Method of mean weighted residuals

About: Method of mean weighted residuals is a research topic. Over the lifetime, 1344 publications have been published within this topic receiving 35146 citations. The topic is also known as: Method of Weighted Residuals.

Concepts and Applications of Finite Element Analysis

Abstract: Notation. Introduction. One-Dimensional Elements, Computational Procedures. Basic Elements. Formulation Techniques: Variational Methods. Formulation Techniques: Galerkin and Other Weighted Residual Methods. Isoparametric Elements. Isoparametric Triangles and Tetrahedra. Coordinate Transformation and Selected Analysis Options. Error, Error Estimation, and Convergence. Modeling Considerations and Software Use. Finite Elements in Structural Dynamics and Vibrations. Heat Transfer and Selected Fluid Problems. Constaints: Penalty Forms, Locking, and Constraint Counting. Solid of Revolution. Plate Bending. Shells. Nonlinearity: An Introduction. Stress Stiffness and Buckling. Appendix A: Matrices: Selected Definition and Manipulations. Appendix B: Simultaneous Algebraic Equations. Appendix C: Eigenvalues and Eigenvectors. References. Index.

Tests for Specification Errors in Classical Linear Least-Squares Regression Analysis

Abstract: SUMMARY The effects on the distribution of least-squares residuals of a series of model mis-specifications are considered. It is shown that for a variety of specification errors the distributions of the least-squares residuals are normal, but with non-zero means. An alternative predictor of the disturbance vector is used in developing four procedures for testing for the presence of specification error. The specification errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity. THE objectives of this paper are two. The first is to derive the distributions of the classical linear least-squares residuals under a variety of specification errors. The errors considered are omitted variables, incorrect functional form, simultaneous equation problems and heteroskedasticity. It is assumed that the disturbance terms are independently and normally distributed. It will be shown that the effect of the specification errors considered above is, with the exception of the error of heteroskedasticity, to yield residuals which though normally distributed do not have zero means, so that the distribution of the squared residuals is non-central x2. The second objective is to derive procedures to test for the presence of the specification errors considered in the first part of the paper. The tests are developed by comparing the distribution of residuals under the hypothesis that the specification of the model is correct to the distribution of the residuals yielded under the alternative hypothesis that there is a specification error of one of the types considered in the first part of the paper. As a preliminary step to deriving the test procedures the classical least-squares residual vector is transformed to a sub-vector which has more desirable properties for testing the null hypothesis that the specification of the model is correct. Also, under certain assumptions, with respect to the alternative hypothesis, it is shown that the mean vector of the residuals can be approximated by a linear sum of vectors qj,

Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems

Abstract: 1. Introduction.- 1.1 Problem classification.- 1.2 Problem formulation and exact definition of the subject.- 1.2.1 Application of the different models.- 1.2.2 Remarks on the term model.- 1.2.3 Objective and structure of this book.- 2. Fundamental principles of conceptual modeling.- 2.1 Preliminary remarks.- 2.1.1 General remarks.- 2.1.2 Definitions and fundamental terms.- 2.2 System properties.- 2.2.1 Mass and mole fractions.- 2.2.2 Density.- 2.2.3 Viscosity.- 2.2.4 Specific enthalpy, specific internal energy.- 2.2.5 Surface tension.- 2.2.6 Specific heat capacity.- 2.3 Phase state, phase transition, phase change.- 2.3.1 Phase state.- 2.3.2 Phase transition, phase change.- 2.4 Capillarity.- 2.4.1 Microscopic capillarity.- 2.4.2 Macroscopic capillarity.- 2.4.3 Capillarity in fractures.- 2.5 Hysteresis.- 2.6 Definition of different saturations.- 2.7 Relative permeability.- 2.7.1 Permeability.- 2.7.2 Relative permeability at the micro scale.- 2.7.3 Relative permeability at the macro scale.- 2.7.4 Relative permeability-saturation relation in fractures.- 2.7.5 Fracture-matrix interaction.- 2.8 Pressure and temperature dependence of porosity.- Mathematical modeling.- 3.1 General balance equation.- 3.1.1 Preconditions and assumptions.- 3.1.2 The Reynolds transport theorem in integral form.- 3.1.3 Derivation of the general balance equation.- 3.1.4 Initial and boundary conditions.- 3.1.5 Choice of the primary variables.- 3.2 Continuity equation per phase.- 3.2.1 Time derivative.- 3.3 Momentum equation and Darcy's law.- 3.3.1 General remarks.- 3.3.2 Darcy's law of single-phase flow.- 3.3.3 Generalization of Darcy's law for multiphase flow.- 3.4 General form of the multiphase flow equation.- 3.4.1 Pressure formulation.- 3.4.2 Pressure-saturation formulation.- 3.4.3 Saturation formulation.- 3.4.4 Mathematical modeling for three-phase infiltration and remobilization processes.- 3.5 Transport equation.- 3.5.1 Basic transport equation.- 3.5.2 Transport in a multiphase system.- 3.5.3 Description of the mass transfer between phases.- 3.5.4 Multicomponent transport processes in the gas phase.- 3.6 Energy equation.- 3.7 Multiphase/multicomponent system.- 4. Numerical modeling.- 4.1 Classification.- 4.1.1 Problem and special solution methods.- 4.1.2 Fundamentals of discretization.- 4.1.3 Conservative discretization.- 4.1.4 Weighted residual method.- 4.2 Finite element and finite volume methods.- 4.2.1 Spatial discretization.- 4.2.2 Choice of element types.- 4.2.3' Galerkin finite element method.- 4.2.4 Sub domain collocation - finite volume method.- 4.2.5 Time discretization.- 4.3 Linearization of the multiphase problem.- 4.3.1 Weak nonlinearities.- 4.3.2 Strong nonlinearities.- 4.3.3 Handling of the nonlinearities.- 4.3.4 Example: Linearized two-phase equation.- 4.4 Discussion of the instationary hyperbolic (convective) transport equation.- 4.4.1 Classification of hyperbolic differential equations.- 4.4.2 A linear hyperbolic transport equation.- 4.4.3 A quasilinear hyperbolic transport equation - Buckley-Levereit equation.- 4.4.4 Analytical solutions for the Buckley-Lev ereit problem.- 4.5 Special discretization methods.- 4.5.1 Motivation.- 4.5.2 Upwind method - finite difference method.- 4.5.3 Explicit upwind method of first order - Fully Upwind.- 4.5.4 Multidimensional upwind method of first order.- 4.5.5 Explicit upwind method of higher order - TVD techniques.- 4.5.6 Implicit upwind method of first order - Fully Upwind.- 4.5.7 Petrov-Galerkin finite element method.- 4.5.8 Additional remarks on conservative discretization.- 4.5.9 Flux-corrected method.- 4.5.10 Mixed-hybrid finite element methods.- 5. Comparison of the different discretization methods.- 5.1 Discretization.- 5.1.1 Finite element Galerkin method.- 5.1.2 Sub domain collocation finite volume method (box method).- 5.2 Boundedness principle - discussion of a monotonic solution.- 5.3 Comparative study of the different methods in homogeneous porous media.- 5.3.1 Multiphase flow without capillary pressure effects - Buckley-Lev ereit problem.- 5.3.2 Multiphase flow with capillary pressure effects - McWhorter problem.- 5.4 Heterogeneity effects.- 5.5 Comparative study of the methods for flow in heterogeneous porous media.- 5.6 Five-spot waterflood problem.- 6. Test problems - applications.- 6.1 DNAPL-Infiltration.- 6.2 LNAPL-Infiltration.- 6.3 Non-isothermal multiphase/multicomponent flow.- 6.3.1 Heat pipe.- 6.3.2 Study of bench-scale experiments.- 7. Final remarks.

Randomized Quantile Residuals

Abstract: In this article we give a general definition of residuals for regression models with independent responses Our definition produces residuals that are exactly normal, apart from sampling variability in the estimated parameters, by inverting the fitted distribution function for each response value and finding the equivalent standard normal quantile Our definition includes some randomization to achieve continuous residuals when the response variable is discrete Quantile residuals are easily computed in computer packages such as SAS, S-Plus, GLIM, or LispStat, and allow residual analyses to be carried out in many commonly occurring situations in which the customary definitions of residuals fail Quantile residuals are applied in this article to three example data sets