Recently,inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications.After the discretization,many of inverse problems are reduced...Recently,inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications.After the discretization,many of inverse problems are reduced to linear systems.Due to the typical ill-posedness of inverse problems,the reduced linear systems are often illposed,especially when their scales are large.This brings great computational difficulty.Particularly,a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution.Therefore,regularization methods should be adopted for stable solutions.In this paper,a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems.An iterative scheme becomes a regularization method only when the iteration is early terminated.And a Morozov’s discrepancy principle is applied for the stop criterion.Compared with the conventional Landweber iteration,the new methods have acceleration effect,and can be compared to the well-known acceleratedν-method and Nesterov method.From the numerical results,it is observed that using appropriate discretization schemes,the proposed methods even have better behavior when comparing withν-method and Nesterov method.展开更多
For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is ...For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is not invertible in/3. The first half of this paper proves that if/3 is Banach then the resolvent set PC（A） consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.展开更多
The authors first prove a convergence result on the Ka(?)anov method for solving generalnonlinear variational inequalities of the second kind and then apply the Kacanov method tosolve a nonlinear variational inequalit...The authors first prove a convergence result on the Ka(?)anov method for solving generalnonlinear variational inequalities of the second kind and then apply the Kacanov method tosolve a nonlinear variational inequality of the second kind arising in elastoplasticity. In additionto the convergence result, an a posteriori error estimate is shown for the Kacanov iterates. Ineach step of the Ka(?)anov iteration, one has a (linear) variational inequality of the secondkind, which can be solved by using a regularization technique. The Ka(?)anov iteration andthe regularization technique together provide approximations which can be readily computednumerically. An a posteriori error estimate is derived for the combined effect of the Ka(?)anoviteration and the regularization.展开更多
基金supported by the Natural Science Foundation of China (Nos. 11971230, 12071215)the Fundamental Research Funds for the Central Universities(No. NS2018047)the 2019 Graduate Innovation Base(Laboratory)Open Fund of Jiangsu Province(No. Kfjj20190804)
文摘Recently,inverse problems have attracted more and more attention in computational mathematics and become increasingly important in engineering applications.After the discretization,many of inverse problems are reduced to linear systems.Due to the typical ill-posedness of inverse problems,the reduced linear systems are often illposed,especially when their scales are large.This brings great computational difficulty.Particularly,a small perturbation in the right side of an ill-posed linear system may cause a dramatical change in the solution.Therefore,regularization methods should be adopted for stable solutions.In this paper,a new class of accelerated iterative regularization methods is applied to solve this kind of large-scale ill-posed linear systems.An iterative scheme becomes a regularization method only when the iteration is early terminated.And a Morozov’s discrepancy principle is applied for the stop criterion.Compared with the conventional Landweber iteration,the new methods have acceleration effect,and can be compared to the well-known acceleratedν-method and Nesterov method.From the numerical results,it is observed that using appropriate discretization schemes,the proposed methods even have better behavior when comparing withν-method and Nesterov method.
基金supported by National Natural Science Foundation of China(Grant No.11101079)and China Scholarship Council
文摘For a tuple A = （A1, A2,..., An） of elements in a unital algebra/3 over C, its projective spectrum P（A） or p（A） is the collection of z ∈ Cn, or respectively z ∈ pn-1 such that A（z） = z1A1＋z2A2＋…＋znAn is not invertible in/3. The first half of this paper proves that if/3 is Banach then the resolvent set PC（A） consists of domains of holomorphy. The second half computes the projective spectrum for the generating vectors of a Clifford algebra. The Chern character of an associated kernel bundle is shown to be nontrivial.
基金Project supported by the ONR grant N00014-90-J-1238
文摘The authors first prove a convergence result on the Ka(?)anov method for solving generalnonlinear variational inequalities of the second kind and then apply the Kacanov method tosolve a nonlinear variational inequality of the second kind arising in elastoplasticity. In additionto the convergence result, an a posteriori error estimate is shown for the Kacanov iterates. Ineach step of the Ka(?)anov iteration, one has a (linear) variational inequality of the secondkind, which can be solved by using a regularization technique. The Ka(?)anov iteration andthe regularization technique together provide approximations which can be readily computednumerically. An a posteriori error estimate is derived for the combined effect of the Ka(?)anoviteration and the regularization.
