共查询到20条相似文献,搜索用时 10 毫秒
1.
In this article, we consider the problem of finding a solution to ill-posed problems for abstract wave equations in a Hilbert
space, of the form
when A is a general linear selfadjoint operator. We study issues like existence, uniqueness and continuance dependance of data and
stability for this problem. Under precise constraint conditions on T, we make such problems well posed and in effect, generalize known results about these equations.
相似文献
2.
3.
G. Klein 《Journal of Computational and Applied Mathematics》1979,5(4):259-264
Interpolating natural splines are used for the algebraization and smoothing regularization of linear Fredholm integral equations of the first kind. A simplified version of statistical regularization is presented and, in turn, applied to data graduation by smoothing natural splines. 相似文献
4.
O. A. Liskovets 《Journal of Mathematical Sciences》1986,34(3):1656-1696
One gives a survey of the fundamental directions and results up to 1981, inclusively, regarding the topic mentioned in the title.Translated from Itogi Nauki i Tekhniki, Seriya Matematicheskii Analiz, Vol. 20, pp. 116–178, 1982. 相似文献
5.
S. G. Solodkii 《Ukrainian Mathematical Journal》1996,48(8):1263-1275
We consider the problem of finite-dimensional approximation for solutions of equations of the first kind and propose a modification of the projective scheme for solving ill-posed problems. We show that this modification allows one to obtain, for many classes of equations of the first kind, the best possible order of accuracy for the Tikhonov regularization by using an amount of information which is far less than for the standard projective technique. 相似文献
6.
We introduce a new family of semiiterative schemes for the solution of ill-posed linear equations with selfadjoint and indefinite operators. These schemes avoid the normal equation system and thus benefit directly from the structure of the problem. As input our method requires an enclosing interval of the spectrum of the indefinite operator, based on some a priori knowledge. In particular, for positive operators the schemes are mathematically equivalent to the so-called -methods of Brakhage. In a way, they can therefore be seen as appropriate extensions of the -methods to the indefinite case. This extension is achieved by substituting the orthogonal polynomials employed by Brakhage in the definition of the -methods by appropriate kernel polynomials. We determine the rate of convergence of the new methods and establish their regularizing properties. 相似文献
7.
Carl M Bender Fred Cooper G.S Guralnik Eric Mjolsness Harvey A Rose David H Sharp 《Advances in Applied Mathematics》1980,1(1):22-36
By replacing a differential equation boundary-layer problem by its discrete lattice equivalent we are able to treat the resulting equation as a regular perturbation problem. We obtain the solution on the lattice as a regular perturbation series in inverse powers of the lattice spacing. To obtain the answer to the continuum problem we extrapolate the solution to the lattice problem to zero lattice spacing. This extrapolation, which is a Padé-like procedure, yields good numerical results for a wide range of problems. 相似文献
8.
Jean-Michel Loubes 《Journal of Mathematical Analysis and Applications》2008,344(1):260-273
We consider the linear inverse problem of reconstructing an unknown finite measure μ from a noisy observation of a generalized moment of μ defined as the integral of a continuous and bounded operator Φ with respect to μ. Motivated by various applications, we focus on the case where the operator Φ is unknown; instead, only an approximation Φm to it is available. An approximate maximum entropy solution to the inverse problem is introduced in the form of a minimizer of a convex functional subject to a sequence of convex constraints. Under several assumptions on the convex functional, the convergence of the approximate solution is established. 相似文献
9.
We consider a Bayesian nonparametric approach to a family of linear inverse problems in a separable Hilbert space setting with Gaussian noise. We assume Gaussian priors, which are conjugate to the model, and present a method of identifying the posterior using its precision operator. Working with the unbounded precision operator enables us to use partial differential equations (PDE) methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centered on the true solution. Our methods assume a relatively weak relation between the prior covariance, noise covariance and forward operator, allowing for a wide range of applications. 相似文献
10.
Martin Hanke 《Numerical Algorithms》1996,11(1):203-213
The paper reviews the impact of modern orthogonal polynomial theory on the analysis of numerical algorithms for ill-posed problems. Of major importance are uniform bounds for orthogonal polynomials on the support of the weight function, the growth of the extremal zeros, and asymptotics of the Christoffel functions. 相似文献
11.
Pao-Liu Chow Ildar A. Ibragimov Rafail Z. Khasminskii 《Probability Theory and Related Fields》1999,113(3):421-441
For linear partial differential equations, some inverse source problems are treated statistically based on nonparametric
estimation ideas. By observing the solution in a small Gaussian white noise, the kernel type of estimators is used to estimate
the unknown source function and its partial derivatives.. It is proved that such estimators are consistent as the noise intensity
tends to zero. Depending on the principal part of the differential operator, the optimal asymptotic rate of convergence is
ascertained within a wide class of risk functions in a minimax sense.
Received: 5 May 1997 / Revised version: 18 June 1998 相似文献
12.
Straightforward solution of discrete ill-posed least-squares problems with error-contaminated data does not, in general, give meaningful results, because propagated error destroys the computed solution. Error propagation can be reduced by imposing constraints on the computed solution. A commonly used constraint is the discrepancy principle, which bounds the norm of the computed solution when applied in conjunction with Tikhonov regularization. Another approach, which recently has received considerable attention, is to explicitly impose a constraint on the norm of the computed solution. For instance, the computed solution may be required to have the same Euclidean norm as the unknown solution of the error-free least-squares problem. We compare these approaches and discuss numerical methods for their implementation, among them a new implementation of the Arnoldi–Tikhonov method. Also solution methods which use both the discrepancy principle and a solution norm constraint are considered. 相似文献
13.
In this paper, we study the weak linear bilevel programming problems. For such problems, under some conditions, we first conclude that there exists a solution which is a vertex of the constraint region. Based on the classical Kth-Best algorithm, we then present a solution approach. Finally, an illustrative example shows that the proposed approach is feasible. 相似文献
14.
V. V. Vasin 《Mathematical Notes》1970,7(3):161-165
A study is made of the relationship among three known methods for the approximate solution of linear operator equations of the first kind.Translated from Matematicheskie Zametki, Vol. 7, No. 3, pp. 265–272, March, 1970.I wish to express my deep gratitude to V. K. Ivanov for his interest in my work and for valuable remarks. 相似文献
15.
We consider a new version of the projection-iterative method for the solution of operator equations of the first kind. We show that it is more economical in the sense of amount of used discrete information. 相似文献
16.
The truncated singular value decomposition is a popular method for the solution of linear ill-posed problems. The method requires the choice of a truncation index, which affects the quality of the computed approximate solution. This paper proposes that an L-curve, which is determined by how well the given data (right-hand side) can be approximated by a linear combination of the first (few) left singular vectors (or functions), be used as an aid for determining the truncation index. 相似文献
17.
A semismooth equation approach to the solution of nonlinear complementarity problems 总被引:20,自引:0,他引:20
In this paper we present a new algorithm for the solution of nonlinear complementarity problems. The algorithm is based on
a semismooth equation reformulation of the complementarity problem. We exploit the recent extension of Newton's method to
semismooth systems of equations and the fact that the natural merit function associated to the equation reformulation is continuously
differentiable to develop an algorithm whose global and quadratic convergence properties can be established under very mild
assumptions. Other interesting features of the new algorithm are an extreme simplicity along with a low computational burden
per iteration. We include numerical tests which show the viability of the approach. 相似文献
18.
A mollification method for ill-posed problems 总被引:3,自引:0,他引:3
Summary.
A mollification method for a class of ill-posed
problems is suggested. The idea of the method is very simple and
natural: if the data are given inexactly then we try to find a
sequence of ``mollification operators" which map the improper
data into well-posedness classes of the problem (mollify the
improper data). Within these mollified data our problem becomes
well-posed. And when these facts are in hand we try to obtain
error estimates and optimal or ``quasi-optimal" mollification
parameters. The
method is working not only for problems in Hilbert spaces, but
also for problems in Banach spaces. Applications of the
method to concrete problems, like numerical differentiation,
parabolic equations backwards in time, the Cauchy problem for the
Laplace equation,
one- and multidimensional non-characteristic Cauchy problems for
parabolic equations (in infinite or finite domains),... give us
very sharp stability estimates of H\"older continuous type. In
these cases the method is optimal in the sense that it gives the
same order of H\"older continuous dependence on the data as for
the regularized problems. Furthermore, the method may be
implemented numerically using fast Fourier transforms. For the
first time a uniform stability estimate of H\"older continuous
type of the solution of the heat equation backwards in time in the
space for all could
be established by our mollification method. A new simple sharp
pointwise estimate of H\"older type for the weak solution of a
non-characteristic Cauchy problem for parabolic equations in a
finite domain is established.
Received June 25, 1993 / Revised version received February
18, 1994 相似文献
19.
20.
A. S. Leonov 《Numerical Analysis and Applications》2012,5(1):68-83
A new scheme of a posteriori accuracy estimation for approximate solutions to ill-posed inverse problems is presented along with an algorithm of calculating this estimation. A new notion of extra-optimal regularizing algorithmis introduced as a method for solving ill-posed inverse problems having optimal in order a posteriori accuracy estimation. Sufficient conditions of extra-optimality are formulated and an example of extra-optimal regularizing algorithm is given. The developed theory is illustrated by numerical experiments. 相似文献