Gromacs is a high performance simulation engine primarily for solving Newtonian dynamics (it also does normal mode analysis, structure minimization and mixed molecular mechanics-quantum mechanics simulations.) It was an industry leader in terms of raw single processor performance for many years, until Desmond from D.E. Shaw Research took over with their super-scalable algorithms (I’ve written about this before.) With Gromacs 4.0, they’ve fixed the scalability problems and with a variety of other algorithmic fixes, they are the top dog once again. Disclaimer: these are all claims by relevant parties and I have not verified them myself, though I’d love to do so unencumbered. Though the Gromacs 4.0 paper is published, I’ll only be writing about it when the actual product is released.

The focus of the workshop was on algorithms, though there were some applications too. I’m sure an applications person would have felt out of place, but I felt I had something to contribute in almost every topic that was discussed. I’m archiving the list of topics here for posterity:

• The new domain decomposition parallelization in Gromacs 4.0, with some tips & tricks to get the most out of your hardware
• Different methods to perform free energy calculations. Slow-growth, perturbations, Bennett Acceptance Ratio. Which protocol is most efficient, and what new things will be in Gromacs 4.0?
• QM/MM. How do you mix Quantum Mechanics with Gromacs?
• Virtual sites for hydrogen motion removal and long time-steps
• Membrane protein simulations
• Replica exchange, and extracting kinetic data from it
• Local pressure extensions to Gromacs
• Gromacs source code walk-through

The take home message: strong coupling between various pieces of the algorithm is anti-thesis to parallel scalability. The CPU industry seems to have hit a brick wall in terms of improving raw computational speed: the future is in multi-core. Therefore, remove the coupling with better algorithms and you are on your way to highly scalable and by definition superbly fast algorithms.

The timestep used in an integrator while solving a set of equations inherently determines the speed of the algorithm. Big timesteps will make the algorithm unstable as you your trajectory will not be able to follow the phase space manifold accurately (as a side note Euler-type integrators also become unstable as you make the timestep smaller, but this is the least of your worries with a non-symplectic integrator.) The Nyquist theorem determines the sampling rate, so removing fast (or high frequency) degrees of motion such as hydrogen bond vibrations with constraints on them is required for a big timestep. Usual constraints algorithms are coupled leading to undesirable non-scalable algorithms. The Gromacs developers have solved this with a new constraints algorithm called P-LINCS.

The above is an actual plot that I created for some Hall measurements I was doing. I was supposed for find functional relationships between temperature and majority charge carriers, which in my case were electrons because of the n-type doping. The simple case was a least squares fit: scipy.optimize.leastsq to the rescue. The more complicated part was solving a non-linear equation for roots and then doing a least squares fit. The root-finding module in scientific python provides lots of options. At this point, I can confidently say that this environment has more features than Octave.

Just today, I wanted to use the Fourier method on a differential equation (plug: the advantages of which are here) and numerical python with fft, fftshift and fftfreq are exact substitutes for their Matlab equivalents. You can also put actual LaTeX equations on plots, which is a major plus.

That is all.

All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.).

Cryptography has generated number theory, algebraic geometry over finite fields, algebra (the creator of modern algebra, Viete, was the cryptographer of King Henry IV of France), combinatorics and computers.

Hydrodynamics procreated complex analysis, partial derivative equations, Lie groups and algebra theory, cohomology theory and scientific computing.

Celestial mechanics is the origin of dynamical systems, linear algebra, topology, variational calculus and symplectic geometry.

– Vladimir I. Arnold. Polymathematics: is mathematics a single science or a set of arts? In Mathematics: Frontiers and Perspectives. American Mathematical Society, 2000, pp. 403-416.

Vladimir Arnold is one of my favorite authors. He along-with Professor Jerrold E. Marsden must have written some of the best books on mechanics (this one and this one.) I’m writing the section connecting geodesics, metrics and Euler-Lagrange equations and I wasn’t sure how to introduce the material and looked to these books for inspiration.

Poincaré group

The Special Euclidean group combines rotations and translations in Euclidean space. The SE group is an isometry, which means it preserves angles and distances.

In special relativity, the 4-vectors are three space variables and one time variable. Rotations in this space can be generalized to the Poincaré group. Instead of having sines and cosines in the rotation part as in SE, they have hyperbolic sines and cosines.

The norm of a vector is defined through its inner product:

which in special relativity is

A friend pointed out that the Lorentz transformation was only recently proved to be linear.

Geodesics and Metrics

One of the most beautiful things I learnt in the last few weeks is the connection between geodesics (shortest paths) and metrics in that space. If is a geodesic path and is the velocity of the path:

then the metric is defined by

Moreover, the path satisfies the Euler-Lagrange equation. This was first shown by Arnold in a hydrodynamical context, and subsequently borrowed by control systems and computer vision folks.

A Poor Joke

Two functions and were going in a car. looked ahead and said, “Oh shit! There comes a differential operator.” says with a smirk, “It can’t do nothing to me!” On approaching the differential operator, it says “Haha! I’m d/dy.”

As narrated to me by a random math major last week…

• The Fast Fourier Transform for Experimentalists – Part I
• Physlets for Quantum Mechanics
• The Physical Basis of Computability
• Ten Good Practices in Scientific Programming
• The Metropolis Algorithm

One of the many things I learnt from my nonlinear physics project is the pseudospectral method. I’m going to try explaining this method the way I understand it.

Any function can be expressed as an infinite sum of the set of basis functions:

For the purpose of an example, if we take the polynomial basis of order 2, then the function can be approximated as

The coefficients can be solved for by building a system of equations. The three unknowns require three points on the interval where we know the exact value of the function. Let these three points be x0, x1 and x2 and the value of the function at these points be b0, b1 and b3.

The system of equations is now,

written compactly as

The system can be solved by inverting the matrix A, which is of complexity . The function p(x) is an interpolating polynomial in the interval of interest. As you can see, using a higher order polynomial quickly becomes prohibitively expensive.

Instead of picking the polynomial basis, you can pick any other basis. The Chebyshev basis is commonly used. A friend of mine picked the Fourier-Bessel basis. Depends on the application.

Fourier basis

The Fourier basis is fundamental to linear systems theory. Linear operators become diagonal in the Fourier basis. This fact comes handy in numerical schemes such as exponential integrators, where you’ll have to take the exponential of a matrix.

The second advantage falls out of the uncertainty principle. Increasing the number of modes in the Fourier domain is equivalent to shrinking the spacing between steps in the space domain. This leads to an exponential convergence in the number of points required as opposed to the usual polynomial convergence with finite differences.

The third key point in the use of the Fourier basis is that the FFT lets us do the equivalent of a matrix inversion in . This is very fast.

Summary

In summary, three random facts come together to form a beautiful theory:

• The FFT is much faster than a matrix inversion.
• Linear operators are diagonal in the Fourier basis.
• Exponential convergence of spectral methods.

Hope you enjoyed reading this post as much as I enjoyed writing it.

where M is the mass and q is the state vector. These equations arise in pretty much every single physics engine (astrophysics and molecular dynamics for example.) Traditionally, solvers such as the forward/backward Euler have been used, but these solvers do not respect the manifold of the configuration system. This led to the development of solvers such as the Verlet (and Velocity Verlet) which are symplectic in nature and follow the geometry of the problem. The Velocity Verlet specifically is the workhorse of molecular dynamics.

By geometry, I mean the invariants that arise due to symmetry in the system. For example, rotational and translational symmetry in the system give rise to conservation of angular and linear momentum. Above, the equations of motion for a simple pendulum were integrated with four different solvers. The symplectic nature of the problem does not fall out naturally from the solvers because these “local” methods still look at differential changes in momenta and position.

Variational methods, on the other hand, directly deal with equations arising out of Hamilton’s action principle. The Lagrangian is defined as

and the action functional is the integral of the Lagrangian along a path q(t).

First order variations to compute the stationary action leads to the Euler-Lagrange differential equation. A similar derivation can be done for discrete variables yielding the Discrete Euler Lagrange (DEL) equation.

This is very attractive because physical invariants that arise from the variational principle are guaranteed to be maintained in the discrete situation as well. This is also true for constraints on the system: constraint versions of Verlet and Velocity Verlet, SHAKE and RATTLE are natural in the DEL equation setting.

The only roadblock to the rapid adoption of these algorithms is the computational expense of solving the DEL. Each integration timestep requires the solution of a set of implicit nonlinear equations (usually by the use of Newton’s method.) This is a problem for molecular dynamics where you are trying to run the simulation for a few nanoseconds and your timestep is in femtoseconds.

Granular media, like sand, rice or cereal, also happen to form interesting patterns when vibrated on a membrane. This was first discovered and published in Nature by Umbanhowar, Melo & Swinney from the Center for Nonlinear Dynamics at the University of Texas at Austin. This was the same team that had created the very popular Faraday waves video posted on YouTube.

The physical phenomena is similar to Faraday waves seen in liquids almost a hundred and thirty years ago. The key difference is that while liquid faraday waves can be derived from a continnum model such as the Navier-Stokes equations, no such model exists for granular media as far as my limited knowledge goes. Therefore, the analytical derivations for the purposes of a project is limited.

The phase-plot is in terms of a dimensionless amplitude given by

where A is the driving amplitude, g is the acceleration due to gravity, and f is the driving frequency. Some of these have been experimentally verified by means of a molecular dynamics simulation. Once again, the computational part wasn’t sufficient to fulfill the requirements because the equations are “simple” non-linear coupled ordinary differential equations. Whereas systems we have studied in class were described by partial differential equations.

The term “oscillon” was coined by the paper mentioned above. In recent years, there has been a lot of research into these structures. The picture above is oscillons in a nonlinear field model and the author has some really cool videos on his site.

I spent most of last week looking for an idea for my nonlinear physics course project. We were given a few pointers to topics that could become potential projects: various kinds of pattern formation in Fluid Mechanics, Complex fluids, Biological systems, Chemical systems, Optics, Combustion and Hydrodynamics.

I came up with four topics (all four of them weren’t on the initial list of things suggested.) Somebody else in the class had exactly the same proposal as mine, including references, so that was out of the picture. One turned out to be computationally infeasible (couple of hours for a single step of a Finite Element Method), the other turned out to be extremely hard and outside the scope of the course.

I finally settled on “Global feedback methods for the subcritical Complex Ginzburg-Landau Equation.” If that didn’t make any sense, I’ve tried to explain what each word means in my proposal.

(picture courtesy of Shankar Venkataramani.)

Parabolic excitation

Working on a homework problem for my non-linear physics class has been a huge timesink. This week, we were asked to solve the general low viscosity Mathieu equation

for a parabolic excitation signal. My professor, Dr. Bechhoefer had published a paper to describe parametrically excited surface waves with delta and triangle excitation signals. We were asked to extend this. At the end, you get a complicated implicit equation for the threshold condition. I had to learn how to solve implicit equations numerically in Matlab (numerically because Maple couldn’t do it analytically.)

Geometry of Diffusion Tensors

As I’m working a lot with Diffusion Tensors, I wanted to get a better feel for the computations on them. I found this paper(pdf) by Dr. Joshi titled “Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors” really helpful. Ofcourse, I couldn’t understand some of the mathematical terms, so it’s more reading for me. I also realized that not all graduate students are on top of their game — they sometimes can mislead you. It’s better to mislead yourself than to be misled.

Geodesic Shooting

As part of my weekly reading, a friend and I are trying to understand this paper titled “Geodesic Shooting for Computational Anatomy.” Amidst all the complicated math, the basic idea they are trying to show is that flows on the deformation diffeomorphism conserve momentum. And because of this conservation law, you only need the velocity at t=0 to completely determine the flow. In the grand scheme of things, you do not and cannot average images just by averaging pixel/voxel intensities. This is because these images do not form a vector space and simple linear averaging does not respect the curved aspect of the space. The space of the velocity vectors form a vector space, and we can average those instead.

MRI scan

I also had my first MRI scan last week. I had to go all the way to the UBC hospital for that. A graduate friend of mine is doing some research on parameter optimization on the MR machine, but with about 400 controls you can tune, this is a very difficult problem. It was an interesting experience for me, though I slept through most of it. To take your mind off the heavy beating (and for claustrophobic people I guess,) they give you headphones. I wondered for a long time how electronic headphones worked in presence of a strong magnetic field (MR machines have a magnetic field strength of 1.5 to 3.0 tesla, in contrast to the earth’s field of 30 to 60 microteslas.) I’ll leave that question unanswered because it makes me feel too dumb.

That’s all for now. I’ll save the rest for later.