Convex Optimization Methods In Study Of A Class Of Inverse Problem

With the development of science and technology, sensor networks more and more get theattention of people. Typically, the basic task in sensor networks is the information collection,data transmission and signal recovery. Due to the increasing required information, which leadto sensor takes time to collect more information. Therefore, how to efficient use of sensor isparticularly important. Some of the data collection in sensor network is not easy toimplement，Collect data excessively not only affects the efficiency of the transmission of thesensor, but also lead to a waste of resources. Therefore, for the inverse problem of restoringthe original signal, the study of the number of required measurement value is very necessary..Meanwhile, in order to analysis sensor networks more simple, by analysis the character of thegraphical model, decompose complex system into a number of simpler component partsusing graphical model knowledge. We can also construct the corresponding graph by knowngraphical model characteristics. These are necessary for sensor network analysis.Convex optimization method is more popular in recent years, more and more attentionare paid for researchers. The inverse of the number of measured values required for therestoration of signal problems and graph decomposition and generation problems, how to usethe basic knowledge of convex analysis to analysis and solve these problems are interested forus. In this paper, the main work is as follows:(1) This paper studies how to transform vector to be estimated recovery problem into aconvex optimization problem based on measure values. Restore ratio of vector to be estimateddepends on the measured values dimension. So through basic analysis of the convex, we cantransform the problem of measurement dimension into the problem of calculating the tangentcone Gaussian width that induced by atomic norm. And the problem of solve Gaussian widthmainly use dual structure. Eventually solving the dimension of the measurement depends onsolving the problem of dual cone Gaussian width. At the same time, this paper describes the atomic norm representing problems, and when the norm is not easy to calculate atomicapproximate relaxations is presented. Finally, this paper solves the sparse vector and thelow-rank matrix through computer simulation software. Verify the validity of the number ofdimensions of the measurements that determined.(2) This paper put forward two questions based on graph model：composite graphdecomposition based on same node and generate graph using some known character. In thispaper, in order to obtain the solution of the issues raised, we use the structural characteristicsand nature of the graph model, combined with knowledge of convex analysis and studyconvex graph invariants, given the variety of convex graph invariants and the invariantconvex sets. Finally, through the analysis of convex graph invariants obtained a generallyconvex optimization constructor, and solve the problems above effectively.