方程近似解的計算復雜性.pdf

1、The problem on optimization of approximate solution of operator equations is mainly to determine the exact orders of error and complexity of algorithms,and construct the optimal algorithm realizing the orders.Complexity

2、of operations is also called the cost of operation.Therefore,the problem of the ε-complexity of approximate solution of operator equations,roughing speaking,is the minimal cost among all algorithms which solve the proble

3、m with error at most ε,and it has widely practical backgrounds.Generally speaking,the optimization of a problem can be done in various setting,such as worst case setting,average case setting,probability case setting.The

4、worst case setting is in common use,but the others are also of interest.To now,as far as the optimization of approximate solution of operator equations is concerned,many works have been done in the worst case setting.In

5、the average case setting,however,the result about this problem has hardly ever been seen.In Chapter 1 we have considered the problem of optimization of approximate solution of integral equations of several variables in w

6、orst case.We determine the exact error order and construct the optimal error algorithm.In Chapter 2 The problem of ε-complexity of integral equations have been considered in worst case.We determine the exact order of the

7、 complexity and construct the optimal algorithm realizing the order.In Chapter 3 we have considered the problem of the optimization of approximate solution of operator equations in the average case and its application.To

8、 solve above mentioned problems we have used the classical methods and skills in approximate theory,especially the profound results on width and approximate by Fourier sum.Moreover,some new ideas and skills in modern app

9、roximate theory have also played very important roles.Generally speaking,to synthesize traditional methods and modern mathematical idea and methods in functional analysis,linear algebra,probabilistic theory may provide n