In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called numerically stable. One of the common tasks of numerical analysis is to try to select algorithms which are robust – that is to say, do not produce a wildly different result for very small change in the input data. An opposite phenomenon is instability. Typically, an algorithm involves an approximative method, and in some cases one could prove that the algorithm would approach the right solution in some limit (when using actual real numbers, not floating point numbers). Even in this case, there is no guarantee that it would converge to the correct solution, because the floating-point round-off or truncation errors can be magnified, instead of damped, causing the deviation from the exact solution to grow exponentially. (Wikipedia).
Lecture: Numerical Differentiation Methods
From simple Taylor series expansions, the theory of numerical differentiation is developed.
From playlist Beginning Scientific Computing
Eva Darulova : Programming with numerical uncertainties
Abstract : Numerical software, common in scientific computing or embedded systems, inevitably uses an approximation of the real arithmetic in which most algorithms are designed. Finite-precision arithmetic, such as fixed-point or floating-point, is a common and efficient choice, but introd
From playlist Mathematical Aspects of Computer Science
Numerical Aperture in Fourier Optics
https://www.patreon.com/edmundsj If you want to see more of these videos, or would like to say thanks for this one, the best way you can do that is by becoming a patron - see the link above :). And a huge thank you to all my existing patrons - you make these videos possible. In this video
From playlist Fourier Optics
What is the definition of absolute value
http://www.freemathvideos.com In this video playlist you will learn how to solve and graph absolute value equations and inequalities. When working with absolute value equations and functions it is important to understand that the absolute value symbol represents the absolute distance from
From playlist Solve Absolute Value Equations
Alessio Figalli - Quantitative Stability in Geometric and Functional Inequalities - IPAM at UCLA
Recorded 08 February 2022. Alessio Figalli of ETH Zurich presents "Quantitative Stability in Geometric and Functional Inequalities" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: Geometric and functional inequalities play a crucial role in several problems
From playlist Workshop: Calculus of Variations in Probability and Geometry
Deep Differential System Stability - Learning advanced computations from examples (Paper Explained)
Determining the stability properties of differential systems is a challenging task that involves very advanced symbolic and numeric mathematical manipulations. This paper shows that given enough training data, a simple language model with no underlying knowledge of mathematics can learn to
From playlist Papers Explained
Von Neumann Stability Analysis of the FTCS Scheme | Lecture 70 | Numerical Methods for Engineers
A stability analysis of the forward-time centered-space scheme for solving the one-dimensional diffusion equation. Join me on Coursera: https://www.coursera.org/learn/numerical-methods-engineers Lecture notes at http://www.math.ust.hk/~machas/numerical-methods-for-engineers.pdf Subscr
From playlist Numerical Methods for Engineers
Thermodynamic-consistent multiple-relaxation-time lattice Boltzmann equation model
Zhonghua Qiao The Hong Kong Polytechnic University, Hong Kong
From playlist 2018 Modeling and Simulation of Interface Dynamics in Fluids/Solids and Their Applications
On effective numerical methods for phase-field models – Tao Tang – ICM2018
Numerical Analysis and Scientific Computing Invited Lecture 15.10 On effective numerical methods for phase-field models Tao Tang Abstract: In this article, we overview recent developments of modern computational methods for the approximate solution of phase-field problems. The main diffi
From playlist Numerical Analysis and Scientific Computing
Xiaolei Zhao: The MMP for deformations of Hilbert schemes of points on projective plane
Abstract: Hilbert schemes of points on projective plane admit deformations, which were constructed by Nevins and Stafford. I will explain this construction, and report on my recent joint work with Li, in which we study the birational models of these deformations using wall crossing in Brid
From playlist HIM Lectures: Junior Trimester Program "Algebraic Geometry"
ME564 Lecture 17: Numerical solutions to ODEs (Forward and Backward Euler)
ME564 Lecture 17 Engineering Mathematics at the University of Washington Numerical solutions to ODEs (Forward and Backward Euler) Notes: http://faculty.washington.edu/sbrunton/me564/pdf/L17.pdf Matlab code: * http://faculty.washington.edu/sbrunton/me564/matlab/L17_pend.m * http:
From playlist Engineering Mathematics (UW ME564 and ME565)
What is the max and min of a horizontal line on a closed interval
👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points
From playlist Extreme Value Theorem of Functions
F-measure is a harmonic mean of recall and precision. Think of it as accuracy, but without the effect of true negatives (which made accuracy meaningless for evaluating search algorithms). F-measure can also be interpreted as the Dice coefficient between the relevant set and the retrieved s
From playlist IR13 Evaluating Search Engines
Stability of the Schwarzschild solution by Amitabh Virmani
Date: 23 February 2017 VENUE: Ramanujan Lecture Hall, ICTS, Bengaluru This one-day discussion meeting celebrates the life and work of the pioneering black hole physicist C. V. Vishveshwara. The morning session will feature four scientific talks summarizing various aspects of Vishveshwara’
From playlist Remembering C. V. Vishveshwara
Jingwei Hu: New stability and convergence proof of the Fourier-Galerkin spectral method for the...
CIRM VIRTUAL EVENT Recorded during the meeting "Kinetic Equations: from Modeling, Computation to Analysis" the March 22, 2021 by the Centre International de Rencontres Mathématiques (Marseille, France) Filmmaker: Guillaume Hennenfent Find this video and other talks given by worldwide m
From playlist Virtual Conference
Convergent Evolving Surface Finite Element Algorithms for Geometric Evolution Equations
Professor Christian Lubich University of Tübingen, Germany
From playlist Distinguished Visitors Lecture Series
Lecture: Error and Stability of Time-stepping Schemes
The accuracy and stability of time-stepping schemes are considered and compared on various time-stepping algorithms.
From playlist Beginning Scientific Computing
Invariant Measures for Non-autonomous systems.... by Sergey Kryzhevich
PROGRAM SMOOTH AND HOMOGENEOUS DYNAMICS ORGANIZERS: Anish Ghosh, Stefano Luzzatto and Marcelo Viana DATE: 23 September 2019 to 04 October 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Ergodic theory has its origins in the the work of L. Boltzmann on the kinetic theory of gases.
From playlist Smooth And Homogeneous Dynamics
MIT RES.TLL-004 Concept Vignettes View the complete course: http://ocw.mit.edu/RES-TLL-004F13 Instructor: Anthony Patera This video introduces students to stability of equilibria. A temperature example is explored using an energy argument, and then the typical linear stability analysis fr
From playlist MIT STEM Concept Videos
Why we can't take "dt" to 0 in a computer: Sources of error in numerical differentiation
We have seen that the error of numerical differentiation typically scales with the time step dt. So why can't we just reduce the time step arbitrarily small to control the error? This video describes how numbers are stored in a computer and how small roundoff errors are amplified by very
From playlist Engineering Math: Differential Equations and Dynamical Systems