Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Accuracy and Stability: Algorithms Performance

In course

PHYS-403: Computer simulation of physical systems I

The two main topics covered by this course are classical molecular dynamics and the Monte Carlo method. ...

Description

This lecture covers issues related to accuracy and stability in computational algorithms, focusing on the Verlet and Gear algorithms. The Verlet algorithm's stability is discussed in the context of a harmonic oscillator, highlighting the challenges of divergence and imaginary solutions. The Gear algorithm is introduced as a solution, emphasizing its predictor and corrector steps. Performance comparisons between various algorithms, including Euler, Verlet, and Gear, are presented through tests on a harmonic oscillator, evaluating accuracy, root-mean-square deviation, and energy drift.

Instructor

Alfredo Pasquarello

Alfredo Pasquarello studied physics at the Scuola Normale Superiore of Pisa and at the University of Pisa, obtaining their respective degrees in 1986. He obtained a doctoral degree at the EPFL in 1991 with a thesis on Multiphoton Transitions in Solids . Then, he moved to Bell Laboratories at Murray Hill (New Jersey), where he carried out postdoctoral research on the magnetic properties of carbon fullerenes. In 1993, he joined the Institute for Numerical Research in the Physics of Materials (IRRMA), where his activity involved first-principles simulation methods. In 1998, he was awarded the EPFL Latsis Prize for his research work on disordered silica materials. Succeeding in grant programs of the Swiss National Science Foundation, he then set up his own research group at IRRMA. In July 2003, he is appointed Professor in Theoretical Condensed Matter Physics at EPFL. Currently, he leads the Chair of Atomic Scale Simulation.

Official source

Related concepts (28)

Algorithm

In mathematics and computer science, an algorithm (ˈælɡərɪðəm) is a finite sequence of rigorous instructions, typically used to solve a class of specific problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning), achieving automation eventually.

Verlet integration

Verlet integration (vɛʁˈlɛ) is a numerical method used to integrate Newton's equations of motion. It is frequently used to calculate trajectories of particles in molecular dynamics simulations and computer graphics. The algorithm was first used in 1791 by Jean Baptiste Delambre and has been rediscovered many times since then, most recently by Loup Verlet in the 1960s for use in molecular dynamics. It was also used by P. H. Cowell and A. C. C.

Euler method

In mathematics and computational science, the Euler method (also called the forward Euler method) is a first-order numerical procedure for solving ordinary differential equations (ODEs) with a given initial value. It is the most basic explicit method for numerical integration of ordinary differential equations and is the simplest Runge–Kutta method. The Euler method is named after Leonhard Euler, who first proposed it in his book Institutionum calculi integralis (published 1768–1870).

Symplectic integrator

In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, molecular dynamics, discrete element methods, accelerator physics, plasma physics, quantum physics, and celestial mechanics. Symplectic integrators are designed for the numerical solution of Hamilton's equations, which read where denotes the position coordinates, the momentum coordinates, and is the Hamiltonian.

Computational complexity theory

In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by a computer. A computation problem is solvable by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used.

Related lectures (31)

Belief Propagation: Key Methods and Analysis

Covers Belief Propagation, a key method for both analysis and algorithm.

Solving Parity Games in Practice

Explores practical aspects of solving parity games, including winning strategies, algorithms, complexity, determinism, and heuristic approaches.

Numerical Methods: Exercises

Covers exercises related to numerical methods, focusing on computational algorithms and problem-solving strategies.

Complexity & Induction: Algorithms & Proofs

Covers worst-case complexity, algorithms, and proofs including mathematical induction and recursion.

Explicit Stabilised Methods: Applications to Bayesian Inverse Problems