Recent research has used deep learning to develop partial differential equation (PDE) models in science and engineering. The functional form of the PDE is determined by a neural network, network parameters are calibrated available data. Calibration embedded can be performed optimizing over PDE. Motivated these applications, we rigorously study optimization class linear elliptic PDEs with terms....

In this note we present a multigrid preconditioning method for solving quadratic optimization problems constrained by fractional diffusion equation. Multigrid methods within the all-at-once approach to solve first order optimality Karush–Kuhn–Tucker (KKT) systems are widely popular, but their development have relied on underlying being sparse. On other hand, most discretizations, matrix represe...