Iterative method for invariant imbedding

In this paper, we consider two-point linear boundary-value problems which depend continuously on many parameters, and we prove that, under certain condition, there exists an iterative scheme for solving the invariant-imbedding equations with properties that save some storage and reduce computation times. An application to linear transport theory in slab geometry is discussed.     

Stochastic invariant imbedding

We study stochastic differential equations of the type:

Identification of heat-conduction coefficient via method of variational imbedding

The inverse parabolic problem of coefficient identification from over-posed data is embedded into a fourth order in space and second order in time elliptic boundary value problem. The latter is well posed for redundant data at boundaries. The equivalence of the two problems is demonstrated. A difference scheme of splitting-type is employed, and featuring examples are elaborated numerically.     

Partitioning of invariant imbedding systems

The choice of partitioning the system matrix for a system of N linear ordinary differential equations may determine the ease or difficulty of obtaining a solution by the invariant imbedding method of Scott. This paper shows how the configuration and partitioning of the system matrix is reflected in the fundamental matrix. The partitioning of the fundamental matrix is the key to the ease or difficulty of obtaining a solution. If the fundamental matrix is known for a given system matrix configuration and partitioning, then the fundamental matrix associated with a new system matrix configuration may be derived by the same row and column interchanges that transformed the old system matrix into the new system matrix. The fundamental matrix for the new system matrix does not have to be recalculated anew from the Kronecker delta initial conditions.     

Obtaining scattering kernels using invariant imbedding

The symplectic group structure associated with the Riccati equation is exploited to derive a unifying group-theoretical framework encompassing a large number of well-known results, such as partitioning, doubling, and Chandrasekhar algorithms, the algebraic Riccati equation, and the discrete-time case. It is also used to derive a new integration-free Chandrasekhar type algorithm, and to present a fairly complete characterization of the solutions of the Riccati equation in the periodic case.     

Asymptotic behavior of the quadratic variation for trajectories of processes with independent increments

One considers a certain characteristic, similar to the quadratic variation, of processes with independent increments. One investigates its almost surely behavior as t.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 130, pp. 78–88, 1983.     

On the formulation of invariant imbedding problems

The formulation of an invariant imbedding problem from a given linear, two-point boundary-value problem is not unique. In this paper, we illustrate how the formulation of the problem by partitioning the original vectory(z) into [u(z),v(z)], can affect the numerical accuracy. In fact, the partitioning, the choice of theR, O system orS, T system of equations in Scott's method, the location and number of switch points, and the switching procedure, all influence the numerical results and the ease of obtaining solutions. A new method of switching and the appropriate formulas are described, namely, the repeated switching from theR, Q system to theR, Q system of equations or from theS, T system to theS, T system of equations.     

Positive top Lyapunov exponent via invariant cones: Single trajectories

We establish criteria for the positivity of the top Lyapunov exponent of a nonautonomous dynamics in terms of invariant cone families, both for maps and flows. The families of cones are associated with quadratic forms of type (k,p−k)$(k,p-k)$ with k arbitrary. Our work can be seen as a counterpart of results in the context of ergodic theory, where the positivity of the top Lyapunov exponent is obtained for almost all trajectories although saying nothing about each specific trajectory.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

Invariant imbedding and a class of variational problems

Consider the minimization of an integral $$I = \int_a^T {[\frac{1}{2}\dot w^2 + F(w,y)]} dy$$ withw(a)=c andw(T)=free. An initial-value problem for the optimizing function is derived directly from the variational problem. It is shown that the solution of the initial-value problem satisfies the usual Euler equation. The Bellman-Hamilton-Jacobi partial differential equation is also treated.     

Singular quadratic variational problems

The purpose of this paper is to show that the general theory of quadratic forms developed earlier by the author is applicable to singular variational problems as well as to nonsingular ones. In particular, this general theory is applicable to the singular variational problems associated with Legendre polynomials, associated Legendre polynomials, Jacobi polynomials, and Tchebysheff polynomials.     

Conservation laws for conformally invariant variational problems

We succeed in writing 2-dimensional conformally invariant non-linear elliptic PDE (harmonic map equation, prescribed mean curvature equations,..., etc.) in divergence form. These divergence-free quantities generalize to target manifolds without symmetries the well known conservation laws for weakly harmonic maps into homogeneous spaces. From this form we can recover, without the use of moving frame, all the classical regularity results known for 2-dimensional conformally invariant non-linear elliptic PDE (see [Hel]). It enables us also to establish new results. In particular we solve a conjecture by E. Heinz asserting that the solutions to the prescribed bounded mean curvature equation in arbitrary manifolds are continuous and we solve a conjecture by S. Hildebrandt [Hil1] claiming that critical points of continuously differentiable elliptic conformally invariant Lagrangian in two dimensions are continuous.     

Characterization of the finite weak singularities of quadratic systems via invariant theory

This article is about weak singularities of quadratic differential systems, that is, non-degenerate singular points with traces of the corresponding linearized systems at such points equal to zero. These could be foci, centers or saddles. Necessary and sufficient conditions for a real quadratic system to possess a fixed number of weak singularities of a specific order are given. The conditions are stated in terms of affine invariant polynomials in the 12-dimensional space of the coefficients.     

Discrete invariant imbedding for singular boundary value problems with a regular singularity

We describe a discrete invariant imbedding method for solving a two point boundary value problem in the interval [0,b] for a linear second order ordinary differential equation with a singularity of the first kind at x = 0. By employing the series expansion on (0, δ], where δ is near the singularity, we first replace it by a regular problem on some interval [δ, b]. The discrete invariant imbedding method is then described to solve the problem over the reduced interval. The stability analysis of the method is discussed. Some model problems are solved, and the numerical results compared with those of other methods.     

Solution of nonlinear heat transfer problems by variational imbedding

Summary In this paper attention has been drawn to the variational imbedding method for solving a broad class of nonlinear heat transfer problems. Specific illustrations considered in this paper include the variable thermal properties heat conduction problem, the wedge flow problem, and the well known Bénard problem with variable properties, all of which are not amenable to analytical solutions by conventional techniques.

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Zusammenfassung In dieser Arbeit wird auf eine Klasse von nichtlinearen Wärmeleitungsproblemen hingewiesen, die durch variational imbedding gelöst werden können. Die behandelten Beispiele umfassen das Wärmeleitungsproblem mit veränderlichen thermischen Eigenschaften, das Problem der Keilströmung, und das bekannte Bénard-Problem mit veränderlichen Stoffwerten; alle diese Probleme können nicht mit konventionellen analytischen Methoden behandelt werden.

     

Optimal design via variational principles: the three-dimensional case

The analysis of some three-dimensional optimal design problems leads us to study variational principles under curl-free and divergence-free constraints simultaneously. We explicitly exploit the relationship between curl-free and div-free restrictions in order to take advantage of the accumulated experience in the classical curl case by the introduction of potentials. Our discussion takes place in the three-dimensional situation. This is a first contribution in the sense that we only deal with the most basic issues.