

Efficient First Order Methods for Linear Composite Regularizers

A wide class of regularization problems in machine learning and statistics employ a regularization term which is obtained by composing a simple convex function \omega with a linear transformation. This setting includes Group Lasso methods, the Fused Lasso and other total variation methods, multi-task learning methods and many more. In this paper, we present a general approach for computing the proximity operator of this class of regularizers, under the assumption that the proximity operator of the function \omega is known in advance. Our approach builds on a recent line of research on optimal first order optimization methods and uses fixed point iterations for numerically computing the proximity operator. It is more general than current approaches and, as we show with numerical simulations, computationally more efficient than available first order methods which do not achieve the optimal rate. In particular, our method outperforms state of the art O(1/T) methods for overlapping Group Lasso and matches optimal O(1/T^2) methods for the Fused Lasso and tree structured Group Lasso.Comment: 19 pages, 8 figure

Split-Bregman iteration for framelet based image inpainting

AbstractImage inpainting plays a significant role in image processing and has many applications. Framelet based inpainting methods were introduced recently by Cai et al. (2007, 2009) [6,7,9] under an assumption that images can be sparsely approximated in the framelet domain. By analyzing these methods, we present a framelet based inpainting model in which the cost functional is the weighted ℓ1 norm of the framelet coefficients of the underlying image. The split-Bregman iteration is exploited to derive an iterative algorithm for the model. The resulting algorithm assimilates advantages while avoiding limitations of the framelet based inpainting approaches in Cai et al. (2007, 2009) [6,7,9]. The convergence analysis of the proposed algorithm is presented. Our numerical experiments show that the algorithm proposed here performs favorably

Integral equation models for image restoration: high accuracy methods and fast algorithms

Discrete models are consistently used as practical models for image restoration. They are piecewise constant approximations of true physical (continuous) models, and hence, inevitably impose bottleneck model errors. We propose to work directly with continuous models for image restoration aiming at suppressing the model errors caused by the discrete models. A systematic study is conducted in this paper for the continuous out-of-focus image models which can be formulated as an integral equation of the first kind. The resulting integral equation is regularized by the Lavrentiev method and the Tikhonov method. We develop fast multiscale algorithms having high accuracy to solve the regularized integral equations of the second kind. Numerical experiments show that the methods based on the continuous model perform much better than those based on discrete models, in terms of PSNR values and visual quality of the reconstructed images.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/85415/1/ip10_4_045006.pd