index.md

RegularizedLeastSquares.jl

Solvers for Linear Inverse Problems using Regularization Techniques

Introduction

RegularizedLeastSquares.jl is a Julia package for solving large linear systems using various types of algorithms. Ill-conditioned problems arise in many areas of practical interest. Regularisation techniques and nonlinear problem formulations are often used to solve these problems. This package provides implementations for a variety of solvers used in areas such as MPI and MRI. In particular, this package serves as the optimization backend of the Julia packages MPIReco.jl and MRIReco.jl.

The implemented methods range from the $l^2_2$-regularized CGNR method to more general optimizers such as the Alternating Direction of Multipliers Method (ADMM) or the Split-Bregman method.

For convenience, implementations of popular regularizers, such as $l_1$-regularization and TV regularization, are provided. On the other hand, hand-crafted regularizers can be used quite easily.

Depending on the problem, it becomes unfeasible to store the full system matrix at hand. For this purpose, RegularizedLeastSquares.jl allows for the use of matrix-free operators. Such operators can be realized using the interface provided by the package LinearOperators.jl. Other interfaces can be used as well, as long as the product *(A,x) and the adjoint adjoint(A) are provided. A number of common matrix-free operators are provided by the package LinearOperatorColection.jl.

Features

• Variety of optimization algorithms optimized for least squares problems
• Support for matrix-free operators
• GPU support

Usage

• See Getting Started for an introduction to using the package

See also

Packages:

• ProximalAlgorithms.jl
• ProximalOperators.jl
• Krylov.jl
• RegularizedOptimization.jl

Organizations:

• JuliaNLSolvers
• JuliaSmoothOptimizers
• JuliaFirstOrder