Aggregation-based aggressive coarsening with polynomial smoothing

This paper develops an algebraic multigrid preconditioner for the graph Laplacian. The proposed approach uses aggressive coarsening based on the aggregation framework in the setup phase and a polynomial smoother with sufficiently large degree within a (nonlinear) Algebraic Multilevel Iteration as a preconditioner to the flexible Conjugate Gradient iteration in the solve phase. We show that by combining these techniques it is possible to design a simple and scalable algorithm. Results of the algorithm applied to graph Laplacian systems arising from the standard linear finite element discretization of the scalar Poisson problem are reported

A rational deferred correction approach to parabolic optimal control problems

The accurate and efficient solution of time-dependent PDE-constrained optimization problems is a challenging task, in large part due to the very high dimension of the matrix systems that need to be solved. We devise a new deferred correction method for coupled systems of time-dependent PDEs, allowing one to iteratively improve the accuracy of low-order time stepping schemes. We consider two variants of our method, a splitting and a coupling version, and analyze their convergence properties. We then test our approach on a number of PDE-constrained optimization problems. We obtain solution accuracies far superior to that achieved when solving a single discretized problem, in particular in cases where the accuracy is limited by the time discretization. Our approach allows for the direct reuse of existing solvers for the resulting matrix systems, as well as state-of-the-art preconditioning strategies

Fast interior point solution of quadratic programming problems arising from PDE-constrained optimization

Interior point methods provide an attractive class of approaches for solving linear, quadratic and nonlinear programming problems, due to their excellent efficiency and wide applicability. In this paper, we consider PDE-constrained optimization problems with bound constraints on the state and control variables, and their representation on the discrete level as quadratic programming problems. To tackle complex problems and achieve high accuracy in the solution, one is required to solve matrix systems of huge scale resulting from Newton iteration, and hence fast and robust methods for these systems are required. We present preconditioned iterative techniques for solving a number of these problems using Krylov subspace methods, considering in what circumstances one may predict rapid convergence of the solvers in theory, as well as the solutions observed from practical computations

Embedded, micro-interdigitated flow fields in high areal-loading intercalation electrodes towards seawater desalination and beyond

Faradaic deionization (FDI) is a promising technology for energy-efficient water desalination using porous electrodes containing redox-active materials. Herein, we demonstrate for the first time the capability of a symmetric FDI flow cell to produce freshwater (<17.1 mM NaCl) from concentrated brackish water (118mM), to produce effluent near freshwater salinity (19.1 mM) from influent with seawater-level salinity (496 mM), and to reduce the salinity of hypersaline brine from 781 mM to 227 mM. These remarkable salt-removal levels were enabled by using flow-through electrodes with high areal-loading of nickel hexacyanoferrate (NiHCF) Prussian Blue analogue intercalation material. The pumping energy consumption due to flow-through electrodes was mitigated by embedding an interdigitated array of <100 $\mu$m wide channels in the electrodes using laser micromachining. The micron-scale dimensions of the resulting embedded, micro-interdigitated flow fields (e$\mu$-IDFFs) facilitate flow-through electrodes with high apparent permeability while minimizing active-material loss. Our modeling shows that these e$\mu$-IDFFs are more suitable for our intercalation electrodes because they have >100X lower permeability compared to common redox-flow battery electrodes, for which millimetric flow-channel widths were used exclusively in the past. Total desalination thermodynamic energy efficiency (TEE) was improved by more than ten-fold relative to unpatterned electrodes: 40.0% TEE for brackish water, 11.7% TEE for hypersaline brine, and 7.4% TEE for seawater-salinity feeds. Water transport between diluate and brine streams and charge efficiency losses resulting from (electro)chemical effects are implicated as limiting energy efficiency and water recovery, motivating their investigation for enhancing future FDI performance.Comment: 70 pages, 23 figures. Energy Environ. Sci. (2023

On Block Triangular Preconditioners for the Interior Point Solution of PDE-Constrained Optimization Problems

We consider the numerical solution of saddle point systems of equations resulting from the discretization of PDE-constrained optimization problems, with additional bound constraints on the state and control variables, using an interior point method. In particular, we derive a Bramble-Pasciak Conjugate Gradient method and a tailored block triangular preconditioner which may be applied within it. Crucial to the usage of the preconditioner are carefully chosen approximations of the (1,1)-block and Schur complement of the saddle point system. To apply the inverse of the Schur complement approximation, which is computationally the most expensive part of the preconditioner, one may then utilize methods such as multigrid or domain decomposition to handle individual sub-blocks of the matrix system