J. W. Craggs

Bio: J. W. Craggs is an academic researcher. The author has contributed to research in topics: Mathematical sciences. The author has an hindex of 1, co-authored 1 publications receiving 4285 citations.

A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model

Abstract: We propose a new multiphase level set framework for image segmentation using the Mumford and Shah model, for piecewise constant and piecewise smooth optimal approximations. The proposed method is also a generalization of an active contour model without edges based 2-phase segmentation, developed by the authors earlier in T. Chan and L. Vese (1999. In Scale-Space'99, M. Nilsen et al. (Eds.), LNCS, vol. 1682, pp. 141–151) and T. Chan and L. Vese (2001. IEEE-IP, 10(2):266–277). The multiphase level set formulation is new and of interest on its own: by construction, it automatically avoids the problems of vacuum and overlaps it needs only log n level set functions for n phases in the piecewise constant cases it can represent boundaries with complex topologies, including triple junctionss in the piecewise smooth case, only two level set functions formally suffice to represent any partition, based on The Four-Color Theorem. Finally, we validate the proposed models by numerical results for signal and image denoising and segmentation, implemented using the Osher and Sethian level set method.

A Posteriori Error Estimation in Finite Element Analysis

Abstract: This monograph presents a summary account of the subject of a posteriori error estimation for finite element approximations of problems in mechanics. The study primarily focuses on methods for linear elliptic boundary value problems. However, error estimation for unsymmetrical systems, nonlinear problems, including the Navier-Stokes equations, and indefinite problems, such as represented by the Stokes problem are included. The main thrust is to obtain error estimators for the error measured in the energy norm, but techniques for other norms are also discussed.

Nonlinear dispersive equations : local and global analysis

Abstract: Ordinary differential equations Constant coefficient linear dispersive equations Semilinear dispersive equations The Korteweg de Vries equation Energy-critical semilinear dispersive equations Wave maps Tools from harmonic analysis Construction of ground states Bibliography.

The Variational Approach to Fracture

Abstract: 1 Introduction 2 Going variational 2.1 Griffith's theory 2.2 The 1-homogeneous case - A variational equivalence 2.3 Smoothness - The soft belly of Griffith's formulation 2.4 The non 1-homogeneous case - A discrete variational evolution 2.5 Functional framework - A weak variational evolution 2.6 Cohesiveness and the variational evolution 3 Stationarity versus local or global minimality - A comparison 3.1 1d traction 3.1.1 The Griffith case - Soft device 3.1.2 The Griffith case - Hard device 3.1.3 Cohesive case - Soft device 3.1.4 Cohesive case - Hard device 3.2 A tearing experiment 4 Initiation 4.1 Initiation - The Griffith case 4.1.1 Initiation - The Griffith case - Global minimality 4.1.2 Initiation - The Griffith case - Local minimality 4.2 Initiation - The cohesive case 4.2.1 Initiation - The cohesive 1d case - Stationarity 4.2.2 Initiation - The cohesive 3d case - Stationarity 4.2.3 Initiation - The cohesive case - Global minimality 5 Irreversibility 5.1 Irreversibility - The Griffith case - Well-posedness of the variational evolution 5.1.1 Irreversibility - The Griffith case - Discrete evolution 5.1.2 Irreversibility - The Griffith case - Global minimality in the limit 5.1.3 Irreversibility - The Griffith case - Energy balance in the limit 5.1.4 Irreversibility - The Griffith case - The time-continuous evolution 5.2 Irreversibility - The cohesive case 6 Path 7 Griffith vs. Barenblatt 8 Numerics and Griffith 8.1 Numerical approximation of the energy 8.1.1 The first time step 8.1.2 Quasi-static evolution 8.2 Minimization algorithm 8.2.1 The alternate minimization algorithm 8.2.2 The backtracking algorithm 8.3 Numerical experiments 8.3.1 The 1D traction (hard device) 8.3.2 The Tearing experiment 8.3.3 Revisiting the 2D traction experiment on a fiber reinforced matrix 9 Fatigue 9.1 Peeling Evolution 9.2 The limitfatigue law when d tends to 0 9.3 A variational formulation for fatigue 9.3.1 Peeling revisited 9.3.2 Generalization Appendix Glossary References.