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  • 1
    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
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  • 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
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    Paper (German National Licenses)
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  • 3
    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
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  • 4
    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
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    Paper (German National Licenses)
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