基于离散数据的非线性抛物型方程反问题

Research on Inverse Problems of a Nonlinear Parabolic Equation Based on Discrete Measurement Data

考虑了一类利用离散数据进行线性插值作为终端观测值重构二阶非线性抛物型方程系数的反问题,它在自然科学和工程技术的很多领域都有重要应用.基于最优控制理论框架,先将原问题转化为一个非线性最优控制问题,并导出了最优解所满足的变分不等式.在插值步长趋于零时,利用正问题所满足的一些先验估计结果和变分不等式,证明了极小元的收敛性.

The inverse problem of identifying the coefficient of a nonlinear parabolic equation was discussed by using discrete measurement data.Problems of this type have important applications in many fields of natural sciences and engineering.Based on the optimal control framework,the original problems were transformed into a nonlinear optimization problem and the variational inequality which must be satisfied by the optimal solution was deduced.When the interpolation step goes to zero,the convergence of minimizer was proved by utilizing some prior estimates and variational inequalities of the direct problem.