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L2 Norm (l2 + norm)

Distribution by Scientific Domains
Mathematics and Statistics72%

Selected Abstracts

SEMINONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF CONDITIONAL MOMENT RESTRICTION MODELS,

INTERNATIONAL ECONOMIC REVIEW, Issue 4 2007
Chunrong Ai
This article studies estimation of a conditional moment restriction model with the seminonparametric maximum likelihood approach proposed by Gallant and Nychka (Econometrica 55 (March 1987), 363,90). Under some sufficient conditions, we show that the estimator of the finite dimensional parameter , is asymptotically normally distributed and attains the semiparametric efficiency bound and that the estimator of the density function is consistent under L2 norm. Some results on the convergence rate of the estimated density function are derived. An easy to compute covariance matrix for the asymptotic covariance of the , estimator is presented. [source]

A posteriori error estimation for extended finite elements by an extended global recovery

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 8 2008
Marc Duflot
Abstract This contribution presents an extended global derivative recovery for enriched finite element methods (FEMs), such as the extended FEM along with an associated error indicator. Owing to its simplicity, the proposed scheme is ideally suited to industrial applications. The procedure is based on global minimization of the L2 norm of the difference between the raw strain field (C,1) and the recovered (C0) strain field. The methodology engineered in this paper extends the ideas of Oden and Brauchli (Int. J. Numer. Meth. Engng 1971; 3) and Hinton and Campbell (Int. J. Numer. Meth. Engng 1974; 8) by enriching the approximation used for the construction of the recovered derivatives (strains) with the gradients of the functions employed to enrich the approximation employed for the primal unknown (displacements). We show linear elastic fracture mechanics examples, both in simple two-dimensional settings, and for a three-dimensional structure. Numerically, we show that the effectivity index of the proposed indicator converges to unity upon mesh refinement. Consequently, the approximate error converges to the exact error, indicating that the error indicator is valid. Additionally, the numerical examples suggest a novel adaptive strategy for enriched approximations in which the dimensions of the enrichment zone are first increased, before standard h - and p -adaptivities are applied; we suggest to coin this methodology e-adaptivity. Copyright © 2008 John Wiley & Sons, Ltd. [source]

Inverse design of directional solidification processes in the presence of a strong external magnetic field

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2001
Rajiv Sampath
Abstract A computational method for the design of directional alloy solidification processes is addressed such that a desired growth velocity ,f under stable growth conditions is achieved. An externally imposed magnetic field is introduced to facilitate the design process and to reduce macrosegregation by the damping of melt flow. The design problem is posed as a functional optimization problem. The unknowns of the design problem are the thermal boundary conditions. The cost functional is taken as the square of the L2 norm of an expression representing the deviation of the freezing interface thermal conditions from the conditions corresponding to local thermodynamic equilibrium. The adjoint method for the inverse design of continuum processes is adopted in this work. A continuum adjoint system is derived to calculate the adjoint temperature, concentration, velocity and electric potential fields such that the gradient of the L2 cost functional can be expressed analytically. The cost functional minimization process is realized by the conjugate gradient method via the FE solutions of the continuum direct, sensitivity and adjoint problems. The developed formulation is demonstrated with an example of designing the boundary thermal fluxes for the directional growth of a germanium melt with dopant impurities in the presence of an externally applied magnetic field. The design is shown to achieve a stable interface growth at a prescribed desired growth rate. Copyright © 2001 John Wiley & Sons, Ltd. [source]

Multigrid methods for the symmetric interior penalty method on graded meshes

NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2009
S. C. Brenner
Abstract The symmetric interior penalty (SIP) method on graded meshes and its fast solution by multigrid methods are studied in this paper. We obtain quasi-optimal error estimates in both the energy norm and the L2 norm for the SIP method, and prove uniform convergence of the W -cycle multigrid algorithm for the resulting discrete problem. The performance of these methods is illustrated by numerical results. Copyright © 2009 John Wiley & Sons, Ltd. [source]

A CFL-free explicit characteristic interior penalty scheme for linear advection-reaction equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 3 2010
Kaixin Wang
Abstract We develop a CFL-free, explicit characteristic interior penalty scheme (CHIPS) for one-dimensional first-order advection-reaction equations by combining a Eulerian-Lagrangian approach with a discontinuous Galerkin framework. The CHIPS method retains the numerical advantages of the discontinuous Galerkin methods as well as characteristic methods. An optimal-order error estimate in the L2 norm for the CHIPS method is derived and numerical experiments are presented to confirm the theoretical estimates. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010 [source]

Modification of upwind finite difference fractional step methods by the transient state of the semiconductor device

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 2 2008
Yirang Yuan
Abstract The mathematical model of the three-dimensional semiconductor devices of heat conduction is described by a system of four quasi-linear partial differential equations for initial boundary value problem. One equation of elliptic form is for the electric potential; two equations of convection-dominated diffusion type are for the electron and hole concentration; and one heat conduction equation is for temperature. Upwind finite difference fractional step methods are put forward. Some techniques, such as calculus of variations, energy method multiplicative commutation rule of difference operators, decomposition of high order difference operators, and the theory of prior estimates and techniques are adopted. Optimal order estimates in L2 norm are derived to determine the error in the approximate solution.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008 [source]

Alternating direction finite volume element methods for 2D parabolic partial differential equations

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008
Tongke Wang
Abstract On the basis of rectangular partition and bilinear interpolation, this article presents alternating direction finite volume element methods for two dimensional parabolic partial differential equations and gives three computational schemes, one is analogous to Douglas finite difference scheme with second order splitting error, the second has third order splitting error, and the third is an extended locally one dimensional scheme. Optimal L2 norm or H1 semi-norm error estimates are obtained for these schemes. Finally, two numerical examples illustrate the effectiveness of the schemes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source]

Error analysis of the L2 least-squares finite element method for incompressible inviscid rotational flows,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004
Chiung-Chiou Tsai
Abstract In this article we analyze the L2 least-squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity-vorticity-pressure formulation. The least-squares functional is defined in terms of the sum of the squared L2 norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first-order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H1 norm for velocity and pressure and a suboptimal rate of convergence in the L2 norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source]

The upwind finite difference fractional steps methods for two-phase compressible flow in porous media

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2003
Yirang Yuan
Abstract The upwind finite difference fractional steps methods are put forward for the two-phase compressible displacement problem. Some techniques, such as calculus of variations, multiplicative commutation rule of difference operators, decomposition of high-order difference operators, and prior estimates, are adopted. Optimal order estimates in L2 norm are derived to determine the error in the approximate solution. This method has already been applied to the numerical simulation of seawater intrusion and migration-accumulation of oil resources. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 67,88, 2003 [source]

Error analysis of the L2 least-squares finite element method for incompressible inviscid rotational flows,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 6 2004
Chiung-Chiou Tsai
Abstract In this article we analyze the L2 least-squares finite element approximations to the incompressible inviscid rotational flow problem, which is recast into the velocity-vorticity-pressure formulation. The least-squares functional is defined in terms of the sum of the squared L2 norms of the residual equations over a suitable product function space. We first derive a coercivity type a priori estimate for the first-order system problem that will play the crucial role in the error analysis. We then show that the method exhibits an optimal rate of convergence in the H1 norm for velocity and pressure and a suboptimal rate of convergence in the L2 norm for vorticity. A numerical example in two dimensions is presented, which confirms the theoretical error estimates. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source]

Lagrange interpolation and finite element superconvergence,

NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2004
Bo Li
Abstract We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d -dimensional Qk -type elements with d , 1 and k , 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H1 norm. For d -dimensional Pk -type elements, we consider the standard Lagrange interpolation,the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d , 2 and k , d + 1 that such interpolation and the finite element solution are not superclose in both H1 and L2 norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33,59, 2004. [source]