Abstract
Various iterative methods are available for the approximate solution of non-smooth minimization problems. For a popular non-smooth minimization problem arising in image processing, we discuss the suitable application of three prototypical methods and their stability. The methods are compared experimentally with a focus on choice of stopping criteria, influence of rough initial data, step sizes as well as mesh sizes. An overview of existing algorithms is given.
Funding source: Deutsche Forschungsgemeinschaft
Award Identifier / Grant number: SPP 1748: BA 2268/2-1
Funding statement: The authors acknowledge financial support by the Deutsche Forschungsgemeinschaft for the project “Finite Element Approximation of Functions of Bounded Variation and Application to Model of Damage, Fracture and Plasticity” (BA 2268/2-1) via the priority program “Reliable Simulation Techniques in Solid Mechanics, Development of Non-Standard Discretization Methods, Mechanical and Mathematical Analysis” (SPP 1748), and by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.
A Roots of Quartic Equations
Having (3.2) in mind we consider the problem of finding the roots of a quartic equation
with
with
Our goal is to produce a perfect square on the right-hand side. The following procedure is due to Lodovico Ferrari. We introduce a new variable z, which is to be specified later, within the square on the left-hand side and obtain
Now the right-hand side is a perfect square with respect to y if and only if the associated discriminant vanishes, i.e., if and only if
The cubic polynomial in (A.5) is called cubic resolvent. We see that with z being any root of the cubic resolvent the term in (A.3) simplifies to a perfect square. Indeed, using (A.4), we have
We can take z to be, for instance, a real root of (A.5) and obtain
so we have two quadratic algebraic equations in y and can compute the roots of (A.2). We obtain the roots
B Roots of Cubic Equations
In order to obtain the roots of (A.5) we briefly discuss how to compute the roots of a cubic algebraic equation. Given an equation of the form
we set
with
Comparing coefficients yields
According to Vieta’s formulas,
Hence we have
The cube roots u and v have to be chosen such that
The roots
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