Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

93
Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...
93
Root-Locus Method01:19

Root-Locus Method

198
A cruise control system in a car is designed to maintain a specified speed automatically by adjusting the gas pedal. The system continuously measures the vehicle's speed and makes fine adjustments to the pedal to achieve this goal. The root locus method is particularly useful for understanding how the cruise control system's behavior changes under varying conditions, such as when the car goes uphill, downhill, or faces strong wind resistance.
This system can be represented by a block...
198
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

113
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
113
Fast Decoupled and DC Powerflow01:24

Fast Decoupled and DC Powerflow

256
The fast decoupled power flow method addresses contingencies in power system operations, such as generator outages or transmission line failures. This method provides quick power flow solutions, essential for real-time system adjustments. Fast decoupled power flow algorithms simplify the Jacobian matrix by neglecting certain elements, leading to two sets of decoupled equations:
256
Elevation of Intermediate Points on Vertical Curves01:20

Elevation of Intermediate Points on Vertical Curves

59
Vertical curves are essential in roadway design because they provide smooth transitions between varying roadway grades. Designing vertical curves involves calculating intermediate elevations and identifying the curve's highest or lowest point, which is essential for optimal roadway performance.Intermediate elevations on a vertical curve are determined using the tangent offset method. This method considers the initial elevation at the start of the curve, the grades, and the curve's geometry. The...
59
Properties of the Root Locus01:05

Properties of the Root Locus

153
The root locus method is an invaluable tool for analyzing higher-order systems without needing to factor the denominator of the transfer function. A pole of the system is identified when the characteristic polynomial in the transfer function's denominator equals zero.
To determine if a point lies on the root locus, the criterion involves the sum of angles contributed by all poles and zeros to that point. Specifically, this sum must be an odd multiple of 180 degrees. The gain at any point on...
153

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

On the convergence rate of the Kačanov scheme for shear-thinning fluids.

Calcolo·2021
Same journal

Newton-type algorithms for inverse optimization: weighted bottleneck Hamming distance and <math><msub><mi>ℓ</mi> <mi>∞</mi></msub></math> -norm objectives.

Optimization letters·2025
Same journal

A necessary condition for the guarantee of the superiorization method.

Optimization letters·2025
Same journal

Strategy investments in zero-sum games.

Optimization letters·2024
Same journal

A unified analysis of convex and non‑convex <math></math>‑ball projection problems.

Optimization letters·2024
Same journal

Multi-objective home health care routing: a variable neighborhood search method.

Optimization letters·2023
Same journal

On the firefighter problem with spreading vaccination for maximizing the number of saved nodes: the IP model and LP rounding algorithms.

Optimization letters·2023
See all related articles

Related Experiment Video

Updated: Aug 14, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

1.7K

A link between the steepest descent method and fixed-point iterations.

Pascal Heid1

  • 1Mathematical Institute, University of Oxford, Woodstock Road, Oxford, OX2 6GG UK.

Optimization Letters
|January 9, 2023
PubMed
Summary
This summary is machine-generated.

We link steepest descent for unconstrained minimization to fixed-point iterations for its Euler-Lagrange equation. This connection reveals the preconditioned algebraic conjugate gradient method for discretized problems, shown via numerical experiments.

Keywords:
Fixed-point iterationsPreconditioned conjugate gradient methodPreconditioning operatorSobolev gradientSteepest descent method

More Related Videos

Evolution of Staircase Structures in Diffusive Convection
07:28

Evolution of Staircase Structures in Diffusive Convection

Published on: September 5, 2018

6.6K
Experimental and Data Analysis Workflow for Soft Matter Nanoindentation
13:04

Experimental and Data Analysis Workflow for Soft Matter Nanoindentation

Published on: January 18, 2022

4.1K

Related Experiment Videos

Last Updated: Aug 14, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator
06:45

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

1.7K
Evolution of Staircase Structures in Diffusive Convection
07:28

Evolution of Staircase Structures in Diffusive Convection

Published on: September 5, 2018

6.6K
Experimental and Data Analysis Workflow for Soft Matter Nanoindentation
13:04

Experimental and Data Analysis Workflow for Soft Matter Nanoindentation

Published on: January 18, 2022

4.1K

Area of Science:

  • Numerical Analysis
  • Optimization Theory
  • Computational Mathematics

Background:

  • Unconstrained minimization problems are fundamental in applied mathematics and machine learning.
  • Steepest descent and fixed-point iterations are established numerical methods for solving such problems.
  • The Euler-Lagrange equation provides a critical link between optimization and differential equations.

Purpose of the Study:

  • To establish a novel theoretical connection between steepest descent and fixed-point iterations.
  • To demonstrate how this connection leads to the preconditioned algebraic conjugate gradient method.
  • To validate the practical implications through a numerical experiment.

Main Methods:

  • Formulating the unconstrained minimization problem.
  • Deriving the corresponding Euler-Lagrange equation.
  • Applying fixed-point iteration techniques to the Euler-Lagrange equation.
  • Connecting these concepts to the steepest descent method.
  • Discretizing the problem and applying the conjugate gradient method.

Main Results:

  • A direct mathematical link is established between steepest descent and fixed-point iterations for the Euler-Lagrange equation.
  • The preconditioned algebraic conjugate gradient method is shown to be a natural outcome of this connection for discretized problems.
  • Numerical experiments confirm the efficacy and benefits of this unified perspective.

Conclusions:

  • The unified framework offers a deeper understanding of the relationship between different optimization algorithms.
  • This approach provides a systematic way to derive and understand preconditioned conjugate gradient methods.
  • The findings have implications for the development of more efficient numerical solvers in optimization and related fields.