Numerical Algorithm for Inverse Problems and its Visualization.

在这个研究项目中，我们设计并分析了反问题的数值算法。该问题最初来源于紧算子的反演，其典型例子是第一类Fredholm积分方程。我们的主题包括发展稳定的数值算法并给出它的数学基础。并将该算法扩展到多媒体信息系统中，提出了一种新的数学模型，实现了数据项的语义解释。正则化的基本操作是根据约束选择合适的子空间。在积分方程的情况下，约束条件是解的平滑性。我们将此操作推广到语义解释，其中约束对应于上下文。我们可以用这个模型来处理意义，即内容和上下文的关系。该模型应用于实际的信息系统，包括元数据库系统和多媒体数据的关联搜索。

In this research project, we designed and analyzed numerical algorithm for inverse problems. The problems originally comes from inversion of compact operators whose typical example is Fredholm integral equations of the first kind. Our main theme includes developing stable numerical algorithm and giving its mathematical foundations. Moreover we extended our algorithms to multimedia information system and proposed new mathematical model which enables semantic interpretation of data items.The essential operation of regularization is that selecting an appropriate subspace according to the constraint. In the case of the integral equation the constraint is the smoothness of the solution. We generalized this operation to semantic interpretation where the constraint corresponds to context. We can deal with meaning which is the relation of content and the context using the model. This model is applied to actual information systems including meta-database systems and associative search for multimedia data.