Approximate Solution of Operator Equations

One of the most important chapters in modern functional analysis is the theory of approximate methods for solution of various mathematical problems. Besides providing considerably simplified approaches to numerical methods, the ideas of functional analysis have also given rise to essentially new computation schemes in problems of linear algebra, differential and integral equations, nonlinear analysis, and so on. The general theory of approximate methods includes many known fundamental results. We refer to the classical work of Kantorovich; the investigations of projection methods by Bogolyubov, Krylov, Keldysh and Petrov, much furthered by Mikhlin and Pol'skii; Tikho­ nov's methods for approximate solution of ill-posed problems; the general theory of difference schemes; and so on. During the past decade, the Voronezh seminar on functional analysis has systematically discussed various questions related to numerical methods; several advanced courses have been held at Voronezh Uni­ versity on the application of functional analysis to numerical mathe­ matics. Some of this research is summarized in the present monograph. The authors' aim has not been to give an exhaustive account, even of the principal known results. The book consists of five chapters.

1 Successive approximations.- §1. Existence of the fixed point of a contraction operator.- 1.1. Contraction operators.- 1.2. Use of an equivalent norm.- 1.3. Relative uniqueness of the solution.- 1.4. Spectral radius of a linear operator.- 1.5. Operators which commute with contraction operators.- 1.6. Case of a compact set.- 1.7. Estimating the Lipschitz constant.- 1.8. Equations with uniform contraction operators.- 1.9. Local implicit function theorem.- §2. Convergence of successive approximations.- 2.1. Successive approximations.- 2.2. Equations with a contraction operator.- 2.3. Linear equations.- 2.4. Fractional convergence.- 2.5. Nonlinear equations.- 2.6. Effect of the initial approximation on estimate of convergence rate.- 2.7. Acceleration of convergence.- 2.8. Distribution of errors.- 2.9. Effect of round-off errors.- §3. Equations with monotone operators.- 3.1. Statement of the problem.- 3.2. Cones in Banach spaces.- 3.3. Solvability of equations with monotone operators.- 3.4. Non-trivial positive solutions.- 3.5. Equations with concave operators.- 3.6. Use of powers of operators.- 3.7. The contracting mapping principle in metric spaces.- 3.8. On special metric.- 3.9. Equations with power nonlinearities.- 3.10. Equations with uniformly concave operators.- §4. Equations with nonexpansive operators.- 4.1. Nonexpansive operators.- 4.2. Example.- 4.3. Selfadjoint nonexpansive operators.- 2 Linear equations.- §5. Bounds for the spectral radius of a linear operator.- 5.1. Spectral radius.- 5.2. Positive linear operators.- 5.3. Indecomposable operators.- 5.4. Comparison of the spectral radii of different operators.- 5.5. Lower bounds for the spectral radius.- 5.6. Upper bounds for the spectral radius.- 5.7. Strict inequalities.- 5.8. Examples.- §6. The block method for estimating the spectral radius..- 6.1. Fundamental theorem.- 6.2. Examples.- 6.3. Conic norm.- 6.4. Generalized contracting mapping principle.- §7. Transformation of linear equations.- 7.1. General scheme.- 7.2. Chebyshev polynomials.- 7.3. Equations with selfadjoint operators in Hilbert spaces.- 7.4. Equations with compact operators.- 7.5. Seidel’s method.- 7.6. Remark on convergence.- §8. Method of minimal residuals.- 8.1. Statement of the problem.- 8.2. Convergence of the method of minimal residuals.- 8.3. The moment inequality.- 8.4. ?-processes.- 8.5. Computation scheme.- 8.6. Application to equations with nonselfadjoint operators.- §9. Approximate computation of the spectral radius.- 9.1. Use of powers of an operator.- 9.2. Positive operators.- 9.3. Computation of the greatest eigenvalue.- 9.4. Approximate computation of the eigenvalues of selfadjoint operators.- 9.5. The method of normal chords.- 9.6. The method of orthogonal chords.- 9.7. Simultaneous computation of several iterations.- §10. Monotone iterative processes.- 10.1. Statement of the problem.- 10.2. Choice of initial approximations.- 10.3. Acceleration of convergence.- 10.4. Equations with nonpositive operators.- 3 Equations with smooth operators.- §11. The Newton-Kantorovich method.- 11.1. Linearization of equations.- 11.2. Convergence.- 11.3. Further investigation of convergence rate.- 11.4. Global convergence condition.- 11.5. Simple zeros.- §12. Modified Newton-Kantorovich method.- 12.1. The modified method.- 12.2. Fundamental theorem.- 12.3. Uniqueness ball.- 12.4. Modified method with perturbations.- 12.5. Equations with compact operators.- 12.6. Equations with nondifferentiable operators.- 12.7. Remark on the Newton-Kantorovich method.- §13. Approximate solution of linearized equations.- 13.1. Statement of the problem.- 13.2. Convergence theorem.- 13.3. Application of the method of minimal residuals.- 13.4. Choice of initial approximations.- §14. A posteriori error estimates.- 14.1. Error estimates and existence theorems.- 14.2. Linearization of the equation.- 14.3. Rotation of a finite-dimensional vector field.- 14.4. Rotation of a compact vector field.- 14.5. Index of a fixed point.- 14.6. Index of a fixed point and a posteriori error estimates.- 14.7. Special case.- 14.8. Relaxing the compactness condition.- 4 Projection methods.- § 15. General theorems on convergence of projection methods..- 15.1. Projection methods.- 15.2. Fundamental convergence theorem.- 15.3. The aperture of subspaces of a Hilbert space.- 15.4. The Galerkin method for equations of the second kind.- 15.5. Supplements to Theorem 15.3..- 15.6. Derivation of an equation of the second kind.- 15.7. The collocation method.- 15.8. The factor method.- § 16. The Bubnov-Galerkin and Galerkin-Petrov methods.- 16.1. Convergence of the Bubnov-Galerkin method for equations of the second kind.- 16.2. Necessary and sufficient conditions for convergence of the Galerkin-Petrov method for equations of the second kind.- 16.3. Convergence of the Bubnov-Galerkin method for equations with a positive definite principal part.- 16.4. Lemma on similar operators.- 16.5. Convergence of the Bubnov-Galerkin method.- 16.6. Regular operators.- 16.7. Rate of convergence of the Galerkin-Petrov method.- 16.8. The method of moments.- §17. The Galerkin method with perturbations and the general theory of approximate methods.- 17.1. Statement of the problem.- 17.2. Lemma on invertibility of the operators I - ?T and I - ?Tn.- 17.3. Convergence theorem.- 17.4. Relation to Kantorovich’s general theory of approximate methods.- 17.5. Perturbation of the Galerkin approximation.- 17.6. Bases in subspaces of a Hilbert space.- 17.7. Lemma on Gram matrices.- 17.8. Stability of the Galerkin-Petrov method.- 17.9. Convergent quadrature processes.- 17.10. The method of mechanical quadratures for integral equations.- §18. Projection methods in the eigenvalue problem.- 18.1. The eigenvalue problem.- 18.2. Convergence of the perturbed Galerkin method.- 18.3. Rate of convergence.- 18.4. The Bubnov-Galerkin method; eigenvalue estimates.- 18.5. Estimates for eigenelements.- 18.6. Improved eigenvalue estimates.- 18.7. The Galerkin-Petrov method.- 18.8. Case of unbounded operators.- 18.9. Operators in real spaces.- §19. Projection methods for solution of nonlinear equations.- 19.1. Statement of the problem.- 19.2. Solvability of nonlinear equations.- 19.3. Convergence of the perturbed Galerkin method.- 19.4. A posteriori error estimate.- 19.5. Rotation of a compact vector field.- 19.6. Second convergence proof.- 19.7. The method of mechanical quadratures for nonlinear integral equations.- 19.8. The method of finite differences.- 19.9. The Galerkin method.- 19.10. The Galerkin-Petrov method. Stability.- 19.11. Projection methods assuming one-sided estimates.- 19.12. Minimum problems for functional.- 19.13. The factor method for nonlinear equations.- 5 Small solutions of operator equations.- §20. Approximation of implicit functions.- 20.1. Fundamental implicit function theorem.- 20.2. Multilinear operators and Taylor’s formula.- 20.3. Differentiation of an implicit function.- 20.4. Method of undetermined coefficients.- 20.5. Successive approximations.- 20.6. Asymptotic approximations to implicit functions.- 20.7. Formal series and formal implicit functions.- 20.8. Analyticity of an implicit function.- §21. Finite systems of equations.- 21.1. Review of ring theory.- 21.2. Rings of power series.- 21.3. The Newton diagram.- 21.4. Branching of solutions of scalar analytic equations.- 21.5. The method of elimination.- 21.6. Structurally stable case.- 21.7. Systems of analytic equations.- §22. Branching of solutions of operator equations.- 22.1. Statement of the problem.- 22.2. Decomposable operators.- 22.3. Matrix representations of decomposable operators.- 22.4. Branching equations.- 22.5. Asymptotic approximations of branching equations.- 22.6. Equations in formal power series.- 22.7. Equations in analytic operators.- § 23. Simple solutions and the method of undetermined coefficients.- 23.1. Simple solutions.- 23.2. Asymptotic approximations to simple solutions.- 23.3. Simple solutions and branching equations.- 23.4. Quasi-solutions of operator equations.- 23.5. The method of undetermined coefficients.- 23.6. Equations with analytic operators.- §24. The problem of bifurcation points.- 24.1. Statement of the problem.- 24.2. Equations with compact operators.- 24.3. Use of the branching equation.- 24.4. Amplitude curves.