Research On Partial Differential Equation Inverse Problem Based On Evolutionary Computation

It is well known that many problems founded can be described with partial differential equation in physics, mechanics and engineer and so on due to its connection of many characters of natural phenomenon tightly and straightly. With the development of scientific technology, partial differential equation, on one hand, have absorbed new methods from its development and renewed its contents, on the other hand, it also have improved the level of mathematical development. In the field of nature and engineering, inverse problems of undetermined parameter are often met. When some originally known condition of partial differential equation definite problem become unknown, we must define these unknown quantity by employing the definite condition and some additional condition on equation, though original equation function is possible unknown or some information on this unknown function is acquired. As can be seen from above, we regard these problems as partial differential equation inverse problem.Evolutionary computation based on natural selection has been advance to overcome above drawbacks proposed. ECs express each sort of complex model by working on a set of code solutions. It guides learn and determines search direction based on simple genetic algorithm and natural selection by a set of code solutions. This feature also helps to induce a large amount of implicit parallelism in the computational procedure. They express their ability by efficiently exploiting the historical information to speculate on new offspring with expected improved performance. ECs are executed iteratively on a set of coded solutions, called a population, with three basic operators: selection, crossover, and mutation. Evolution computation do not need any sort of auxiliary information and any limit of search condition. This feature which provided with ensures ECs simple, universal used, efficiency, easy-manipulated.This paper is to propose essential theory and original algorithm based on partial differential equations inverse problem. Due to its intrinsic ill-posed nature, we must use stabilizing measures to deal with regularization. Although study on partial differential equation inverse problem has made great progress in theory and onmethods and have advanced some usefully mathematical methods, theory or methods which need positive problem's mathematical expression based on analytical search methods will be not fit for some positive problems which is so much complex that they can not be computed based on analytical methods. Moreover, mathematical methods have much chance to get stuck in local minima, unless a good starting point is available. Clearly classical methods operate locally are not intrinsically parallel. So we bring ECs into partial differential equations inverse problem due to its character. In this paper we address the parameter identification inverse problem using GA and GP. In the third chapter of this paper, we identify the unknown discontinue coefficients of the partial differential equation with GA and use H1 norm regularization to avoid ill-posed of partial differential equations inverse problem, While identifying the unknown continue coefficients, we adopt regularization with piecewise H1 semi-norms. In the fourth chapter, we build the model of coefficient on one dimension partial differential equation and adopt finite element method to discrete partial differential equation and in the fifth chapter, we employ GP(genetic program) to identify and build the model of right-side function and adopt over-relaxation or alternating direction implicit method to solute difference equation.