Joint Bidiagonalization Algorithms For The Computation Of Partial GSVDs And Discrete Ill-Posed Problems With General-form Regularization

Discrete ill-posed problems with general-form regularization are closely related to the generalized singular value decomposition(GSVD)of a matrix pair,and these two kinds of large problems can be solved using algorithms based on the joint bidiagonalization(JBD)process.In this thesis,we analyze several critical problems when we use the JBD based algorithms to solve the above two kinds of problems.We present a rounding error analysis of the JBD process of a matrix pair {A,L},which establishes connections between the JBD process and the two joint Lanczos bidiagonalizations in finite precision arithmetic.We investigate the loss of orthogonality of the Lanczos vectors computed by the JBD process.Based on the results of rounding error analysis,we investigate the convergence and accuracy of the approximate generalized singular values and vectors.The results show that semiorthogonality of the Lanczos vectors is enough to guarantee the accuracy and convergence of the approximate generalized singular values,which is a guidance for designing an efficient semiorthogonalization strategy for the JBD process.We also investigate the residual norm appeared in the partial GSVD computation of {A,L},and prove that its upper bound can be used as a stopping criterion.We use several numerical examples to illustrate our results.We propose a semiorthogonalization strategy for the JBD process such that the orthogonality level of the Lanczos vectors is maintained at O((?)),where ? is the roundoff unit.Our rounding error analysis establishes connections between the JBD process with the semiorthonalization strategy and the Lanczos bidiagonalization process in finite precision arithmetic.We prove that the JBD process with the semiorthogonalization strategy can avoid the delay of the convergence of the computed quantities and the final accuracy is not far from O(?).Based on the semiorthogonalization strategy,we develop the joint bidiagonalization process with partial reorthogonalization(JBDPRO)algorithm.The JBDPRO algorithm can save much unnecessary reorthogonalization work compared with the algorithm with full reorthogonalization.Several numerical examples are used to confirm our theory and show the numerical behavior of our algorithm.We analyze the solution accuracy requirement on the inner iteration of the JBD based algorithms for solving discrete ill-posed problems with general-form regularization.We investigate the properties of the JBD process with its inner iteration solved inexactly,and propose a modified process denoted by JBD(?)and an algorithm denoted by JBDQR(?),where ? is the solution accuracy of the inner iteration.We make a preliminary analysis of the accuracy of the regularized solution obtained by the JBDQR(?)and design a criterion for the choice of ?.Our results indicate that an appropriate choice of ? depends on the noise level and the property of a given ill-posed problem.The numerical experiments show that if the noise level is not too small,the inner iteration of the algorithm can be performed with considerable relaxed accuracy,and thus the overall efficiency of the algorithm can be improved.