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

Bernoulli's Equation: Problem Solving01:16

Bernoulli's Equation: Problem Solving

1.5K
A Venturi meter is essential for measuring fluid flow rates in pipelines. It utilizes the relationship between fluid velocity and pressure described by Bernoulli's equation. When installed in a sewage system, the Venturi meter accurately determines the wastewater flow rate by measuring pressure differences.
The first step is to compute the cross-sectional areas of the pipe and the Venturi throat to analyze the pressure difference indicated by the pressure gauge. Next, the continuity...
1.5K
Navier–Stokes Equations01:28

Navier–Stokes Equations

1.1K
For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
1.1K
Divergence and Stokes' Theorems01:06

Divergence and Stokes' Theorems

2.8K
The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
2.8K
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

150
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...
150
Gaussian Elimination: Problem Solving01:30

Gaussian Elimination: Problem Solving

48
Systems of linear equations in several variables are pivotal in modeling complex scenarios involving multiple unknowns and constraints. Such systems are widely used in various fields to represent relationships where several conditions must be simultaneously satisfied. Each variable in the system corresponds to an unknown quantity, while each equation imposes a linear constraint, leading to a structured approach for analyzing and solving real-world problems.A system of three equations with three...
48
Second Derivatives and Laplace Operator01:22

Second Derivatives and Laplace Operator

2.2K
The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or vector function yield second order expressions. Second-order expressions play a very important role in mathematics and physics. Some second order expressions include the divergence and curl of a gradient function, the divergence and curl of a curl function, and the gradient of a divergence function.
Consider a scalar function. The curl of its...
2.2K

You might also read

Related Articles

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

Sort by
Same author

Applications of Artificial Intelligence, Deep Learning, and Machine Learning to Support the Analysis of Microscopic Images of Cells and Tissues.

Journal of imaging·2025
Same author

Using machine learning on new feature sets extracted from three-dimensional models of broken animal bones to classify fragments according to break agent.

Journal of human evolution·2024
Same author

Artificial Intelligence for Classifying the Relationship between Impacted Third Molar and Mandibular Canal on Panoramic Radiographs.

Life (Basel, Switzerland)·2023
Same author

An Overview of In Vitro Assays of <sup>64</sup>Cu-, <sup>68</sup>Ga-, <sup>125</sup>I-, and <sup>99m</sup>Tc-Labelled Radiopharmaceuticals Using Radiometric Counters in the Era of Radiotheranostics.

Diagnostics (Basel, Switzerland)·2023
Same author

Anti-Arthritic and Anti-Cancer Activities of Polyphenols: A Review of the Most Recent In Vitro Assays.

Life (Basel, Switzerland)·2023
Same author

Radar-Based Shape and Reflectivity Reconstruction Using Active Surfaces and the Level Set Method.

IEEE transactions on pattern analysis and machine intelligence·2022
Same journal

Product-of-Gaussian-mixture diffusion models for joint nonlinear MRI reconstruction.

Journal of mathematical imaging and vision·2026
Same journal

Linear Optimal Transport Subspaces for Point Set Classification.

Journal of mathematical imaging and vision·2026
Same journal

Diffusion-Shock PDEs for Deep Learning on Position-Orientation Space.

Journal of mathematical imaging and vision·2026
Same journal

Connected Components on Lie Groups and Applications to Multi-Orientation Image Analysis.

Journal of mathematical imaging and vision·2026
Same journal

CoRRECT: A Deep Unfolding Framework for Motion-Corrected Quantitative R2* Mapping.

Journal of mathematical imaging and vision·2025
Same journal

Stochastic Primal-Dual Hybrid Gradient Algorithm with Adaptive Step Sizes.

Journal of mathematical imaging and vision·2024
See all related articles

Related Experiment Video

Updated: Nov 3, 2025

Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom
06:26

Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom

Published on: February 25, 2022

4.6K

Accelerated Variational PDEs for Efficient Solution of Regularized Inversion Problems.

Minas Benyamin1, Jeff Calder2, Ganesh Sundaramoorthi3

  • 1School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, Georgia.

Journal of Mathematical Imaging and Vision
|June 3, 2021
PubMed
Summary
This summary is machine-generated.

We introduce PDE acceleration, a novel framework generalizing gradient descent for optimization problems. This method enhances numerical algorithms for inverse problems and image processing by using nonlinear wave equations instead of diffusion equations.

Keywords:
35Q9365K1065M0697N40Accelerated gradient descentBeltrami regularizationImage deblurringImage denoisingImage restorationNesterov accelerationNonlinear wave equationsPDE accelerationTotal variation

More Related Videos

A Modeling and Simulation Method for Preliminary Design of an Electro-Variable Displacement Pump
09:04

A Modeling and Simulation Method for Preliminary Design of an Electro-Variable Displacement Pump

Published on: June 1, 2022

3.3K

Related Experiment Videos

Last Updated: Nov 3, 2025

Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom
06:26

Particle Image Velocimetry Investigation of Hemodynamics via Aortic Phantom

Published on: February 25, 2022

4.6K
A Modeling and Simulation Method for Preliminary Design of an Electro-Variable Displacement Pump
09:04

A Modeling and Simulation Method for Preliminary Design of an Electro-Variable Displacement Pump

Published on: June 1, 2022

3.3K

Area of Science:

  • Numerical analysis
  • Optimization
  • Image processing

Background:

  • Calculus of variation problems are crucial for optimization.
  • Existing methods often rely on diffusion equations for solving inverse problems.
  • Gradient descent methods can be slow for complex optimization tasks.

Purpose of the Study:

  • To develop a new framework called PDE acceleration.
  • To create efficient numerical algorithms for optimization problems.
  • To apply this framework to regularized inversion and image processing.

Main Methods:

  • Generalizing momentum (accelerated) gradient descent to the PDE setting.
  • Deriving nonlinear damped wave equations from elliptic problems.
  • Developing explicit and semi-implicit numerical schemes with stability constraints.

Main Results:

  • Achieved improved CFL conditions (Δt ~ Δx) compared to diffusion equations (Δt ~ Δx²).
  • Demonstrated applicability to quadratic, Beltrami, and total variation regularization.
  • Showcased effectiveness in image denoising, deblurring, and inpainting.

Conclusions:

  • PDE acceleration offers a generalized and efficient approach for solving optimization problems.
  • The framework provides significant improvements for regularized inversion and image processing tasks.
  • The proposed numerical schemes are adaptable and effective compared to existing algorithms.