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Two perturbation methods for the non-Markovian quantum state diffusion (NMQSD) equation are investigated. The first perturbation method under investigation is based on a functional expansion of the NMQSD equation, while the second one expands the NMQSD equation in terms of the coupling strength. We have compared the advantages of the two methods based on bipartite systems where the accuracy of both perturbation methods can be examined by comparing the approximations with the exact solutions. Additionally, we provide an analytical solution for a special family of system's initial states, and the entanglement dynamics is discussed based on this solution.

We demonstrate via several examples how the backward error viewpoint can be used in the analysis of solutions obtained by perturbation methods. We show that this viewpoint is quite general and offers several important advantages. Perhaps the most important is that backward error analysis can be used to demonstrate the validity of the solution, however obtained and by whichever method. This includes a nontrivial safeguard against slips, blunders, or bugs in the original computation. We also demonstrate its utility in deciding when to truncate an asymptotic series, improving on the well-known rule of thumb indicating truncation just prior to the smallest term. We also give an example of elimination of spurious secular terms even when genuine secularity is present in the equation. We give short expositions of several well-known perturbation methods together with computer implementations (as scripts that can be modified). We also give a generic backward error based method that is equivalent to iteration (but we believe useful as an organizational viewpoint) for regular perturbation.

Bell System Technical Journal, 43: 3. May 1964 pp 847-884. Perturbation Methods for Satellite Orbits. (Geyling, F.T.)

Bell System Technical Journal, 43: 3. May 1964 pp 847-884. Perturbation Methods for Satellite Orbits. (Geyling, F.T.)

Bell System Technical Journal, 43: 3. May 1964 pp 847-884. Perturbation Methods for Satellite Orbits. (Geyling, F.T.)

In this work, we carefully study the energy eigen-values and splitting of heavy quarkonia as there exist $1/r^3$ and $\delta^3(\vec r)$ singular terms in the potential which make a direct numerical solution of the Schr\"{o}dinger equation impossible. We compare the results obtained in terms of perturbation and variational methods with various treatments.

We investigate the differentiability of minimal average energy associated to the functionals $S_\ep (u) = \int_{\mathbb{R}^d} (1/2)|\nabla u|^2 + \ep V(x,u)\, dx$, using numerical and perturbative methods. We use the Sobolev gradient descent method as a numerical tool to compute solutions of the Euler-Lagrange equations with some periodicity conditions; this is the cell problem in homogenization. We use these solutions to determine the average minimal energy as a function of the slope. We also obtain a representation of the solutions to the Euler-Lagrange equations as a Lindstedt series in the perturbation parameter $\ep$, and use this to confirm our numerical results. Additionally, we prove convergence of the Lindstedt series.

In this work we present a new methodology for orbit propagation, the hybrid perturbation theory, based on the combination of an integration method and a prediction technique. The former, which can be a numerical, analytical or semianalytical theory, generates an initial approximation that contains some inaccuracies derived from the fact that, in order to simplify the expressions and subsequent computations, not all the involved forces are taken into account and only low-order terms are considered, not to mention the fact that mathematical models of perturbations not always reproduce physical phenomena with absolute precision. The prediction technique, which can be based on either statistical time series models or computational intelligence methods, is aimed at modelling and reproducing missing dynamics in the previously integrated approximation. This combination results in the precision improvement of conventional numerical, analytical and semianalytical theories for determining the position and velocity of any artificial satellite or space debris object. In order to validate this methodology, we present a family of three hybrid orbit propagators formed by the combination of three different orders of approximation of an analytical theory and a statistical time series model, and analyse their capability to process the effect produced by the flattening of the Earth. The three considered analytical components are the integration of the Kepler problem, a first-order and a second-order analytical theories, whereas the prediction technique is the same in the three cases, namely an additive Holt-Winters method.

In this work we present a new methodology for orbit propagation, the hybrid perturbation theory, based on the combination of an integration method and a prediction technique. The former, which can be a numerical, analytical or semianalytical theory, generates an initial approximation that contains some inaccuracies derived from the fact that, in order to simplify the expressions and subsequent computations, not all the involved forces are taken into account and only low-order terms are considered, not to mention the fact that mathematical models of perturbations not always reproduce physical phenomena with absolute precision. The prediction technique, which can be based on either statistical time series models or computational intelligence methods, is aimed at modelling and reproducing missing dynamics in the previously integrated approximation. This combination results in the precision improvement of conventional numerical, analytical and semianalytical theories for determining the position and velocity of any artificial satellite or space debris object. In order to validate this methodology, we present a family of three hybrid orbit propagators formed by the combination of three different orders of approximation of an analytical theory and a statistical time series model, and analyse their capability to process the effect produced by the flattening of the Earth. The three considered analytical components are the integration of the Kepler problem, a first-order and a second-order analytical theories, whereas the prediction technique is the same in the three cases, namely an additive Holt-Winters method.

In this work we present a new methodology for orbit propagation, the hybrid perturbation theory, based on the combination of an integration method and a prediction technique. The former, which can be a numerical, analytical or semianalytical theory, generates an initial approximation that contains some inaccuracies derived from the fact that, in order to simplify the expressions and subsequent computations, not all the involved forces are taken into account and only low-order terms are considered, not to mention the fact that mathematical models of perturbations not always reproduce physical phenomena with absolute precision. The prediction technique, which can be based on either statistical time series models or computational intelligence methods, is aimed at modelling and reproducing missing dynamics in the previously integrated approximation. This combination results in the precision improvement of conventional numerical, analytical and semianalytical theories for determining the position and velocity of any artificial satellite or space debris object. In order to validate this methodology, we present a family of three hybrid orbit propagators formed by the combination of three different orders of approximation of an analytical theory and a statistical time series model, and analyse their capability to process the effect produced by the flattening of the Earth. The three considered analytical components are the integration of the Kepler problem, a first-order and a second-order analytical theories, whereas the prediction technique is the same in the three cases, namely an additive Holt-Winters method.

In a previous paper (J. Phys. A 36, 11807 (2003)), we introduced the `asymptotic iteration method' for solving second-order homogeneous linear differential equations. In this paper, we study perturbed problems in quantum mechanics and we use the method to find the coefficients in the perturbation series for the eigenvalues and eigenfunctions directly, without first solving the unperturbed problem.

A reduced drift-diffusion (Smoluchowski-Poisson) equation is found for the electric charge in the high-field limit of the Vlasov-Poisson-Fokker-Planck system, both in one and three dimensions. The corresponding electric field satisfies a Burgers equation. Three methods are compared in the one-dimensional case: Hilbert expansion, Chapman-Enskog procedure and closure of the hierarchy of equations for the moments of the probability density. Of these methods, only the Chapman-Enskog method is able to systematically yield reduced equations containing terms of different order.

We study the problem of existence of surfaces in ${\bf R}^3$ parametrized on the sphere ${\mathbb S}^2$ with prescribed mean curvature $H$ in the perturbative case, i.e. for $H=H_0+\epsilon H_1$, where $H_0$ is a nonzero constant, $H_1$ is a $C^2$ function and $\epsilon$ is a small perturbation parameter.

We extend the perturbation theory of Vishik, Ljusternik and Lidskii for eigenvalues of matrices, using methods of min-plus algebra. We show that the asymptotics of the eigenvalues of a perturbed matrix is governed by certain discrete optimisation problems, from which we derive new perturbation formulae, extending the classical ones and solving cases which where singular in previous approaches. Our results include general weak majorisation inequalities, relating leading exponents of eigenvalues of perturbed matrices and min-plus analogues of eigenvalues.

Book Source: Digital Library of India Item 2015.194121 dc.contributor.author: Kailash C. Patidar dc.date.accessioned: 2015-07-08T03:58:26Z dc.date.available: 2015-07-08T03:58:26Z dc.date.digitalpublicationdate: 2005-08-28 dc.identifier.barcode: 1990010092384 dc.identifier.origpath: /rawdataupload/upload/0092/384 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/194121 dc.description.scannerno: 5 dc.description.scanningcentre: IIIT, Allahabad dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 266 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher: Indian Institute Of Technology Kanpur dc.rights: In Public Domain dc.source.library: Indian Institute Of Technology Kanpur dc.subject.classification: Natural Sciences dc.subject.classification: Mathematics dc.subject.classification: Geometry dc.title: Spline Methods For The Numerical Solution Of Singular Perturbation Problems

The convex feasibility problem (CFP) is at the core of the modeling of many problems in various areas of science. Subgradient projection methods are important tools for solving the CFP because they enable the use of subgradient calculations instead of orthogonal projections onto the individual sets of the problem. Working in a real Hilbert space, we show that the sequential subgradient projection method is perturbation resilient. By this we mean that under appropriate conditions the sequence generated by the method converges weakly, and sometimes also strongly, to a point in the intersection of the given subsets of the feasibility problem, despite certain perturbations which are allowed in each iterative step. Unlike previous works on solving the convex feasibility problem, the involved functions, which induce the feasibility problem's subsets, need not be convex. Instead, we allow them to belong to a wider and richer class of functions satisfying a weaker condition, that we call "zero-convexity". This class, which is introduced and discussed here, holds a promise to solve optimization problems in various areas, especially in non-smooth and non-convex optimization. The relevance of this study to approximate minimization and to the recent superiorization methodology for constrained optimization is explained.

We investigate projected scaled gradient (PSG) methods for convex minimization problems. These methods perform a descent step along a diagonally scaled gradient direction followed by a feasibility regaining step via orthogonal projection onto the constraint set. This constitutes a generalized algorithmic structure that encompasses as special cases the gradient projection method, the projected Newton method, the projected Landweber-type methods and the generalized Expectation-Maximization (EM)-type methods. We prove the convergence of the PSG methods in the presence of bounded perturbations. This resilience to bounded perturbations is relevant to the ability to apply the recently developed superiorization methodology to PSG methods, in particular to the EM algorithm.

The inherently homogeneous stationary-state and time-dependent Schroedinger equations are often recast into inhomogeneous form in order to resolve their solution nonuniqueness. The inhomogeneous term can impose an initial condition or, for scattering, the preferred permitted asymptotic behavior. For bound states it provides sufficient focus to exclude all but one of the homogeneous version's solutions. Because of their unique solutions, such inhomogeneous versions of Schroedinger equations have long been the indispensable basis for a solution scheme of successive perturbational corrections which are anchored by their inhomogeneous term. Here it is noted that every such perturbational solution scheme for an inhomogeneous linear vector equation spins off a nonperturbational continued-fraction scheme. Unlike its representation-independent antecedent, the spin-off scheme only works in representations where all components of the equation's inhomogeneous term are nonzero. But that requirement seems to confer theoretical physics robustness heretofore unknown: for quantum fields the order of the perturbation places a bound on unperturbed particle number, the spin-off scheme contrariwise has only basis elements of unbounded unperturbed particle number. It furthermore is difficult to visualize such a continued-fraction spin-off scheme generating infinities, since its successive iterations always go into denominators.

In this report a single longitudinal tracker is developed for the aircraft for three different flight conditions. The method used is the singular perturbation method applied to fast-sampling, output-feedback digital control. Each flight condition has three command modes: positive pitch pointing, vertical translation and straight climb. A sensitivity study is performed to validate the design and illustrate design parameter influences on system response. A computer-aided-design program, MULTI, is developed to assist in the iterative design process. The program is fully interactive, user-oriented, and provides error protection. The program allows complete design and simulation of three types of control law designs: known-regular plants, known irregular plants, and unknown plants. The report contains a brief but complete summary of each of these control law design methods. A user's manual and a programmer's manual are provided for further development of the program. (Author)

In this report a single longitudinal tracker is developed for the aircraft for three different flight conditions. The method used is the singular perturbation method applied to fast-sampling, output-feedback digital control. Each flight condition has three command modes: positive pitch pointing, vertical translation and straight climb. A sensitivity study is performed to validate the design and illustrate design parameter influences on system response. A computer-aided-design program, MULTI, is developed to assist in the iterative design process. The program is fully interactive, user-oriented, and provides error protection. The program allows complete design and simulation of three types of control law designs: known-regular plants, known irregular plants, and unknown plants. The report contains a brief but complete summary of each of these control law design methods. A user's manual and a programmer's manual are provided for further development of the program. (Author)

Chiral lagrangians describing the interactions of Goldstone bosons in a theory possessing spontaneous symmetry breaking are effective, non-renormalizable field theories in four dimensions. Yet, in a momentum expansion one is able to extract definite, testable predictions from perturbation theory. These techniques have yielded in recent years a wealth of information on many problems where the physics of Goldstone bosons plays a crucial role, but theoretical issues concerning chiral perturbation theory remain, to this date, poorly treated in the literature. We present here a rather comprehensive analysis of the regularization and renormalization ambiguities appearing in chiral perturbation theory at the one loop level. We discuss first on the relevance of dealing with tadpoles properly. We demonstrate that Ward identities severely constrain the choice of regulators to the point of enforcing unique, unambiguous results in chiral perturbation theory at the one-loop level for any observable which is renormalization-group invariant. We comment on the physical implications of these results and on several possible regulating methods that may be of use for some applications.

Two possibile applications of the optimized expansion for the free energy of the quantum-mechanical anharmonic oscillator are discussed. The first method is for the finite temperature effective potential; the second one, for the classical effective potential. The results of both methods show a quick convergence and agree well with the exact free energy in the whole range of temperatures. Postscript figures are available under request to AO email [email protected]

Two possibile applications of the optimized expansion for the free energy of the quantum-mechanical anharmonic oscillator are discussed. The first method is for the finite temperature effective potential; the second one, for the classical effective potential. The results of both methods show a quick convergence and agree well with the exact free energy in the whole range of temperatures. Postscript figures are available under request to AO email [email protected]

Two possibile applications of the optimized expansion for the free energy of the quantum-mechanical anharmonic oscillator are discussed. The first method is for the finite temperature effective potential; the second one, for the classical effective potential. The results of both methods show a quick convergence and agree well with the exact free energy in the whole range of temperatures. Postscript figures are available under request to AO email [email protected]

Two possibile applications of the optimized expansion for the free energy of the quantum-mechanical anharmonic oscillator are discussed. The first method is for the finite temperature effective potential; the second one, for the classical effective potential. The results of both methods show a quick convergence and agree well with the exact free energy in the whole range of temperatures. Postscript figures are available under request to AO email [email protected]

40 p. 28 cm

We study the variation in the Land$\acute{\text{e}}$ g-factor of electron spins induced by both anisotropic gate potentials and magnetic fields in InAs quantum dots for possible implementation towards solid state quantum computing. In this paper, we present analytical expressions and numerical simulations of the variation in the Land$\acute{\text{e}}$ g-factor for both isotropic and anisotropic quantum dots. Using both analytical techniques and numerical simulations, we show that the Rashba spin-orbit coupling has a major contribution in the variation of the g-factor with electric fields before the regime, where level crossing or anticrossing occurs. In particular, the electric field tunability is shown to cover a wide range of g-factor through strong Rashba spin-orbit interaction. Another major result of this paper is that the anisotropic gate potential gives quenching effect in the orbital angular momentum that reduces the variation in the E-field and B-field tunability of the g-factor if the area of the symmetric and asymmetric quantum dots is held constant. We identify level crossings and anticrossings of the electron states in the variation of the Land$\acute{\text{e}}$ g-factor. We model the wavefunctions of electron spins and estimate the size of the anticrossing for the spin states $|0,-1,+1/2>$ and $|0,0,-1/2>$ corresponding to a quantum dot that has been recently studied experimentally (Phys. Rev. Lett. \textbf{104}, 246801 (2010)).

We present a method to automatically derive the Feynman rules for mesonic chiral perturbation theory with a lattice regulator. The Feynman rules can be output both in a human-readable format and in a form suitable for an automated numerical evaluation of lattice Feynman diagrams. The automated method significantly simplifies working with improved or extended actions. Some applications to the study of finite-volume effects will be presented.

We consider the problem of finding optimal parameters under simulation optimization setup. For a $p$-dimensional parameter optimization, the classical Kiefer-Wolfowitz Finite Difference Stochastic Approximation (FDSA) scheme uses $p+1$ or $2p$ simulations of the system feedback for one-sided and two-sided gradient estimates respectively. The dependence on the dimension $p$ makes FDSA impractical for high dimensional problems. An alternative approach for gradient estimation in high dimensional problems is the simultaneous perturbation technique that appears in [1],[2]. The main idea in this approach is to estimate the gradient by using only two settings of the $p$-dimensional parameter being optimized. The two settings of the parameter are obtained by simultaneously perturbing all the components of the parameter by adding a random direction. A drawback of using random directions for the gradient estimate is the very large or possibly infinite range of these random directions (for e.g. $\pm 1$ symmetric Bernoulli perturbations typically used in 1SPSA algorithm has a range of cardinality $2^p$ ). In this article we consider deterministic perturbations with a range of cardinality $p+1$ to improve the convergence of these algorithms. A novel construction of deterministic perturbations based on specially chosen circulant matrix is proposed. Convergence analysis of the proposed algorithms is presented along with numerical experiments.

This study utilizes ultrasonic water tank modeling to examine three- dimensional scattering trends from a random set of parallel grooves, and compares this with theoretical results obtained from two-dimensional finite- difference calculations. Ultrasonic laboratory modeling is carried out using computer-controlled source and receivers with an aluminum bock submerged in a water tank. The block's upper interface is plane for the reference model and grooved for the test model. The grooves measure about one-third the center source wavelength and have a Gaussian distribution with a mean of 1 wavelength and a standard deviation of 1/3 wavelength. This experiment places both the source and receiver at the water's surface with the receiver array in the horizontal plane. The receiver line is then positioned at various angles to grooves. A staggered-grid finite-difference scheme is used for theoretical computations and comparisons with laboratory data. These theoretical results matched experimental data well for both the plane interface and the grooved model. Specifically, this study shows that scattering mechanisms are different for propagation normal to grooves than those parallel to the grooves. In the first case scattering takes place in the form of point diffractors. This causes reduction of the specular reflections. Amplitudes decrease by more than 60%, relative to a plane interface, when the incidence angle exceeds 45 degrees. 'Snapshots' of finite-difference synthetics helped to clarify details of scattering. In the second case, where the wave front is parallel to the grooves, scattering takes e form of guided head waves and continuous diffractions giving rise to constructive and destructive interference. This gives the illusion of 'broken' reflectors at depth.

Variational methods and degenerate perturbation theory

I discuss a recent application of homotopy perturbation and Adomian decomposition methods to the linear and nonlinear Schr\"odinger equations. I propose a generalization of the procedure for the treatment of a wider class of problems.

Continuing the program begun by the authors in a previous paper, we develop an exact low-density expansion for the random minimum spanning tree (MST) on a finite graph, and use it to develop a continuum perturbation expansion for the MST on critical percolation clusters in space dimension d. The perturbation expansion is proved to be renormalizable in d=6 dimensions. We consider the fractal dimension D_p of paths on the latter MST; our previous results lead us to predict that D_p=2 for d>d_c=6. Using a renormalization-group approach, we confirm the result for d>6, and calculate D_p to first order in \epsilon=6-d for d\leq 6 using the connection with critical percolation, with the result D_p = 2 - \epsilon/7 + O(\epsilon^2).

For singular perturbation problems in dynamical systems, various appropriate singular perturbation methods have been proposed to eliminate secular terms appearing in the naive expansion. For example, the method of multiple time scales, the normal form method, center manifold theory, the renormalization group method are well known. In this paper, it is shown that all of the solutions of the reduced equations constructed with those methods are exactly equal to sum of the most divergent secular terms appearing in the naive expansion. For the proof, a method to construct a perturbation solution which differs from the conventional one is presented, where we make use of the theory of Lie symmetry group.

We consider systems governed by partial differential equations with spatially periodic coefficients over unbounded domains. These spatially periodic systems are considered as perturbations of spatially invariant ones, and we develop perturbation methods to study their stability and H2 system norm. The operator Lyapunov equations characterizing the H2 norm are studied using a special frequency representation, and formulae are given for the perturbation expansion of their solution. The structure of these equations allows for a recursive method for solving for the expansion terms. Our analysis provides conditions that capture possible resonances between the periodic coefficients and the spatially invariant part of the system. These conditions can be regarded as useful guidelines when spatially periodic coefficients are to be designed to increase/decrease the H2 norm of a spatially distributed system. The developed perturbation framework also gives simple conditions for checking exponential stability.

Linear problems of central interest in numerical analysis are the solution of linear equations, the construction of the inverse or a generalized inverse of a linear operator, finding the eigenvalues and eigenvectors of a linear operator, and linear programming. A survey is made of methods which apply if the data of a solved linear problem is perturbed by operators and vectors of small norm (analytic perturbation), or by operators of finite rank and vectors belonging to a finite-dimensional subspace (algebraic perturbation). Perturbation methods may be used to extend the theory of linear problems, to obtain economy of effort in the solution of perturbed problems, and to estimate errors due to inaccurate data and computation. (Author)

Progress has been made on understanding the complex behavior of physical processes described by nonlinear ordinary and partial differential equations through the use of singular perturbation methods. Modulation equations for the amplitude and phase of dissipatively perturbed strongly nonlinear oscillators and traveling waves have been derived from the action equation using the usual method of multiple scales. Equivalent results have been obtained using the method of averaging developed for the first time for a nonlinear partial differential equation, the Klein-Gordon equation, describing dispersive waves. In another study, Whitham's averaged Lagrangian principle has been generalized to account for arbitrary perturbations of the initial conditions. In other work, Bourland and Haberman analyzed the slow crossing of an unperturbed homoclinic orbit (separatrix) for dynamical systems. Solutions in the neighborhood of the separatrix are matched to the nonlinear slowly varying oscillations, resulting in the determination of accurate analytic formulas for the boundaries of the basin of attraction and connection formulas across the separatrix for the amplitude and phase. Under current investigation are generalizations of the slow crossing of a separatrix to arbitrary Hamiltonian systems and to nonchaotic situations in which small periodic forcing causes the existence of an infinite sequence of resonance layers that coalesce on the separatrix.

The purpose of this paper is to compare two distinct methods to extract a linear state-space model about a reference flight condition from a nonlinear simulation. The frequency response method uses a time history input which contains the frequencies of interest to drive the simulation. The frequency response method uses a time history input which contains the frequencies of interest to drive the simulation frequency input and the output of the simulation are transformed to the frequency domain, and the desired frequency responses of the simulation are calculated. A linear model is then fit to the frequency responses using system identification techniques. The perturbation method extracts a linear model by perturbing the model states and inputs from the reference flight condition and calculating the resulting model coefficients. Both method were used to extract a fourth order longitudinal state-space model from the V-22 full nonlinear simulation. The time history responses and system matrices of the extracted model were compared. The comparison showed that both methods are effective means to reduce a nonlinear simulation to a linear state-space model.

The purpose of this paper is to compare two distinct methods to extract a linear state-space model about a reference flight condition from a nonlinear simulation. The frequency response method uses a time history input which contains the frequencies of interest to drive the simulation. The frequency response method uses a time history input which contains the frequencies of interest to drive the simulation frequency input and the output of the simulation are transformed to the frequency domain, and the desired frequency responses of the simulation are calculated. A linear model is then fit to the frequency responses using system identification techniques. The perturbation method extracts a linear model by perturbing the model states and inputs from the reference flight condition and calculating the resulting model coefficients. Both method were used to extract a fourth order longitudinal state-space model from the V-22 full nonlinear simulation. The time history responses and system matrices of the extracted model were compared. The comparison showed that both methods are effective means to reduce a nonlinear simulation to a linear state-space model.

The purpose of this paper is to compare two distinct methods to extract a linear state-space model about a reference flight condition from a nonlinear simulation. The frequency response method uses a time history input which contains the frequencies of interest to drive the simulation. The frequency response method uses a time history input which contains the frequencies of interest to drive the simulation frequency input and the output of the simulation are transformed to the frequency domain, and the desired frequency responses of the simulation are calculated. A linear model is then fit to the frequency responses using system identification techniques. The perturbation method extracts a linear model by perturbing the model states and inputs from the reference flight condition and calculating the resulting model coefficients. Both method were used to extract a fourth order longitudinal state-space model from the V-22 full nonlinear simulation. The time history responses and system matrices of the extracted model were compared. The comparison showed that both methods are effective means to reduce a nonlinear simulation to a linear state-space model.

This report summarizes work performed during the period April 1, 1979 to January 31. The work is concerned with certain topics on perturbation theory, transonic flow theory, numerical analysis, and adaptive grid generation.

This report summarizes work performed during the period April 1, 1979 to January 31. The work is concerned with certain topics on perturbation theory, transonic flow theory, numerical analysis, and adaptive grid generation.

This report summarizes work performed during the period April 1, 1979 to January 31. The work is concerned with certain topics on perturbation theory, transonic flow theory, numerical analysis, and adaptive grid generation.

In this paper, we give explicit estimates that insure the existence of solutions for first order partial differential operators on compact manifolds, using a viscosity method. In the linear case, an explicit integral formula can be found, using the characteristics curves. The solution is given explicitly on the critical points and the limit cycles of the vector field of the first order term of the operator. In the nonlinear case, a generalization of the Weitzenbock formula provides pointwise estimates that insure the existence of a solution, but the uniqueness question is left open. Nevertheless we prove that uniqueness is stable under a C^{1} perturbation. Finally, we give some examples where the solution fails to exist globally, justifying the need to impose conditions that warrant global existence. The last result reveals that the zero order term in the first order operator is necessary to obtain generically bounded solutions.

We derive a correct first-order perturbation theory in electromagnetism for cases where an interface between two anisotropic dielectric materials is slightly shifted. Most previous perturbative methods give incorrect results for this case, even to lowest order, because of the complicated discontinuous boundary conditions on the electric field at such an interface. Our final expression is simply a surface integral, over the material interface, of the continuous field components from the unperturbed structure. The derivation is based on a "localized" coordinate-transformation technique, which avoids both the problem of field discontinuities and the challenge of constructing an explicit coordinate transformation by taking a limit in which a coordinate perturbation is infinitesimally localized around the boundary. Not only is our result potentially useful in evaluating boundary perturbations, e.g. from fabrication imperfections, in highly anisotropic media such as many metamaterials, but it also has a direct application in numerical electromagnetism. In particular, we show how it leads to a sub-pixel smoothing scheme to ameliorate staircasing effects in discretized simulations of anisotropic media, in such a way as to greatly reduce the numerical errors compared to other proposed smoothing schemes.

A unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on Tsallis entropy is given here. In the case of the variational method, the Bogoliubov inequality is generalized in a very natural way following the Feynman proof for the usual statistical mechanics. The inequality turns out to be form-invariant with respect to the entropic index $q$. The method is illustrated with a simple example in classical mechanics. The formalisms developed here are expected to be useful in the discussion of nonextensive systems.

A unified presentation of the perturbation and variational methods for the generalized statistical mechanics based on Tsallis entropy is given here. In the case of the variational method, the Bogoliubov inequality is generalized in a very natural way following the Feynman proof for the usual statistical mechanics. The inequality turns out to be form-invariant with respect to the entropic index $q$. The method is illustrated with a simple example in classical mechanics. The formalisms developed here are expected to be useful in the discussion of nonextensive systems.

Different perturbation theory treatments of the Ginzburg-Landau phase transition model are discussed. This includes a criticism of the perturbative renormalization group (RG) approach and a proposal of a novel method providing critical exponents consistent with the known exact solutions in two dimensions. The new values of critical exponents are discussed and compared to the results of numerical simulations and experiments.

Singular perturbation equations contain many of the essential difficulties of the Navier-Stokes equations. In this report the weighted-mean scheme for linear equations and the monotone difference scheme for nonlinear equations were adopted. Presented here are fast iterative techniques for solving large systems of equations that result from discretization. Numerical results are also presented for nonlinear cases using Newton's method combined with the minimal residual method. The main conclusions are that minimal residual methods with a preconditioning technique and multigrid methods with a special relaxation scheme have proved to be quite reliable and far more efficient than standard iterative methods. (Author)

Singular perturbation equations contain many of the essential difficulties of the Navier-Stokes equations. In this report the weighted-mean scheme for linear equations and the monotone difference scheme for nonlinear equations were adopted. Presented here are fast iterative techniques for solving large systems of equations that result from discretization. Numerical results are also presented for nonlinear cases using Newton's method combined with the minimal residual method. The main conclusions are that minimal residual methods with a preconditioning technique and multigrid methods with a special relaxation scheme have proved to be quite reliable and far more efficient than standard iterative methods. (Author)