The Method of Adjoints

Another efficient method for the solution of linear two-point boundary-value problems is that of the method of adjoints. A good reference for this material is the book by Roberts and Shipman (1972). In the method of adjoints we define the system equations and the adjoint system equations. 

The basic linear differential equations are the system equations and are given by 

The adjoint system is defined in terms of the homogeneous part of the system equations as 

By multiplying the transpose of the adjoint variable vector, p, by the system equation (7.7.1), we have 

Multiplying the transpose of the system variable vector, x, by the adjoint system equation (7.7.2) we have 

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Three basic numerical techniques are introduced. These are shooting techniques, quasilinearization with the use of the principle of superposition, and the method of adjoints. 

Apply the method of adjoints to solve for the annular reactor profiles of the problem of section 7.5. 

An estimate of the convergence rate for the method of minimal corrections is derived in the case when A is a non-self-adjoint operator and in others situations. 

In some previous papers [3], the author has ivestigated the properties of pairs of adjoint operators T and T+ in general, and the present paper may be considered as an extension of these studies. In the Uppsala group, there has also been a considerable interest in the properties of non-self-adjoint operators in connection with the method of complex scaling [4] with results which may be considered as illustrations of the more general theorems treated here. 

The puzzle depended on the simple fact that most physicists using the method of complex scaling had not realized that the associated operator u - the so-called dilatation operator - was an unbounded operator, and that the change of spectra -e.g. the occurrence of complex eigenvalues - was due to a change of the boundary conditions. Some of these features have been clarified in reference A, and in this paper we will discuss how these properties will influence the Hartree-Fock scheme. The existence of the numerical examples finally convinced us that the Hartree-Fock scheme in the complex symmetric case would not automatically reduce to the ordinary Hartree-Fock scheme in the case when the many-electron Hamiltonian became real and self-adjoint. Some aspects of this problem have been briefly discussed at the 1987 Sanibel Symposium, and a preliminary report has been given in a paper4 which will be referred to as reference D. 

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

How can we compute yi (o>) so that we can get x + (0) This can be achieved by formulating adjoint equations for yi(co) that are solved simultaneously with the xj and xj differential equations. However, the adjoint equations must be initial-value differential equations. The method of doing this... 

The described method can generate a first-order backward or a first-order forward difference scheme depending whether 0 = 0 or 0 = 1 is used. For 9 = 0.5, the method yields a second order accurate central difference scheme, however, other considerations such as the stability of numerical calculations should be taken into account. Stability analysis for this class of time stepping methods can only be carried out for simple cases where the coefficient matrix in Equation (2.106) is symmetric and positive-definite (i.e. self-adjoint problems Zienkiewicz and Taylor, 1994). Obviously, this will not be the case in most types of engineering flow problems. In practice, therefore, selection of appropriate values of 6 and time increment At is usually based on trial and error. Factors such as the nature of non-linearity of physical parameters and the type of elements used in the spatial discretization usually influence the selection of the values of 0 and At in a problem. 

Remark Quite often, the Dirichlet problem is approximated by the method based on the difference approximation at the near-boundary nodes of the Laplace operator on an irregular pattern, with the use of formulae (14) instead of (16) at the nodes x G However, in some cases the difference operator so constructed does not possess several important properties intrinsic to the initial differential equation, namely, the self-adjointness and the property of having fixed sign, For this reason iterative methods are of little use in studying grid equations and will be excluded from further consideration. 

We thus have shown that under conditions (26) inequality (14) is sufficient for the stability of scheme (la) in the space Ha, that is, relation (29) occurs. Let us stress that the requirement of self-adjointness of the operator B is necessary here, while the energy method demands only the positivity of B and no more. 

The general theory of iterative methods is presented in the next sections with regard to an operator equation of the first kind Au — f, where T is a self-adjoint operator in a finite-dimensional Euclidean space. The applications of such theory to elliptic grid equations began to spread to more and more branches as they took on an important place in real-life situations. 

On account of the basic theorem proved in Section 1 of the present chapter Seidel method converges if the operator A is self-adjoint and positive. More specifically, the sufficient stability condition (11) for the convergence of iterations in scheme (3 ) with a non-self-adjoint operator B takes the form... 

Stability theory is the central part of the theory of difference schemes. Recent years have seen a great number of papers dedicated to investigating stability of such schemes. Many works are based on applications of spectral methods and include ineffective results given certain restrictions on the structure of difference operators. For schemes with non-self-adjoint operators the spectral theory guides only the choice of necessary stability conditions, but sufficient conditions and a priori estimates are of no less importance. An energy approach connected with the above definitions of the scheme permits one to carry out an exhaustive stability analysis for operators in a prescribed Hilbert space Hh-... 

It was reported that the convergence of the Krotov iteration method [81, 93] was four or five times faster than that of the gradient-type methods. The formulation of Rabitz and others, [44, 45, 92], designed to improve the convergence of the above algorithm, introduces a further nonlinear propagation step into the adjoint equation (i.e., the equation for the undetermined Lagrange multiplier % t)) and is expressed as... 

To solve a second-order inhomogeneous ordinary differential equation, either the Green s function method or the variation of parameters method can be used. Consider the self-adjoint equation... 

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