Monte Carlo methods have been developed for these computer programs to more effectively choose the most appropriate next move. To do so, the program simulates literally thousands of games from the current state, in the end choosing the best move. This method requires little domain knowledge of Go (knowledge of how to play the game) and thus leads to strong strategy but poor tactics.

To remedy the problem a new search technique was developed, called Upper Confidence Bounds Applied to Trees (UCT) which is a natural extension of Monte-Carlo Go programs. UCT is essentially “player memory” in that it remembers states of previous games and the outcomes of those games based on the chosen moves. It then guides the program towards the randomly generated moves that would yield the most successful result based on past “experience”.

Now beating the computer at Go is harder than ever.

http://en.wikipedia.org/wiki/Computer_Go#Monte-Carlo_methods

http://senseis.xmp.net/?UCT

Into the picture steps a team of computer scientists from UC San Diego with new raytracing algorithm that vastly cuts the computing costs for smoky and foggy images, thus allowing for much more realistic 3-D scenes. Traditional raytracing methods calculate the light at thousands of points, while the new method, called photon mapping, calculates the entire lightray in one shot. One of the co-authors explains, “Instead of computing the light at thousands of discrete points along the ray between the camera and the object, which is the conventional approach, we compute the lighting along the whole length of the ray all at once.”

Although the more costly raytracing methods may be adequate for scenarios such as movies, there simply isn’t enough computer power to effectively render images in near real-time as required in video games. However, the photon mapping method is much more apt for use in video games due to its much higher efficiency.

Below are two images of a lighthouse in fog, both rendered with the same computational resources. The top image, which is clearly more realistic, was rendered with the new photon mapping method.

http://www.jacobsschool.ucsd.edu/news/news_releases/release.sfe?id=731

http://graphics.ucsd.edu/~wjarosz/publications/jarosz08beam.pdf

This process of extracting all this information starts with a probabilistic model for timing deviations and uses tempo tracking algorithms for quantization. Then, the Monte Carlo methods are used to approximate the probabilities of note placement. The paper cited followed a Bayesian modeling approach for the tempo tracking and transcription. This paper then goes on to specifically explore the Markov Chain Monte Carlo (MCMC) and the sequential Monte Carlo (SMC) methods using both fake data and real data. It was found that the SMC method outperformed the MCMC method in terms of the quality of the results. This is reasonable, as SMC is well suited for applications in which observations arrive sequentially.

Monte Carlo Methods for Tempo Tracking
and Rhythm Quantization

www.nuc.berkeley.edu/courses/classes/NE39/Vujic-cancer.ppt

www.mgnet.org/~douglas/Classes/cs521-s01/monte-carlo/application.ppt

Browsing on youtube, I found a game named after the two German mathematicians Carl Runge and Martin Kutta.  The author of the game says he used the Runge Kutta method to simulate the spring in the game.

He says he uses RK4 with a frame timestep of t = t + 0.1, when running at 60fps.  He says he used the Runge Kutta method because RK4 appeared accurate and fast compared to the other methods out there.

By clicking on the link, you will see a quick demo to the game.

Currently, Monte Carlo methods are used in cancer therapy, traffic flow, Dow-Jones forecasting, oil well exploration, and several physics applications. Examples of those applications include stellar evolution, reactor design, and quantum chromo-dynamics. In addition, Monte Carlo is used widely in modeling materials and chemicals.
Expanding on using Monte Carlo in cancer treatment, the U.S. Department of Energy’s Brookhaven National Laboratory has been conducting a trial of an experimental treatment called boron neutron capture therapy (BNCT). Studies have concluded that a BNCT neutron source engineering input deck could complete its calculation in 19.35 minutes using the ORNL Intel Paragon XPS-150 computer. On the other hand, this identical calculation required 4.7 days to execute on a DEC Alpha workstation at INEEL. Overall, the speed-up ratio was 350. This conclusion could prove to be significant in the use of MCNP code. Having the calculations finished in under an hour will be an important factor in cancer treatment, especially for the brain cancer glioblastoma multiforme.
http://www.csm.ornl.gov/ssi-expo/MChist.html

One of the most exiting branches of computational physics is numerical relativity. It is the astrophysical study to simulate objects in space-time universe like stars, black holes, and gravitational waves, based on Einstein’s Theory of General Relativity.

The physical law in space-time can basically be represented by one equation: the Einstein Equation. Therefore, numerical relativity is primarily the study of numerical solution of the Einstein equations:

Gij = 8πTij.

These are16 coupled hyperbolic-elliptic nonlinear partial differential equations (which reduce to 10 by symmetry). They equate the curvature of space-time (Einstein tensor G) with the energy and momentum in the space-time (stress-energy tensor T). As the result, the differential equations evolve time and space derivatives. Simulating the movements of objects in space-time on computer requires the methods to take spatial derivatives and time derivative. I introduce some of the practical methods we learned in class to deal with these problems.

Special derivatives
Einstein equations by themselves cannot be differentiated exactly. Therefore, we need to change them to the forms so that they can be differentiated exactly. Fourier series expansion, expansion in terms of Chebyshev polynomials are used for this purpose. These methods are effective because they have high convergence rates.

Time derivatives
Runge-Kutta equations of fourth-order are often used in this step. As we increase the order of Runge-Kutta equations, we can acquire arbitrary convergence rate.

Numerical Relativity group at the Albert Einstein Institute is expert in these simulations, and I site their resulted images.

General information of Numerical Relativity

Image and Movie Archive of the AEI Numerical Relativity Group

A new technique called multiple importance sampling greatly increases the robustness of Monte Carlo integration. To evaluate an integral, it uses more than one sampling technique and combines these samples in a way that is very close to optimal. Veach finally links all of these ideas together to get new Monte Carlo light transport algorithms. Once of these is the bi-directional path tracing that uses a family of different path sampling techniques and creates some path vertices starting from a from either a light source or a sensor. The algorithm is unbiased, handles arbitrary geometry and materials, and most importantly, it is simple to implement. The second algorithm is called metropolis light transport. This algorithm generates paths by following a random walk through path space so the probability of visiting each path is in proportion to the contribution it makes to the optimal image. This algorithm is also unbiased, it handles arbitrary geometry and materials, and more importantly it requires little storage and can be orders of magnititude more efficient than its predecessors.

This dissertation by Eric Veach of Stanford University has a little relationship with one of the techniques that we have been learning about in CS 322 Scientific Computing. We have learned the basic Monte Carlo integration technique and we see that here it is applied to a model that is used for light transportation simulation and used to create light transport algorithms that generate realistic images by simulating the emission and scattering of light in an artificial environment. There are many applications for which these algorithms are good for that include lighting design, architecture, and computer animation. Related engineering applications include neutron transport and radiative heat transfer. As we can see, the things that we learn in CS 322 are not a waste of our time but have useful applications in this world.

            It is easy to see that this topic ties into the topics of the CS 322 course Scientific Computing. In the course, we learn about a lot of methods and one of these happens to be the least squares method. However, we do not get to see much of the methods we learn in practice. This article shows how the use of the least squares method can be very beneficial for the analysis of Biomedical data. The authors point out that they used the weighted least squares method because it was both computationally efficient and it gave a great approximation to the generalized least squares method. That is exactly what this course is about, methods to guarantee computations that a extremely close to optimal if not optimal, while still taking in consideration computational efficiency. There seems to be a trade-off in the amount of error a method has and its efficiency.

            This article may not be the most exciting read, but it shows that there is a use for the Least Squares method out there. In one of the author’s studies, they analyzed a set of mammalian mitochondrial protein data that included 3414 aligned amino acids from the cow, harbor seal, human, mouse, opossum, and rabit. They used generalized least squares in a program to construct a confidence set of trees to analyze this data. They used similar techniques for studying Hepatitis C, and DNA hybridization data. So in conclusion, there are uses for the least squares method out there and scientific computing is cool.