Professor Heiko Rieger is an internationally well known expert in the physics of disordered systems.
He has numerous publications which have led to important progress of this field. The past few years have witnessed a substantial growth in the number of applications for optimization algorithms in solving problems in the field of physics.
Examples include determining the structure of molecules, estimating the parameters of interacting galaxies, the ground states of electronic quantum systems, the behavior of disordered magnetic materials, and phase transitions in combinatorial optimization problems. This book serves as an introduction to the field, while also presenting a complete overview of modern algorithms.
Optimization Algorithms in Physics
The authors begin with the relevant foundations from computer science, graph theory and statistical physics, before moving on to thoroughly explain algorithms - backed by illustrative examples. They include pertinent mathematical transformations, which in turn are used to make the physical problems tractable with methods from combinatorial optimization. Throughout, a number of interesting results are shown for all physical examples. The final chapter provides numerous practical hints on software development, testing programs, and evaluating the results of computer experiments.
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Share your thoughts with other customers. Write a customer review. Most helpful customer reviews on Amazon. October 3, - Published on Amazon. Traditionally, physicists haven't used many of the algorithms and ideas in computer science. The reason is simple. Computer science deals mostly with discrete items. Whereas most of physics uses continuum methods. In physics, the simple harmonic oscillator is a mechanical system consisting of a particle of mass m connected to a fixed point by means of a spring. The system always tends to stay at the equilibrium position, namely at zero potential energy.
Whenever the spring is stretched or compressed, there will be a force acting on the object that will tend to bring the mass towards the resting position again. Those who are not very familiar with the concept of differential equation ODE can read Neural networks with infinite layers , in which ODEs are briefly summarized.
By compressing the spring and releasing it at point , the system will start oscillating. The solution of the above differential equation gives the position of the particle over time, which is , where is the oscillating frequency.
In this system there are two forms of energy, namely the elastic potential energy given by and the kinetic energy , being the velocity of the object. Under to the well-known law of energy conservation, this system continues to oscillate indefinitely. In fact, the initial potential energy is converted to kinetic energy, which is, in turn, converted back to potential energy and so on, just like in roller-coasters. In the real world, however, such systems do not exist and they behave more like a damped harmonic oscillator , where a frictional force, proportional to the negative velocity of the particle is also present.
Unlike the simple harmonic oscillator, such a system would reach an equilibrium where the particle eventually stops with zero potential energy, due to the dissipation of the kinetic energy caused by the friction. Such a dynamic is described by the following differential equation:. The behavior of the system is determined by the damping ratio :. Depending on the value of , the aforementioned system can be categorized as:. How is all this related to gradient-based algorithms? As a matter of fact, there is an interesting connection between the damped harmonic oscillator and gradient-based optimization algorithms.
In fact, we can interpret the problem of optimization as a system in which a particle is falling inside a given potential. One can think of it as a ball bouncing in a landscape with hills and valleys.
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When it starts bouncing, the ball has high potential energy. Then, this energy gets converted into kinetic energy. In the long term, the ball will settle in a low elevation area and will have small potential energy which will be zero at sea level. The solution to the optimization problem corresponds to the point of zero energy attained when the particle is standing still. In order to show the connection between damped harmonic oscillators and optimization algorithms, here is an example of the gradient descent iteration step. The value at the next iteration is calculated from the current one, following the direction of the negative gradient.
Using the Taylor expansion of the objective function f around , we get. By plugging this equation into the previous one, in the limit , we obtain the following differential equation:. Does this look familiar?
It should, because this is just the damped oscillator system we have seen before. This reasoning can be extended to other variants of gradient descent algorithms, such as the Adam method mentioned in the beginning of this article. In this new perspective, the various gradient descent optimization methods correspond to damped oscillators with different particle mass and damping coefficients, characterized by different dissipation mechanisms of their mechanical energy.
Moreover, the decay rate of the mechanical energy is connected to the convergence rate of the related optimization algorithm.