1 / 7

Introduction to Calculus of Variations

Computer Science Department. Technion-Israel Institute of Technology. Introduction to Calculus of Variations. Ron Kimmel www.cs.technion.ac.il/~ron. Geometric Image Processing Lab. Calculus of Variations.

yoland
Download Presentation

Introduction to Calculus of Variations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Computer Science Department • Technion-Israel Institute of Technology Introduction to Calculus of Variations Ron Kimmel www.cs.technion.ac.il/~ron • Geometric Image Processing Lab

  2. Calculus of Variations Generalization of Calculus that seeks to find the path, curve, surface, etc., for which a given Functional has a minimum or maximum. Goal: find extrema values of integrals of the form It has an extremum only if the Euler-Lagrange Differential Equation is satisfied,

  3. Calculus of Variations Example: Find the shape of the curve {x,y(x)} with shortest length: , given y(x0),y(x1) Solution: a differential equation that y(x) must satisfy,

  4. Extrema points in calculus Gradient descent process

  5. Calculus of variations Gradient descent process

  6. Euler Lagrange Equation Proof. for fixed u(x0),u(x1) : Thus the Euler Lagrange equation is

  7. Euler-Lagrange Conclusions • Gradient descent process Calculus Calculus of variations

More Related