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Related Concept Videos

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

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

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...
Scaling01:26

Scaling

In designing and analyzing filters, resonant circuits, or circuit analysis at large, working with standard element values like 1 ohm, 1 henry, or 1 farad can be convenient before scaling these values to more realistic figures. This approach is widely utilized by not employing realistic element values in numerous examples and problems; it simplifies mastering circuit analysis through convenient component values. The complexity of calculations is thereby reduced, with the understanding that...
Introduction to Scalers01:21

Introduction to Scalers

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, "a class period lasts 50 min," or "the gas tank in my car holds 65 L," or "the distance between the two posts is 100 m." A physical quantity that can be specified completely in this manner is called a scalar quantity. The word "scalar" is a synonym for "number." Time, mass, distance, length, volume, temperature, and energy are some examples of scalar quantities.
Scalar...
Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
Multimachine Stability01:25

Multimachine Stability

Multimachine stability analysis is crucial for understanding the dynamics and stability of power systems with multiple synchronous machines. The objective is to solve the swing equations for a network of M machines connected to an N-bus power system.
In analyzing the system, the nodal equations represent the relationship between bus voltages, machine voltages, and machine currents. The nodal equation is given by:
Ampere-Maxwell's Law: Problem-Solving01:17

Ampere-Maxwell's Law: Problem-Solving

A parallel-plate capacitor with capacitance C, whose plates have area A and separation distance d, is connected to a resistor R and a battery of voltage V. The current starts to flow at t = 0. What is the displacement current between the capacitor plates at time t? From the properties of the capacitor, what is the corresponding real current?
To solve the problem, we can use the equations from the analysis of an RC circuit and Maxwell's version of Ampère's law.
For the first part of the problem,...

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Updated: Jun 24, 2026

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces
06:14

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces

Published on: September 11, 2018

Equation-free multiscale computation: algorithms and applications.

Ioannis G Kevrekidis1, Giovanni Samaey

  • 1Department of Chemical Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA. yannis@princeton.edu

Annual Review of Physical Chemistry
|April 2, 2009
PubMed
Summary
This summary is machine-generated.

This study introduces an equation-free framework for multiscale analysis. It enables macroscopic simulations using only microscopic data, bypassing complex equation derivation for complex systems.

Related Experiment Videos

Last Updated: Jun 24, 2026

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces
06:14

Multiscale Structures Aggregated by Imprinted Nanofibers for Functional Surfaces

Published on: September 11, 2018

Area of Science:

  • Computational Physics
  • Multiscale Modeling
  • Complex Systems Analysis

Background:

  • Traditional modeling relies on macroscopic evolution equations, which are often unavailable for complex systems.
  • Accurate models for complex systems are frequently limited to fine-scale, microscopic descriptions like molecular dynamics.
  • Bridging these scales is crucial for simulating and analyzing complex phenomena.

Purpose of the Study:

  • To present a computer-aided multiscale analysis framework.
  • To enable macroscopic computational tasks using only microscopic simulations.
  • To bypass the need for deriving unavailable macroscopic evolution equations.

Main Methods:

  • Utilizes appropriately initialized microscopic simulations on short time and length scales.
  • Employs an 'equation-free' approach, avoiding explicit macroscopic equation derivation.
  • Applies basic algorithms and underlying principles for multiscale analysis.

Main Results:

  • Demonstrates the feasibility of performing macroscopic computational tasks (simulation, analysis) from microscopic data.
  • Enables analysis over extended spatiotemporal scales without closed-form macroscopic models.
  • Illustrates the approach through representative applications.

Conclusions:

  • The equation-free framework offers a powerful alternative for multiscale analysis when macroscopic models are intractable.
  • This methodology facilitates computational tasks on complex systems by leveraging available microscopic simulations.
  • Further research is needed to address potential difficulties and expand the framework's applicability.