Differential equations | Stochastic processes
In stochastic analysis, a rough path is a generalization of the notion of smooth path allowing to construct a robust solution theory for controlled differential equations driven by classically irregular signals, for example a Wiener process. The theory was developed in the 1990s by Terry Lyons.Several accounts of the theory are available. Rough path theory is focused on capturing and making precise the interactions between highly oscillatory and non-linear systems. It builds upon the harmonic analysis of L.C. Young, the geometric algebra of K.T. Chen, the Lipschitz function theory of H. Whitney and core ideas of stochastic analysis. The concepts and the uniform estimates have widespread application in pure and applied Mathematics and beyond. It provides a toolbox to recover with relative ease many classical results in stochastic analysis (Wong-Zakai, Stroock-Varadhan support theorem, construction of stochastic flows, etc) without using specific probabilistic properties such as the martingale property or predictability. The theory also extends Itô's theory of SDEs far beyond the semimartingale setting. At the heart of the mathematics is the challenge of describing a smooth but potentially highly oscillatory and multidimensional path effectively so as to accurately predict its effect on a nonlinear dynamical system . The Signature is a homomorphism from the monoid of paths (under concatenation) into the grouplike elements of the free tensor algebra. It provides a graduated summary of the path . This noncommutative transform is faithful for paths up to appropriate null modifications. These graduated summaries or features of a path are at the heart of the definition of a rough path; locally they remove the need to look at the fine structure of the path. Taylor's theorem explains how any smooth function can, locally, be expressed as a linear combination of certain special functions (monomials based at that point). Coordinate iterated integrals (terms of the signature) form a more subtle algebra of features that can describe a stream or path in an analogous way; they allow a definition of rough path and form a natural linear "basis" for continuous functions on paths. Martin Hairer used rough paths to construct a robust solution theory for the KPZ equation. He then proposed a generalization known as the theory of regularity structures for which he was awarded a Fields medal in 2014. (Wikipedia).
This came as a surprise. Although it looks like an example with smooth time-stepping, it is not. It is with original, simple time-stepping. I'm not exactly sure what this means. Maybe my smooth time-stepping method is superfluous.
From playlist SmoothLife
Felix Otto: Singular SPDE with rough coefficients
Abstract: We are interested in parabolic differential equations (∂t−a∂2x)u=f with a very irregular forcing f and only mildly regular coefficients a. This is motivated by stochastic differential equations, where f is random, and quasilinear equations, where a is a (nonlinear) function of u.
From playlist Probability and Statistics
Peter Friz: Some examples of homogenization related rough paths
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist SPECIAL 7th European congress of Mathematics Berlin 2016.
David Kelly: Fast slow systems with chaotic noise
Find this video and other talks given by worldwide mathematicians on CIRM's Audiovisual Mathematics Library: http://library.cirm-math.fr. And discover all its functionalities: - Chapter markers and keywords to watch the parts of your choice in the video - Videos enriched with abstracts, b
From playlist Probability and Statistics
Multiple-Scattering Microfacet BSDFs with the Smith Model
The paper "Multiple-Scattering Microfacet BSDFs with the Smith Model" is available here: https://eheitzresearch.wordpress.com/240-2/ Update: it is being added to Blender's Cycles! - https://developer.blender.org/D2002 Modeling multiple scattering in microfacet theory is considered an imp
From playlist Light Transport, Ray Tracing and Global Illumination (Two Minute Papers)
Peter Friz (TU and WIAS Berlin) -- Laplace method on rough path and model space
Laplace's method allows one to obtain precise asymptotics in the large deviation principle. I will review the case of rough paths, then talk about extensions to rough volatility and singular SPDEs. Joint work with Paul Gassiat (Paris), Paolo Pigato (Rom) and Tom Klose (Berlin).
From playlist Columbia SPDE Seminar
Josef Teichmann: An elementary proof of the reconstruction theorem
CIRM VIRTUAL EVENT Recorded during the meeting "Pathwise Stochastic Analysis and Applications" the March 09, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide mathematician
From playlist Virtual Conference
Android Development for Beginners 11
Get the vector art here : http://goo.gl/y8HTCp Best Android Book : http://goo.gl/uPhXFI In this part of my App Inventor tutorial I will start covering Android interface design. I want the weather app we have been creating in parts 9 and 10 of this series to look very nice. Here I'll sho
From playlist Android Development for Beginners
NASA’s Perseverance Mars Rover Team to Discuss Early Science, Sample Collection (News Briefing)
Members from NASA’s Perseverance Mars rover team will discuss early science results from the robotic scientist and its preparations to collect the first-ever Martian samples for planned return to Earth. A key objective for Perseverance’s mission to Mars is astrobiology, including the searc
From playlist Mars
Second SIAM Activity Group on FME Virtual Talk
This is the second in a series of online talks on topics related to mathematical finance and engineering. The series is organized by the SIAM Activity Group on Financial Mathematics and Engineering. Title: A Data-driven Market Simulator for Small Data Environments Abstract: In this talk w
From playlist SIAM Activity Group on FME Virtual Talk Series
Mathworks Excellence in Innovation Project 209 : Autonomous Navigation of Vehicle in Rough terrain
This is a demonstration of the implementation of Autonomous Navigation of a robot using Gazebo simulation and Matlab Simulink models The detailed documentation of this project is available below: https://github.com/mathworks/MathWorks-Excellence-in-Innovation/tree/main/projects/Autonomou
From playlist MathWorks Excellence in Innovation