共查询到20条相似文献,搜索用时 15 毫秒
1.
Andreas Neubauer 《Numerical Functional Analysis & Optimization》2018,39(6):737-762
In this paper, we present a new gradient method for linear and nonlinear ill-posed problems F(x) = y. Combined with the discrepancy principle as stopping rule it is a regularization method that yields convergence to an exact solution if the operator F satisfies a tangential cone condition. If the exact solution satisfies smoothness conditions, then even convergence rates can be proven. Numerical results show that the new method in most cases needs less iteration steps than Landweber iteration, the steepest descent or minimal error method. 相似文献
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AbstractWe provide a modified augmented Lagrange method coupled with a Tikhonov regularization for solving ill-posed state constrained elliptic optimal control problems with sparse controls. We consider a linear quadratic optimal control problem without any additional L2 regularization terms. The sparsity is guaranteed by an additional L1 term. Here, the modification of the classical augmented Lagrange method guarantees us uniform boundedness of the multiplier that corresponds to the state constraints. We present a coupling between the regularization parameter introduced by the Tikhonov regularization and the penalty parameter from the augmented Lagrange method, which allows us to prove strong convergence of the controls and their corresponding states. Moreover, convergence results proving the weak convergence of the adjoint state and weak*-convergence of the multiplier are provided. Finally, we demonstrate our method in several numerical examples. 相似文献
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鉴于Newton型方法在实际计算中计算量可能非常大,因此提出了一种一步Newton结合若干步简化Newton的混合Newton-Tikhonov方法,并且在一定条件下证明了该方法的收敛性和稳定性.数值试验表明,在减少计算量方面该方法相对于经典的Newton方法有明显的改善. 相似文献
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Augmented Lagrangian Homotopy Method for the Regularization of Total Variation Denoising Problems 总被引:1,自引:0,他引:1
L. A. Melara A. J. Kearsley R. A. Tapia 《Journal of Optimization Theory and Applications》2007,134(1):15-25
This paper presents a homotopy procedure which improves the solvability of mathematical programming problems arising from
total variational methods for image denoising. The homotopy on the regularization parameter involves solving a sequence of
equality-constrained optimization problems where the positive regularization parameter in each optimization problem is initially
large and is reduced to zero. Newton’s method is used to solve the optimization problems and numerical results are presented. 相似文献
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E. B. Laneev 《Differential Equations》2018,54(4):476-485
Based on the Tikhonov regularization method, we explicitly construct a Carleman function in an ill-posed mixed problem for the Laplace equation. 相似文献
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Marco Donatelli 《高等学校计算数学学报(英文版)》2012,5(1):43-61
Iterative regularization multigrid methods have been successfully applied to signal/image deblurring problems. When zero-Dirichlet
boundary conditions are imposed the deblurring matrix has a Toeplitz
structure and it is potentially full. A crucial task of a multilevel
strategy is to preserve the Toeplitz structure at the coarse levels
which can be exploited to obtain fast computations. The smoother has
to be an iterative regularization method. The grid transfer operator
should preserve the regularization property of the smoother.
This paper improves the iterative multigrid method proposed in
[11] introducing a wavelet soft-thresholding denoising
post-smoother. Such post-smoother avoids the noise amplification
that is the cause of the semi-convergence of iterative
regularization methods and reduces ringing effects. The resulting
iterative multigrid regularization method stabilizes the iterations
so that the imprecise (over) estimate of the stopping iteration does
not have a deleterious effect on the computed solution. Numerical
examples of signal and image deblurring problems confirm the
effectiveness of the proposed method. 相似文献
9.
V. V. Vasin A. F. Skurydina 《Proceedings of the Steklov Institute of Mathematics》2018,301(1):173-190
For an equation with a nonlinear differentiable operator acting in a Hilbert space, we study a two-stage method of construction of a regularizing algorithm. First, we use the Lavrentiev regularization scheme. Then we apply to the regularized equation either Newton’s method or nonlinear analogs of α-processes: the minimum error method, the minimum residual method, and the steepest descent method. For these processes, we establish the linear convergence rate and the Fejér property of iterations. Two cases are considered: when the operator of the problem is monotone and when the operator is finite-dimensional and its derivative has nonnegative spectrum. For the two-stage method with a monotone operator, we give an error bound, which has optimal order on the class of sourcewise representable solutions. In the second case, the error of the method is estimated by means of the residual. The proposed methods and their modified analogs are implemented numerically for three-dimensional inverse problems of gravimetry and magnetometry. The results of the numerical experiment are discussed. 相似文献
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Andreas Neubauer 《Numerical Functional Analysis & Optimization》2013,34(3-4):405-423
A new regularization method, adaptive grid regularization, has been presented. Numerical results there show in a convincing way that this method is a powerful tool to identify discontinuities of solutions of ill-posed problems. It is the aim of this paper to give a convergence analysis for this new method. 相似文献
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U. Tautenhahn B. Hofmann Y. Shao 《Numerical Functional Analysis & Optimization》2013,34(12):1370-1417
The focus of this article is on conditional stability estimates for ill-posed inverse problems in partial differential equations. Conditional stability estimates have been obtained in related literature by a couple different methods. In this article, we propose a method called interpolation method, which is based on interpolation in variable Hilbert scales. We provide the theoretical background of this method and show that optimal conditional stability estimates are obtained. The capabilities of our method are illustrated by a comprehensive collection of different inverse and ill-posed PDE problems containing elliptic and parabolic problems, one source problem and the problem of analytic continuation. 相似文献
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Abstract We propose a new way to iteratively solve large scale ill-posed problems by exploiting the relation between Tikhonov regularization and multiobjective optimization to obtain, iteratively, approximations to the Tikhonov L-curve and its corner. Monitoring the change of the approximate L-curves allows us to adjust the regularization parameter adaptively during a preconditioned conjugate gradient iteration, so that the desired solution can be reconstructed with a low number of iterations. We apply the technique to an idealized image reconstruction problem in positron emission tomography. 相似文献
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A. Alessandri C. Cervellera M. Sanguineti 《Journal of Optimization Theory and Applications》2007,134(3):445-466
The design of state estimators for nonlinear dynamic systems affected by disturbances is addressed in a functional optimization
framework. The estimator contains an innovation function that has to be chosen within a suitably defined class of functions
in such a way to minimize a cost functional given by the worst-case ratio of the ℒ
p
norms of the estimation error and the disturbances. Since this entails an infinite-dimensional optimization problem that
under general hypotheses cannot be solved analytically, an approximate solution is sought by minimizing the cost functional
over linear combinations of simple “basis functions,” represented by computational units with adjustable parameters. The selection
of the parameters is made by solving a constrained nonlinear programming problem, where the constraints are given by pointwise
conditions that ensure the well-definiteness of the functional and the existence of a solution. Penalty terms are introduced
in the cost function to account for constraints imposed on points that result from sampling the sets to which the trajectories
of the state and of the estimation error belong. To ensure an efficient covering of the sets, low-discrepancy sampling techniques
are exploited that generate samples deterministically spread in a uniform way, without leaving regions of the space undersampled.
Work supported by a PRIN grant from the Italian Ministry of University and Research (Project “New Techniques for the Identification
and Adaptive Control of Industrial Systems”) and by the EU and the Regione Liguria trough the Regional Programs of Innovative
Action of the European Regional Development Fund. 相似文献
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CHRISTIE I.; GRIFFITHS D. F.; MITCHELL A. R.; SANZ-SERNA J. M. 《IMA Journal of Numerical Analysis》1981,1(3):253-266
The term "product approximation" is used to refer to a finiteelement technique for non-linear problems which has appearedseveral times in the literature under different presentations.The aims of this paper are to give a unified approach to theproduct integration technique and to provide new evidence forthe fact that it can be a good alternative to the standard Galerkinapproximation in certain circumstances. 相似文献
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本文的目的是在Hilbert空间中引入和研究了一种新的迭代序列,用以寻求具逆一强单调映象的广义平衡问题的解集与无限簇非扩张映象的不动点集的公共元.在适当的条件下,用黏性逼近法证明了逼近于这一公共元的强收敛定理.应用该结论,我们证明了逼近于平衡问题和变分不等式问题的强收敛定理.所得结果改进和推广了文献的相应结果. 相似文献
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The Problem of finding the roots (eigenvalues) of the equationdet A()=0, where A in an nxn matrix, is studied. There existseveral efficient local iterative methods for this problem.However, no efficient global method is available. We describethe application of the continuation method to this problem andsolve two examples by it. We conclude that the continuationmethod is a practical global strategy for locating eigenvaluesof non-linear matrices. This method is even more effective whenit is combined with an appropriate iterative scheme. 相似文献
18.
Mathematical Notes - The notion of the quality of approximate solutions of ill-posed extremum problems is introduced and a posteriori estimates of quality are studied for various solution methods.... 相似文献
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The Aitken's 2-prediction of Brezinski has already been used by Morandi Cecchi et al. in order to compute a numerical approximation of the solution of a parabolic initial-boundary value problem. This method consists in two consecutive steps: the first one is the approximation with a finite elements method, where the solution of the involved nonlinear system is computed by Gauss–Seidel method; the second one is a prediction of further terms with Aitken's 2-process. By comparison with this method, we use other methods of prediction in another way. First, we consider a generalization of 2-prediction, the so-called -prediction. In this paper, we only use vector prediction which is more stable than the scalar one. Then, the methods of prediction presented can be used in order to predict the starting vector of the Gauss–Seidel method. 相似文献
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Guan-Quan Zhang 《计算数学(英文版)》1984,2(1):10-23
The estimation for solutions for the ill-posed Cauchy problems of the differential equation $\frac{du(t)}{dt}=A(t)u(t)+N(t)u(t),\forall t\in (0,1)$ is discussed, where $A(t)$ is a 2-nd order p.d.o. and $N(t)$ is a uniformly bounded $h-›H$ linear operator. Two estimates of $||u(t)||$ are obtained. 相似文献