Library

X
feed icon rss

Your email was sent successfully. Check your inbox.

An error occurred while sending the email. Please try again.

Proceed reservation?

Export
  • 1
    Book
    Book
    Approximation of nonlinear evolution systems; 164 (1983)
    Boston u.a. :Academic Press,
    Details
    Title: Approximation of nonlinear evolution systems; 164
    Author: Jerome, Joseph W.
    Publisher: Boston u.a. :Academic Press,
    Year of publication: 1983
    Pages: 280 S.
    Series Statement: Mathematics in science and engineering 164
    Type of Medium: Book
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    ZIB Catalog
    Zuse Institute Berlin get availability status
  • 2
    Electronic Resource
    Electronic Resource
    An adaptive Newton algorithm based on numerical inversion: Regularization as postconditioner (1985)
    Springer
    Numerische Mathematik 47 (1985), S. 123-138 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65H10 ; CR: G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We examine a class of approximate inversion processes, satisfying estimates similar to those defined by finite element or truncated spectral approximations; these are to be used as approximate right inverses for Newton iteration methods. When viewed at the operator level, these approximations introduce a defect, or “loss of derivatives”, of order one or more. Regularization is introduced as a form of defect correction. A superlinearly convergent, approximate Newton iteration is thereby obtained by using the numerical inversion adaptively, i.e., with spectral or grid parameters correlated to the magnitude of the current residual in an intermediate norm defined by the defect. This adaptive choice makes possible ascribing an order to the convergent process, and this is identified as essentially optimal for elliptic problems, relative to complexity. The design of the algorithm involves multi-parameter selection, thereby opening up interesting avenues for elliptic problems, relative to complexity. This applies also to the regularization which may be carried out in the Fourier transform space, and is band-limited in the language of Whittaker-Shannon sampling theory. The norms employed in the analysis are of Hölder space type; the iteration is an adaptation of Nash-Moser interation; and, the complexity studies use Vituškin's theory of information processing. Computational experience is described in the final section.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389880
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 3
    Electronic Resource
    Electronic Resource
    Newton's method for gradient equations based upon the fixed point map: Convergence and complexity study (1989)
    Springer
    Numerische Mathematik 55 (1989), S. 619-632 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS):65H10 ; CR:G1.5
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary An approximate Newton method, based upon the fixed point mapT, is introduced for scalar gradient equations. Although the exact Newton method coincides for such scalar equations with the standard iteration, the structure of the fixed point map provides a way of defining anR-quadratically convergent finite element iteration in the spirit of the Kantorovich theory. The loss of derivatives phenomenon, typically experienced in approximate Newton methods, is thereby avoided. It is found that two grid parameters are sssential,h and $$\bar h \approx h^2 $$ . The latter is used to calculate the approximate residual, and is isolated as a fractional step; it is equivalent to the approximation ofT. The former is used to calculate the Newton increment, and this is equivalent to the approximation ofT′. The complexity of the finite element computation for the Newton increment is shown to be of optimal order, via the Vituškin inequality relating metric entropy andn-widths.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01389333
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 4
    Electronic Resource
    Electronic Resource
    L ∞ stability of finite element approximations to elliptic gradient equations (1990)
    Springer
    Numerische Mathematik 57 (1990), S. 561-575 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65N30, 35J60 ; CR: G1.8
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary We examine theL ∞ stability of piecewise linear finite element approximationsU to the solutionu to elliptic gradient equations of the form −∇·[a(x)∇u]+f(x, u)=g(x) wheref is monotonically increasing inu. We identify a prioriL ∞ bounds for the finite element solutionU, which we call “reduced” bounds, and which are marginally weaker than those for the original differential equations. For the general,N-dimensionai, case we identify new conditions on the mesh, such that under the assumption thatf is Lipschitz continuous on a finite interval,U satisfies the “reduced”L ∞ bounds mentioned above. The new,N-dimensional regularity conditions preclude quasi-rectangular meshes. Moreover, we show thatU is stable inL ∞ in two dimensions for a discretization mesh on which −∇·[a(x)∇u] gives rise to anM-matrix, whileU is stable for any mesh in one dimension. The condition that the discretization of −∇·[a(x)∇u] has to be anM-matrix, still allows the inclusion of the important case of triangulating in a quasi-rectangular fashion. The results are valid for either the pure Neumann problem or the general mixed Dirichlet-Neumann boundary value problem, while interfaces may be present. The boundary conditions forU are obtained by use of (nonexpansive) pointwise projection operators.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01386428
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 5
    Electronic Resource
    Electronic Resource
    Linear self-adjoint multipoint boundary value problems and related approximation schemes (1970)
    Springer
    Numerische Mathematik 15 (1970), S. 433-449 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary Linear self-adjoint multipoint boundary value problems are investigated. The case of the homogeneous equation is shown to lead to spline solutions, which are then utilized to construct a Green's function for the case of homogeneous boundary conditions. An approximation scheme is described in terms of the eigen-functions of the inverse of the Green's operator and is shown to be optimal in the sense of then-widths of Kolmogorov. Convergence rates are given and generalizations to more general boundary value problems are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02165513
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 6
    Electronic Resource
    Electronic Resource
    Approximate Newton methods and homotopy for stationary operator equations (1985)
    Springer
    Constructive approximation 1 (1985), S. 271-285 
    Details
    ISSN: 1432-0940
    Keywords: 41A25 ; 47H15 ; 47H17 ; 40A05 ; 65J15 ; 65M15 ; Approximate Newton method ; Bootstrapping lemma ; Continuation, Eigenvalues ; Euler-predictor ; Newton-corrector ; R-quadratic convergence ; Solution set
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A quadratically convergent algorithm based on a Newton-type iteration is defined to approximate roots of operator equations in Banach spaces. Fréchet derivative operator invertibility is not required; approximate right inverses are used in a neighborhood of the root. This result, which requires an initially small residual, is sufficiently robust to yield existence; it may be viewed as a generalized version of the Kantorovich theorem. A second algorithm, based on continuation via single, Euler-predictor-Newton-corrector iterates, is also presented. It has the merit of controlling the residual until the homotopy terminates, at which point the first algorithm applies. This method is capable of yielding existence of a solution curve as well. An application is given for operators described by compact perturbations of the identity.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01890035
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 7
    Electronic Resource
    Electronic Resource
    Convergent approximations in parabolic variational inequalities II: Hamilton-jacobi inequalities (1982)
    Springer
    Applied mathematics & optimization 8 (1982), S. 265-274 
    Details
    ISSN: 1432-0606
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In this paper we consider two-sided parabolic inequalities of the form (li) $$\psi _1 \leqslant u \leqslant \psi _2 , in{\mathbf{ }}Q;$$ (lii) $$\left[ { - \frac{{\partial u}}{{\partial t}} + A(t)u + H(x,t,u,Du)} \right]e \geqslant 0, in{\mathbf{ }}Q,$$ for alle in the convex support cone of the solution given by $$K(u) = \left\{ {\lambda (\upsilon - u):\psi _1 \leqslant \upsilon \leqslant \psi _2 ,\lambda 〉 0} \right\}{\mathbf{ }};$$ (liii) $$\left. {\frac{{\partial u}}{{\partial v}}} \right|_\Sigma = 0, u( \cdot ,T) = \bar u$$ where $$Q = \Omega \times (0,T), \sum = \partial \Omega \times (0,T).$$ Such inequalities arise in the characterization of saddle-point payoffsu in two person differential games with stopping times as strategies. In this case,H is the Hamiltonian in the formulation. A numerical scheme for approximatingu is obtained by the continuous time, piecewise linear, Galerkin approximation of a so-called penalized equation. A rate of convergence tou of orderO(h 1/2) is demonstrated in theL 2(0,T; H 1(Ω)) norm, whereh is the maximum diameter of a given triangulation.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01447762
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 8
    Electronic Resource
    Electronic Resource
    Multistep approximation algorithms: Improved convergence rates through postconditioning with smoothing kernels (1999)
    Springer
    Advances in computational mathematics 10 (1999), S. 1-27 
    Details
    ISSN: 1572-9044
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We show how certain widely used multistep approximation algorithms can be interpreted as instances of an approximate Newton method. It was shown in an earlier paper by the second author that the convergence rates of approximate Newton methods (in the context of the numerical solution of PDEs) suffer from a “loss of derivatives”, and that the subsequent linear rate of convergence can be improved to be superlinear using an adaptation of Nash–Moser iteration for numerical analysis purposes; the essence of the adaptation being a splitting of the inversion and the smoothing into two separate steps. We show how these ideas apply to scattered data approximation as well as the numerical solution of partial differential equations. We investigate the use of several radial kernels for the smoothing operation. In our numerical examples we use radial basis functions also in the inversion step.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1018962112170
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 9
    Electronic Resource
    Electronic Resource
    Newton iteration for partial differential equations and the approximation of the identity (2000)
    Springer
    Numerical algorithms 25 (2000), S. 181-195 
    Details
    ISSN: 1572-9265
    Keywords: Newton methods ; partial differential equations ; approximation of the identity ; Nash iteration
    Source: Springer Online Journal Archives 1860-2000
    Topics: Computer Science , Mathematics
    Notes: Abstract It is known that the critical condition which guarantees quadratic convergence of approximate Newton methods is an approximation of the identity condition. This requires that the composition of the numerical inversion of the Fréchet derivative with the derivative itself approximate the identity to an accuracy calibrated by the residual. For example, the celebrated quadratic convergence theorem of Kantorovich can be proven when this holds, subject to regularity and stability of the derivative map. In this paper, we study what happens when this condition is not evident “a priori” but is observed “a posteriori”. Through an in-depth example involving a semilinear elliptic boundary value problem, and some general theory, we study the condition in the context of dual norms, and the effect upon convergence. We also discuss the connection to Nash iteration.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1023/A:1016609007255
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
  • 10
    Electronic Resource
    Electronic Resource
    Evolution systems in semiconductor device modeling: A cyclic uncoupled line analysis for the gummel map (1987)
    Chichester, West Sussex : Wiley-Blackwell
    Mathematical Methods in the Applied Sciences 9 (1987), S. 455-492 
    Details
    ISSN: 0170-4214
    Keywords: Mathematics and Statistics ; Applied Mathematics
    Source: Wiley InterScience Backfile Collection 1832-2000
    Topics: Mathematics
    Notes: The initial/boundary-value problem for isothermal, lattice, semiconductor device modeling is described and analyzed. This nonlinear elliptic/parabolic system of reaction/diffusion/convection type is determined by a Maxwell equation, relating space/charge and the electric field, and by two continuity equations for the free electron and hole carrier concentrations. The Einstein relations for Brownian motion are not assumed in this analysis, so that the electrostatic potential, u, and the carrier concentrations, n and p, are the fundamental dependent variables of the system. The boundary conditions are Dirichlet conditions for dependent variable values on the contact portions of the device, and homogeneous Neumann conditions, expressing insulation, on the complement. Complicating the analysis are the transition singularity points between the mixed boundary conditions, and the field dependence of the mobility and diffusion coefficients. By means of a physically motivated analysis of the convective current component, we are able to uncouple the system by a cyclic horizontal line analysis, without an unreasonable time step restriction. The corresponding linear equations are solved by a contractive inner iteration. The outer iteration is shown to converge to a unique solution of the system, under singularity classification at the transition points. The definition of this outer iteration follows the steady-state Gummel iteration at discrete time steps. An existence theory is a by-product of the analysis, and is separated from uniqueness theory.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1002/mma.1670090132
    Library Location Call Number Volume/Issue/Year Availability
    BibTip Others were also interested in ...
    Paper (German National Licenses)
    Fulltext
Close ⊗
This website uses cookies and the analysis tool Matomo. More information can be found here...