87 research outputs found
A Unified Approach for Uzawa Algorithms
International audienceWe present a unified approach in analyzing Uzawa iterative algorithms for saddle point problems. We study the classical Uzawa method, the augmented Lagrangian method, and two versions of inexact Uzawa algorithms. The target application is the Stokes system, but other saddle point systems, e.g., arising from mortar methods or Lagrange multipliers methods, can benefit from our study. We prove convergence of Uzawa algorithms and find optimal rates of convergence in an abstract setting on finite-or infinite-dimensional Hilbert spaces. The results can be used to design multilevel or adaptive algorithms for solving saddle point problems. The discrete spaces do not have to satisfy the LBB stability condition
Transformed Primal-Dual Methods with Variable-Preconditioners
This paper introduces a novel Transformed Primal-Dual with
variable-metric/preconditioner (TPDv) algorithm, designed to efficiently solve
affine constrained optimization problems common in nonlinear partial
differential equations (PDEs). Diverging from traditional methods, TPDv
iteratively updates time-evolving preconditioning operators, enhancing
adaptability. The algorithm is derived and analyzed, demonstrating global
linear convergence rates under mild assumptions. Numerical experiments on
challenging nonlinear PDEs, including the Darcy-Forchheimer model and a
nonlinear electromagnetic problem, showcase the algorithm's superiority over
existing methods in terms of iteration numbers and computational efficiency.
The paper concludes with a comprehensive convergence analysis
Towards Fast-Convergence, Low-Delay and Low-Complexity Network Optimization
Distributed network optimization has been studied for well over a decade.
However, we still do not have a good idea of how to design schemes that can
simultaneously provide good performance across the dimensions of utility
optimality, convergence speed, and delay. To address these challenges, in this
paper, we propose a new algorithmic framework with all these metrics
approaching optimality. The salient features of our new algorithm are
three-fold: (i) fast convergence: it converges with only
iterations that is the fastest speed among all the existing algorithms; (ii)
low delay: it guarantees optimal utility with finite queue length; (iii) simple
implementation: the control variables of this algorithm are based on virtual
queues that do not require maintaining per-flow information. The new technique
builds on a kind of inexact Uzawa method in the Alternating Directional Method
of Multiplier, and provides a new theoretical path to prove global and linear
convergence rate of such a method without requiring the full rank assumption of
the constraint matrix
PRECONDITIONERS AND TENSOR PRODUCT SOLVERS FOR OPTIMAL CONTROL PROBLEMS FROM CHEMOTAXIS
In this paper, we consider the fast numerical solution of an optimal control
formulation of the Keller--Segel model for bacterial chemotaxis. Upon
discretization, this problem requires the solution of huge-scale saddle point
systems to guarantee accurate solutions. We consider the derivation of
effective preconditioners for these matrix systems, which may be embedded
within suitable iterative methods to accelerate their convergence. We also
construct low-rank tensor-train techniques which enable us to present efficient
and feasible algorithms for problems that are finely discretized in the space
and time variables. Numerical results demonstrate that the number of
preconditioned GMRES iterations depends mildly on the model parameters.
Moreover, the low-rank solver makes the computing time and memory costs
sublinear in the original problem size.Comment: 23 page
Local Fourier analysis for saddle-point problems
The numerical solution of saddle-point problems has attracted considerable interest in
recent years, due to their indefiniteness and often poor spectral properties that make
efficient solution difficult. While much research already exists, developing efficient
algorithms remains challenging. Researchers have applied finite-difference, finite element,
and finite-volume approaches successfully to discretize saddle-point problems,
and block preconditioners and monolithic multigrid methods have been proposed for
the resulting systems. However, there is still much to understand.
Magnetohydrodynamics (MHD) models the flow of a charged fluid, or plasma, in
the presence of electromagnetic fields. Often, the discretization and linearization of
MHD leads to a saddle-point system. We present vector-potential formulations of
MHD and a theoretical analysis of the existence and uniqueness of solutions of both
the continuum two-dimensional resistive MHD model and its discretization.
Local Fourier analysis (LFA) is a commonly used tool for the analysis of multigrid
and other multilevel algorithms. We first adapt LFA to analyse the properties of
multigrid methods for both finite-difference and finite-element discretizations of the
Stokes equations, leading to saddle-point systems. Monolithic multigrid methods,
based on distributive, Braess-Sarazin, and Uzawa relaxation are discussed. From
this LFA, optimal parameters are proposed for these multigrid solvers. Numerical
experiments are presented to validate our theoretical results. A modified two-level
LFA is proposed for high-order finite-element methods for the Lapalce problem, curing
the failure of classical LFA smoothing analysis in this setting and providing a reliable
way to estimate actual multigrid performance. Finally, we extend LFA to analyze the
balancing domain decomposition by constraints (BDDC) algorithm, using a new choice
of basis for the space of Fourier harmonics that greatly simplifies the application of
LFA. Improved performance is obtained for some two- and three-level variants
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