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

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

281
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,...
281
Poisson Probability Distribution01:09

Poisson Probability Distribution

11.4K
A Poisson probability distribution is a discrete probability distribution. It gives the probability of a number of events occurring in a fixed interval of time or space if these events happen at a known average rate and independently of the time since the last event. For example, a book editor might be interested in the number of words spelled incorrectly in a particular book. It might be that, on average, there are five words spelled incorrectly in 100 pages. The interval is 100 pages.
The...
11.4K
Poisson's And Laplace's Equation01:25

Poisson's And Laplace's Equation

4.1K
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
4.1K
Classification of Systems-II01:31

Classification of Systems-II

430
Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,
430
Transfer Function to State Space01:23

Transfer Function to State Space

691
State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
691
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

232
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...
232

You might also read

Related Articles

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

Sort by
Same author

Machine learning in critical care: Improving prediction of mortality and intensive care unit stay after cardiac surgery.

JTCVS open·2026
Same author

Systemic review of the relative age effect in Scottish schooling: a public health imperative for policy reform and equity.

BMJ public health·2026
Same author

Determination of particle-size distributions from light-scattering measurement using constrained Gaussian process regression.

Physical review. E·2026
Same author

Contextual computation by competitive protein dimerization networks.

Cell·2026
Same author

Targeting a specific subset of neutrophils to mitigate cardiac reperfusion injury.

Research square·2026
Same author

Improving Embedding of Graphs With Missing Data by Soft Manifolds.

IEEE transactions on pattern analysis and machine intelligence·2025

Related Experiment Video

Updated: Dec 25, 2025

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

8.3K

Statistical analysis of differential equations: introducing probability measures on numerical solutions.

Patrick R Conrad1, Mark Girolami1,2, Simo Särkkä3

  • 11Department of Statistics, University of Warwick, Coventry, UK.

Statistics and Computing
|April 1, 2020
PubMed
Summary

This study quantifies uncertainty in numerical solutions for differential equations. We introduce a method to account for numerical method uncertainty in statistical analyses of scientific models.

Keywords:
Inverse problemsNumerical analysisProbabilistic numericsUncertainty quantification

More Related Videos

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

423
Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package
06:37

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

Published on: September 17, 2021

4.9K

Related Experiment Videos

Last Updated: Dec 25, 2025

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level
06:55

Synthesis of Cyclic Polymers and Characterization of Their Diffusive Motion in the Melt State at the Single Molecule Level

Published on: September 26, 2016

8.3K
Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
06:44

Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis

Published on: September 23, 2025

423
Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package
06:37

Analyzing Melts and Fluids from Ab Initio Molecular Dynamics Simulations with the UMD Package

Published on: September 17, 2021

4.9K

Area of Science:

  • Computational Mathematics
  • Numerical Analysis
  • Scientific Computing

Background:

  • Numerical solutions of differential equations introduce inherent uncertainties.
  • Accurately quantifying these uncertainties is crucial for reliable statistical analysis of scientific models.
  • Increasing model complexity necessitates formal methods for uncertainty quantification.

Purpose of the Study:

  • To present a formal method for quantifying uncertainty arising from numerical solutions of differential equations.
  • To enable objective assessment of numerical method uncertainty relative to other error sources.
  • To provide a framework for incorporating numerical uncertainty into statistical model analysis.

Main Methods:

  • Randomization of existing numerical solvers to induce probability measures over solutions.
  • Utilizing the method of modified equations.
  • Application to ordinary differential equations and elliptic partial differential equations.

Main Results:

  • Demonstrated that randomized solvers converge to a Dirac measure around the true solution.
  • Showed convergence rates consistent with underlying deterministic numerical methods.
  • Employed modified equations to achieve enhanced convergence rates with stochastic perturbations.

Conclusions:

  • The proposed method formally quantifies uncertainty from numerical solutions of differential equations.
  • This quantification is vital for robust statistical analysis, especially in large-scale scientific modeling.
  • The approach is applicable to both forward and inverse problems in statistical analysis.