Convergence of adaptive stochastic Galerkin FEM

We propose and analyze novel adaptive algorithms for the numerical solution of elliptic partial differential equations with parametric uncertainty. Four different marking strategies are employed for refinement of stochastic Galerkin finite element approximations. The algorithms are driven by the energy error reduction estimates derived from two-level a posteriori error indicators for spatial approximations and hierarchical a posteriori error indicators for parametric approximations. The focus of this work is on the mathematical foundation of the adaptive algorithms in the sense of rigorous convergence analysis. In particular, we prove that the proposed algorithms drive the underlying energy error estimates to zero

Numerical simulations of mixed states quantum computation

We describe quantum-octave package of functions useful for simulations of quantum algorithms and protocols. Presented package allows to perform simulations with mixed states. We present numerical implementation of important quantum mechanical operations - partial trace and partial transpose. Those operations are used as building blocks of algorithms for analysis of entanglement and quantum error correction codes. Simulation of Shor's algorithm is presented as an example of package capabilities.Comment: 6 pages, 4 figures, presented at Foundations of Quantum Information, 16th-19th April 2004, Camerino, Ital

New Laplace and Helmholtz solvers

New numerical algorithms based on rational functions are introduced that can solve certain Laplace and Helmholtz problems on two-dimensional domains with corners faster and more accurately than the standard methods of finite elements and integral equations. The new algorithms point to a reconsideration of the assumptions underlying existing numerical analysis for partial differential equations

A generic framework for the analysis and specialization of logic programs

The relationship between abstract interpretation and partial deduction has received considerable attention and (partial) integrations have been proposed starting from both the partial deduction and abstract interpretation perspectives. In this work we present what we argüe is the first fully described generic algorithm for efñcient and precise integration of abstract interpretation and partial deduction. Taking as starting point state-of-the-art algorithms for context-sensitive, polyvariant abstract interpretation and (abstract) partial deduction, we present an algorithm which combines the best of both worlds. Key ingredients include the accurate success propagation inherent to abstract interpretation and the powerful program transformations achievable by partial deduction. In our algorithm, the calis which appear in the analysis graph are not analyzed w.r.t. the original definition of the procedure but w.r.t. specialized definitions of these procedures. Such specialized definitions are obtained by applying both unfolding and abstract executability. Our framework is parametric w.r.t. different control strategies and abstract domains. Different combinations of such parameters correspond to existing algorithms for program analysis and specialization. Simultaneously, our approach opens the door to the efñcient computation of strictly more precise results than those achievable by each of the individual techniques. The algorithm is now one of the key components of the CiaoPP analysis and specialization system

Comparative Analysis and Evaluation of Image inpainting Algorithms

Image inpainting refers to the task of filling in the missing or damaged regions of an image in an undetectable manner. There are a large variety of image inpainting algorithms existing in the literature. They can broadly be grouped into two categories such as Partial Differential Equation (PDE) based algorithms and Exemplar based Texture synthesis algorithms. However no recent study has been undertaken for a comparative evaluation of these algorithms. In this paper, we are comparing two different types of image inpainting algorithms. The algorithms analyzed are Marcelo Bertalmio’s PDE based inpainting algorithm and Zhaolin Lu et al’s exemplar based Image inpainting algorithm.Both theoretical analysis and experiments have made to analyze the results of these image inpainting algorithms on the basis of both qualitative and quantitative way. Keywords:Image inpainting, Exemplar based, Texture synthesis, Partial Differential Equation (PDE)

Sticky Brownian Rounding and its Applications to Constraint Satisfaction Problems

Semidefinite programming is a powerful tool in the design and analysis of approximation algorithms for combinatorial optimization problems. In particular, the random hyperplane rounding method of Goemans and Williamson has been extensively studied for more than two decades, resulting in various extensions to the original technique and beautiful algorithms for a wide range of applications. Despite the fact that this approach yields tight approximation guarantees for some problems, e.g., Max-Cut, for many others, e.g., Max-SAT and Max-DiCut, the tight approximation ratio is still unknown. One of the main reasons for this is the fact that very few techniques for rounding semidefinite relaxations are known. In this work, we present a new general and simple method for rounding semi-definite programs, based on Brownian motion. Our approach is inspired by recent results in algorithmic discrepancy theory. We develop and present tools for analyzing our new rounding algorithms, utilizing mathematical machinery from the theory of Brownian motion, complex analysis, and partial differential equations. Focusing on constraint satisfaction problems, we apply our method to several classical problems, including Max-Cut, Max-2SAT, and MaxDiCut, and derive new algorithms that are competitive with the best known results. To illustrate the versatility and general applicability of our approach, we give new approximation algorithms for the Max-Cut problem with side constraints that crucially utilizes measure concentration results for the Sticky Brownian Motion, a feature missing from hyperplane rounding and its generalization