L-curve is a
visualization method used in the field of
regularization
Regularization may refer to:
* Regularization (linguistics)
* Regularization (mathematics)
* Regularization (physics)
* Regularization (solid modeling)
* Regularization Law, an Israeli law intended to retroactively legalize settlements
See also ...
in
numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of ...
and
mathematical optimization
Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...
.
It represents a
logarithmic plot where the norm of a regularized solution is plotted against the norm of the corresponding residual norm. It is useful for picking an appropriate regularization parameter for the given data.
This method can be applied on methods of
regularization of least-square problems, such as
Tikhonov regularization
Ridge regression (also known as Tikhonov regularization, named for Andrey Tikhonov) is a method of estimating the coefficients of multiple- regression models in scenarios where the independent variables are highly correlated. It has been used in m ...
and the Truncated SVD,
and iterative methods of solving
ill-posed inverse problem
An inverse problem in science is the process of calculating from a set of observations the causal factors that produced them: for example, calculating an image in X-ray computed tomography, sound source reconstruction, source reconstruction in ac ...
s, such as the
Landweber algorithm,
Modified Richardson iteration
Modified Richardson iteration is an iterative method for solving a system of linear equations. Richardson iteration was proposed by Lewis Fry Richardson in his work dated 1910. It is similar to the Jacobi and Gauss–Seidel method.
We seek the so ...
and
Conjugate gradient method
In mathematics, the conjugate gradient method is an algorithm for the numerical solution of particular systems of linear equations, namely those whose matrix is positive-semidefinite. The conjugate gradient method is often implemented as an it ...
.
References
*Hanke, Martin. "Limitations of the L-curve method in ill-posed problems." BIT Numerical Mathematics 36.2 (1996): 287-301.
*Engl, Heinz W., and Wilhelm Grever. "Using the L--curve for determining optimal regularization parameters." Numerische Mathematik 69.1 (1994): 25-31.
{{Math-stub
Logarithmic scales of measurement