The Study Of Convex Cone Problem Solving Method For Compressive Sensing And Matrix Completion

As a new idea to break through the limitation of Nyquist sampling theorem,compressive sensing and matrix completion have gradually developed into hot issues of concern to everyone.Especially in the geophysical problems,many experts and scholars have tried to introduce it into the treatment of related problems.However,with the deepening of the research,the actual engineering application has encountered great difficulties after the relevant research of mathematical theory has entered the bottleneck period.On its core elements,the design of observation matrix,the selection of sparse matrix,and the optimization of reconstruction algorithm all need to be designed and optimized based on practical engineering problems.This makes compressive sensing and matrix completion difficult in the practical direction,even in the more mature field of information analysis and image processing,there are often difficult problems such as low solution rate,insufficient solution quality,unstable convergence trend and so on.Especially in the seismic inversion,there are abundant inversion methods already,and the generalization ability of the existing methods is obviously insufficient in dealing with the problem of compressive sensing and matrix completion.Aiming at the problem that the generalization ability of the current solution scheme is insufficient to determine whether the target problem can be solved effectively,this paper proposes a new solution scheme,on the basis of the mature convex optimization algorithm,by designing a reasonable description method for the target inverse problem,the target inverse problem is transformed into a convex optimization program which can be solved stably.Under this idea,the deficiency of the generalization ability of the key elements can be compensated by the reasonable description of the target problem,and the solution scheme with stable convergence is provided for the problem of compressive sensing and matrix completion.In this paper,the essential of compressive sensing and matrix completion is analyzed firstly.It is pointed out that both are under the framework of sparse constraint problem,and the sparse constraint underdetermined equation and the low rank constraint underdetermined equation are formed according to the respective constraints.Through the detailed analysis of the current reconstruction algorithm and the theoretical model verification of the reconstruction of the high-dimensional data,the correlation analysis of different subspace,the feasibility of the convex optimization problem under different constraints is analyzed and the feasibility is demonstrated.Finally,by combining different forms of constraint conditions,the target inverse problem is described as conical convex optimization problem,and then the convex optimization solution is carried out by means of duality and smoothing.In order to further verify the universality of the method,this paper uses the conventional first-order optimization algorithm to solve and analyze the different types of sparse reconstruction problems.Then the image denoising and de-blurring processing experiments are carried out based on twodimensional images,and the denoising,defect data completion and super-resolution reconstruction are tested based on seismic data.This paper classifies and summarizes the optimization problems of sparse constraints and low rank constraints,puts forward a general convex conical solution template,and strengthens the generalization ability of the target problem description in order to make up for the shortcoming of the lack of generalization ability of the existing reconstruction method.In addition,aiming at the convex optimization problem of complex constraints,this paper proposes a composite conical smoothing function model to solve the optimization solution,and to improve the performance of the conventional optimization algorithm for convex optimization problems under complex constraints.In the actual data test,this paper reconstructs the seismic data effectively,improves the SNR of the data,and reconstructs the CRP channel set with superresolution,which effectively improves the image quality of seismic data and the resolution of related attributes.