On a Conjugate Gradient/Newton/Penalty Method for the Solution of Obstacle Problems. Application to the Solution of an Eikonal System with Dirichlet Boundary Conditions

On the numerical simulation of Bingham visco-plastic flow: Old and new results

“…a stronger convergence result than the one given by (23) in Section 5. We refer to [12] concerning the practical implementation of the above approach (which is no more complicated to implement than the one based on (17)).…”

“…[22][23][24] to the solution of time dependent parabolic variational inequalities of the obstacle type. Let ε be a small positive parameter; we approximate the dual problem (81) by the following one:…”

On the Numerical Simulation of Viscoplastic Fluid Flow

“…The dual problem (13.30) is essentially (see Section 13.3) an obstacle problem associated with the point-wise constraint (13.51) An alternative to the projection methods discussed so far is provided by a variant of the penalty-Newton-conjugate gradient method applied in Glowinski, Kuznetsov and Pan [2003], Dacorogna, Glowinski, Kuznetsov and Pan [2004], Glowinski, Shiau, Kuo and Nasser [2006] to the solution of time-dependent variational inequalities of the obstacle type. Let ε be a small positive parameter; we approximate the dual problem (13.30) by…”

A Penalization-Regularization-Operator Splitting Method for Eikonal Based Traveltime Tomography

“…So the question is, does the penalization-regularization approach proposed here simulate first-arrival traveltimes to fit the given data? In [5] and [8] an analogous approach for solving Dirichlet problems for eikonal equations is shown to yield a (kind of) viscosity solution in a Stokes sense (at least in two dimensions). Moreover, our current numerical examples strongly indicate that the new approach also simulates first-arrival traveltimes to fit the given data, so the corresponding solution for the eikonal equation may be a viscosity solution in the sense of Crandall and Lions [4].…”

“…Following [5] and [8], we advocate for the solution of the constrained minimization problem (2.6) a methodology combining penalization, regularization, and operator-splitting. Let us introduce first = { 1 , 2 , 3 }, with i 's being small positive numbers.…”