The Moreau-Yosida Regularization: Difference between revisions
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==Motivation== | |||
(to be filled in) | |||
==Definitions== | ==Definitions== | ||
Let <math>(X,d)</math> be a metric space. A function <math>g : X \to (-\infty,+\infty]</math> is said to be '''proper''' if it is not identically equal to <math>+\infty</math>, that is, if there exists <math>x \in X</math> such that <math>g(x) < +\infty</math>. | |||
Given a function <math>g : X \to (-\infty,+\infty]</math>, its '''Moreau-Yosida regularization''' <math>g_k : X \to (-\infty,+\infty]</math> is given by | |||
:<math>g_k(x) := \inf\limits_{y \in X} \left[ g(y) + k d(x,y) \right],</math> | |||
where <math>k \geq 0</math>. | |||
==Examples== | ==Examples== | ||
(to be filled in, hopefully with pictures!) | |||
==References== | |||
Possible list of references, will fix accordingly | |||
== | Bauschke-Combette Ch 12.<ref name="BC" />; Santambrogio (6)<ref name="S" />; Ambrosio-Gigli-Savare (59-61)<ref name="AGS" /> | ||
<references> | <references> | ||
<ref name="BC">Bauschke, Heinz H. and Patrick L. Combettes. ''Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd Ed.'' Ch. 12. Springer, 2017.</ref> | |||
<ref name="S">Santambrogio, Filippo. ''Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling'' Ch. 1.1. Birkhäuser, 2015.</ref> | |||
<ref name="AGS">Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. ''Gradient Flows in Metric Spaces and in the Space of Probability Measures.'' Ch. 3.1. Birkhäuser, 2005.</ref> | |||
</references> | </references> | ||
Revision as of 20:40, 21 January 2022
(to be filled in)
Motivation
(to be filled in)
Definitions
Let be a metric space. A function is said to be proper if it is not identically equal to , that is, if there exists such that .
![{\displaystyle g:X\to (-\infty ,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb30feface8bba531b0a2ad5b5dffc066ed4acbd)
Given a function , its Moreau-Yosida regularization is given by
![{\displaystyle g_{k}:X\to (-\infty ,+\infty ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b718f4b10f7cd1014c845ca702eafc9eab1aa223)
![{\displaystyle g_{k}(x):=\inf \limits _{y\in X}\left[g(y)+kd(x,y)\right],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf9bc0ecf8391b5098d36534b47ef4200cbaaf5b)
where .
Examples
(to be filled in, hopefully with pictures!)
References
Possible list of references, will fix accordingly
Bauschke-Combette Ch 12.[1]; Santambrogio (6)[2]; Ambrosio-Gigli-Savare (59-61)[3]
- ↑ Bauschke, Heinz H. and Patrick L. Combettes. Convex Analysis and Monotone Operator Theory in Hilbert Spaces, 2nd Ed. Ch. 12. Springer, 2017.
- ↑ Santambrogio, Filippo. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling Ch. 1.1. Birkhäuser, 2015.
- ↑ Ambrosio, Luigi, Nicola Gigli, and Giuseppe Savaré. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Ch. 3.1. Birkhäuser, 2005.