Practical aspects of primal-dual nonlinear rescaling method with dynamic scaling parameter update

Primal-dual nonlinear rescaling method with dynamic scaling parameter update (PDNRD) is an optimization method from a class of nonlinear rescaling techniques. Previous work does not discuss practical aspects of PDNRD method such as the explanation and the setting of the parameters. To complete this framework, the parameters were described. Moreover, PDNRD method was applied on two quadratic programming problems with quadratic constraints and recommendations about the setting of the parameters were made

Acceleration of the PDHGM on strongly convex subspaces

We propose several variants of the primal-dual method due to Chambolle and Pock. Without requiring full strong convexity of the objective functions, our methods are accelerated on subspaces with strong convexity. This yields mixed rates, $O(1/N^2)$ with respect to initialisation and $O(1/N)$ with respect to the dual sequence, and the residual part of the primal sequence. We demonstrate the efficacy of the proposed methods on image processing problems lacking strong convexity, such as total generalised variation denoising and total variation deblurring

Fast convergence of primal-dual dynamics and algorithms with time scaling for linear equality constrained convex optimization problems

We propose a primal-dual dynamic with time scaling for a linear equality constrained convex optimization problem, which consists of a second-order ODE for the primal variable and a first-order ODE for the dual variable. Without assuming strong convexity, we prove its fast convergence property and show that the obtained fast convergence property is preserved under a small perturbation. We also develop an inexact primal-dual algorithm derived by a time discretization, and derive the fast convergence property matching that of the underlying dynamic. Finally, we give numerical experiments to illustrate the validity of the proposed algorithm

On-the-fly adaptivity for nonlinear twoscale simulations using artificial neural networks and reduced order modeling

A multi-fidelity surrogate model for highly nonlinear multiscale problems is proposed. It is based on the introduction of two different surrogate models and an adaptive on-the-fly switching. The two concurrent surrogates are built incrementally starting from a moderate set of evaluations of the full order model. Therefore, a reduced order model (ROM) is generated. Using a hybrid ROM-preconditioned FE solver, additional effective stress-strain data is simulated while the number of samples is kept to a moderate level by using a dedicated and physics-guided sampling technique. Machine learning (ML) is subsequently used to build the second surrogate by means of artificial neural networks (ANN). Different ANN architectures are explored and the features used as inputs of the ANN are fine tuned in order to improve the overall quality of the ML model. Additional ANN surrogates for the stress errors are generated. Therefore, conservative design guidelines for error surrogates are presented by adapting the loss functions of the ANN training in pure regression or pure classification settings. The error surrogates can be used as quality indicators in order to adaptively select the appropriate -- i.e. efficient yet accurate -- surrogate. Two strategies for the on-the-fly switching are investigated and a practicable and robust algorithm is proposed that eliminates relevant technical difficulties attributed to model switching. The provided algorithms and ANN design guidelines can easily be adopted for different problem settings and, thereby, they enable generalization of the used machine learning techniques for a wide range of applications. The resulting hybrid surrogate is employed in challenging multilevel FE simulations for a three-phase composite with pseudo-plastic micro-constituents. Numerical examples highlight the performance of the proposed approach