A Video Game Based on Elementary Differential Equations

In this paper a prey-predator video game is presented. In the video game two predators chase a prey that tries to avoid the capture by the predators and to reach a location in space (i.e. its “home”). The prey is animated by a human player (using a joypad), the predators are automated players whose behaviour is decided by the video game engine. The purpose of the video game is to show how to use mathematical models to build a simple prey-predator dynamics representing a physical system where the movements of the game actors satisfy Newton’s dynamical principle and the behaviour of the automated players simulates a simple form of intelligence. The game is based on a simple set of ordinary differential equations. These differential equations are used in classical mechanics to describe the dynamics of a set of point masses subject to a force chosen by the human player, elastic forces and friction forces (i.e. viscous damping). The software that implements the video game is written in C++ and Delphi. The video game can be downloaded from: http://www.ceri.uniroma1.

A Video Game Based on Optimal Control and Elementary Statistics

The video game presented in this paper is a prey-predator game where two preys (human players) must avoid three predators (automated players) and must reach a location in the game field (the computer screen) called preys’ home. The game is a sequence of matches and the human players (preys) must cooperate in order to achieve the best perform- ance against their opponents (predators). The goal of the predators is to capture the preys, which are the predators try to have a “rendez vous” with the preys, using a small amount of the “resources” available to them. The score of the game is assigned following a set of rules to the prey team, not to the individual prey. In some situations the rules imply that to achieve the best score it is convenient for the prey team to sacrifice one of his components. The video game pursues two main purposes. The first one is to show how the closed loop solution of an optimal control problem and elementary sta- tistics can be used to generate (game) actors whose movements satisfy the laws of classical mechanics and whose be- haviour simulates a simple form of intelligence. The second one is “educational”, in fact the human players in order to be successful in the game must understand the restrictions to their movements posed by the laws of classical mechanics and must cooperate between themselves. The video game has been developed having in mind as players for children aged between five and thirteen years. These children playing the video game acquire an intuitive understanding of the basic laws of classical mechanics (Newton’s dynamical principle) and enjoy cooperating with their teammate. The video game has been experimented on a sample of a few dozen children. The children aged between five and eight years find the game amusing and after playing a few matches develop an intuitive understanding of the laws of classical me- chanics. They are able to cooperate in making fruitful decisions based on the positions of the preys (themselves), of the predators (their opponents) and on the physical limitations to the movements of the game actors. The interest in the game decreases when the age of the players increases. The game is too simple to interest a teenager. The game engine consists in the solution of an assignment problem, in the closed loop solution of an optimal control problem and in the adaptive choice of some parameters. At the beginning of each match, and when necessary during a match, an assign- ment problem is solved, that is the game engine chooses how to assign to the predators the preys to chase. The resulting assignment implies some cooperation among the predators and defines the optimal control problem used to compute the strategies of the predators during the match that follows. These strategies are determined as the closed loop solution of the optimal control problem considered and can be thought as a (first) form of artificial intelligence (AI) of the preda- tors. In the optimal control problem the preys and the predators are represented as point masses moving according to Newton’s dynamical principle under the action of friction forces and of active forces. The equations of motion of these point masses are the constraints of the control problem and are expressed through differential equations. The formula- tion of the decision process through optimal control and Newton’s dynamical principle allows us to develop a game where the effectiveness and the goals of the automated players can be changed during the game in an intuitive way sim- ply modifying the values of some parameters (i.e. mass, friction coefficient, ...). In a sequence of game matches the predators (automated players) have “personalities” that try to simulate human behaviour. The predator personalities are determined making an elementary statistical analysis of the points scored by the preys in the game matches played and consist in the adaptive choice of the value of a parameter (the mass) that appears in the differential equations that define the movements of the predators. The values taken by this parameter determine the behaviour of the predators and their effectiveness in chasing the preys. The predators personalities are a (second) form of AI based on elementary statistics that goes beyond the intelligence used to chase the preys in a match. In a sequence of matches the predators using this second form of AI adapt their behaviour to the preys’ behaviour. The video game can be downloaded from the website: http://www.ceri.uniroma1.it/ceri/zirilli/w10/.

High Performance Algorithms Based on a New Wavelet Expansion for Time Dependent Acoustic Obstacle Scattering

This paper presents a highly parallelizable numerical method to solve time dependent acoustic obstacle scattering problems.The method proposed is a generalization of the“operator expansion method”developed by Recchioni and Zirilli[SIAM J.Sci.Comput.,25(2003),1158-1186].The numerical method proposed reduces,via a perturbative approach,the solution of the scattering problem to the solution of a sequence of systems of first kind integral equations.The numerical solution of these systems of integral equations is challenging when scattering problems involving realistic obstacles and small wavelengths are solved.A computational method has been developed to solve these challenging problems with affordable computing resources.To this aim a new way of using the wavelet transform and new bases of wavelets are introduced,and a version of the operator expansion method is developed that constructs directly element by element in a fully parallelizable way.Several numerical experiments involving realistic obstacles and“small”wavelengths are proposed and high dimensional vector spaces are used in the numerical experiments.To evaluate the performance of the proposed algorithm on parallel computing facilities,appropriate speed up factors are introduced and evaluated.