Multiscale theory and computation

Numerical analysis of highly oscillatory ordinary differential equations

Oscillatory systems constitute a broad and active field of scientific computations. One of the typical numerical challenges arises when the frequency of the oscillations is high compared to either the time or the spatial scale of interest. In this case, the cost for computations can typically become exceedingly expensive due to the need of sampling oscillations adequately by numerical discretizations over a relatively large domain.

My research focuses on building efficient multiscale numerical methods that only sample the fast oscillations for short periods The sampled information is used to describe an effective time evolution for the system at longer time scales. The general approach underlying these methods come from the theory of averaging.

Movie: the inverted pendulum. How do you balance a pendulum pointing up. This even works with a wire!

Movie: the drift of a vibrating alarm clock

· G. Ariel, B. Engquist, S. Kim, Y. Lee and R. Tsai, A Multiscale Method for Highly Oscillatory Dynamical

Systems Using a Poincaré Map Type Technique, J. Sci. Comput. 54, 247-268 (2013). [pdf]

· Gil Ariel, Bjorn Engquist and Richard Tsai, Oscillatory Systems with Three Separated Time Scales - Analysis and Computation. Chapter in "Numerical Analysis and Multiscale Computations", Lect. Notes Comput. Sci. Eng., Volume 82, 23-45, Springer (2011). [pdf]

· Gil Ariel, Bjorn Engquist and Richard Tsai, A reversible multiscale integration method, Comm. Math. Sci. 7, 595-610 (2009) [pdf]

· Gil Ariel, Bjorn Engquist and Richard Tsai, A multiscale method for weakly coupled oscillators, Multiscale Modeling and Simulations 7, 1387-1404 (2009) [pdf]

· Gil Ariel, Bjorn Engquist and Richard Tsai, A multiscale method for stiff ordinary differential equations with resonance, Math. Comp 78, 929-956 (2009). [pdf]

· Gil Ariel, Bjorn Engquist, Heinz-Otto Kreiss and Richard Tsai, Numerical methods for highly oscillatory ordinary differential equations, In Lecture Notes in Computational Science and Engineering 66, Bjorn Engquist, Per Lotstedt and Olof Runborg, editors, Springer (2008). [pdf]

High frequency waves

A superposition of ten Gaussian beams with random coefficients.

(a)-(b) the real part of the field in position and a weighted Fourier spaces.

(c)-(d) the energy landscapes.

(e)-(f) following smoothing with a Gaussian kernel.

A probabilistic Expectation-Maximization method is used to approximate

the smoothed energy landscape with real, positive Gaussians.

This fit is used to approximate the initial field as a superposition

of Gaussian beams.

· Gil Ariel, Bjorn Engquist, Nicolay M. Tanushev and Richard Tsai, Gaussian beam decomposition of high frequency wave fields using expectation-maximization, Journal of Computational Physics 230:2303-2321 (2011) [pdf].

Models motivated by statistical physics

We study simplified models motivated by more complex systems in statistical physics. Both analytic and computational aspects are considerd.

· Gil Ariel and Eric Vanden-Eijnden, A strong limit theorem in the Kac-Zwanzig model, Nonlinearity 22, 145 (2009) [pdf]

· Gil Ariel and Eric Vanden-Eijnden, Accelerated simulation of a hard sphere in a gas of elastic spheres, Multiscale Modeling and Simulations 7, 349 (2008) [pdf]

· Gil Ariel and Eric Vanden-Eijnden, Testing transition state theory on Kac-Zwanzig model, The Journal of Statistical Physics 126, 43 (2007) [pdf]