Getting Started

In this example we will go through a simple random inverse problem to get familiar with RegularizedLeastSquares.jl.

Installation

To install RegularizedLeastSquares.jl, you can use the Julia package manager. Open a Julia REPL and run the following command:

using Pkg
Pkg.add("RegularizedLeastSquares")

This will download and install the RegularizedLeastSquares.jl package and its dependencies. To install a different version, please consult the Pkg documentation.

Once the installation is complete, you can import the package with the using keyword:

using RegularizedLeastSquares

RegularizedLeastSquares aims to solve inverse problems of the form:

\[\begin{equation} \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \mathbf{R(x)} . \end{equation}\]

where $\mathbf{A}$ is a linear operator, $\mathbf{b}$ is the measurement vector, and $\mathbf{R(x)}$ is an (optional) regularization term. The goal is to retrieve an approximation of the unknown vector $\mathbf{x}$. In this first exampel we will just work with simple random arrays. For more advanced examples, please refer to the examples.

A = rand(32, 16)
x = rand(16)
b = A*x;

To approximate x from b, we can use the Conjugate Gradient Normal Residual (CGNR) algorithm. We first build the corresponding solver:

solver = createLinearSolver(CGNR, A; iterations=32);

and apply it to our measurement vector

x_approx = solve!(solver, b)
isapprox(x, x_approx, rtol = 0.001)

true

Usually the inverse problems are ill-posed and require regularization. RegularizedLeastSquares.jl provides a variety of regularization terms. The CGNR algorithm can solve optimzation problems of the form:

\[\begin{equation} \underset{\mathbf{x}}{argmin} \frac{1}{2}\vert\vert \mathbf{A}\mathbf{x}-\mathbf{b} \vert\vert_2^2 + \lambda\vert\vert\mathbf{x}\vert\vert^2_2 . \end{equation}\]

The corresponding solver can be built with the $l^2_2$-regularization term:

solver = createLinearSolver(CGNR, A; reg = L2Regularization(0.0001), iterations=32);
x_approx = solve!(solver, b)
isapprox(x, x_approx, rtol = 0.001)

true

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