Showing papers in "SIAM Journal on Numerical Analysis in 1979"
TL;DR: This work studies two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators with real-time requirements.
Abstract: Splitting algorithms for the sum of two monotone operators.We study two splitting algorithms for (stationary and evolution) problems involving the sum of two monotone operators. These algorithms ar...
1,939 citations
TL;DR: In this paper, an algorithm for computing the condition number of the matrix of coefficients of a linear system with O(n 2 ) arithmetic operations is described, which gives a reliable indication of the order of magnitude of condition number.
Abstract: It is important in practice when solving linear systems to have an economical method for estimating the condition number $\kappa (A)$ of the matrix of coefficients. An algorithm involving $O(n^2 )$ arithmetic operations is described; it gives a reliable indication of the order of magnitude of $\kappa (A)$.
328 citations
TL;DR: In this paper, a comparison is made of two stability criteria, i.e., A-stability and Bstability, and it is shown that under certain mild conditions these two concepts are equivalent.
Abstract: A comparison is made of two stability criteria. The first is a modification to nonautonomous problems of A-stability and the second is a similar modification of B-stability. It is shown that under certain mild conditions these two concepts are equivalent. A number of examples are given of methods that satisfy these new stability properties and it is also shown that the growth of errors can be estimated by an extension of this theory.
277 citations
TL;DR: In this paper, the concept of affine invariance associated with Newton's method is extended to S-invariance for related iterative methods where S is some set of linear transformations not changing the iterates.
Abstract: For a given starting point, the sequence of Newton iterates is well known to be invariant under affine transformation of the operator equation to be solved. This property, however, is not sufficiently reflected in most convergence theorems that are presently in common use. For this reason, certain affine invariant convergence theorems are given in this paper. The new theorems can be understood as refined versions of the Newton–Mysovskii theorem, the Newton–Kantorovitch theorem (including optimal error bounds) and a convergence theorem for approximate Newton processes. In addition, the concept of affine invariance associated with Newton’s method is extended to S-invariance associated with related iterative methods where S is some set of linear transformations not changing the iterates. As an application of these considerations, a new convergence theorem for a class of generalized Gauss–Newton methods is given. This theorem is S-invariant with an S containing all unitary transformations. Unlike previous com...
252 citations
TL;DR: In this article, the authors compared eleven different methods for accelerating the convergence of sequences and series, including both linearly and logarithmically convergent series, monotone and alternating series, and found that the only methods that accelerate linear convergence across the board were the u and v transforms of Levin and theta algorithm of Brezinski.
Abstract: Eleven different methods for accelerating convergence of sequences and series have been tested and compared on a wide range of test problems, including both linearly and logarithmically convergent series, monotone and alternating series. All but one of these methods are already in the literature, and they include both linear and nonlinear methods. The only methods found to accelerate convergence across the board were the u and v transforms of Levin and the theta algorithm of Brezinski. The paper gives detailed comparisons of all the tested methods on the basis of number of correct digits in the answer as a function of number of terms of the series used. A theorem of Germain-Bonne states that methods of a certain form which are exact on geometric series will accelerate linear convergence. The theorem applies to theta sub 2, and we have extended it to apply to Levin's transforms. No corresponding theorem is known for logarithmic convergence, but u, v, and theta are exact on certain large classes of logarithmic series, and all tested methods lacking this property failed to accelerate some logarithmically convergent series.
207 citations
TL;DR: A sequence of points has been constructed that very uniformly fill the multi-dimensional cube, successfully used for systematic crude searching, as starting points for global search algorithms to optimize functions of several variables, for computer aided statement of optimum design problems.
Abstract: A sequence of points has been constructed that very uniformly fill the multi-dimensional cube. These points are successfully used for systematic crude searching, as starting points for global search algorithms to optimize functions of several variables, for computer aided statement of optimum design problems.
207 citations
TL;DR: A priori error estimates for Galerkin methods for numerical approximation of the coupled quasilinear system for $c = c(x,t) and $p = p(x-t)$ given by as mentioned in this paper.
Abstract: A priori error estimates for Galerkin methods for numerical approximation of the coupled quasilinear system for $c = c(x,t)$ and $p = p(x,t)$ given by \[\begin{gathered}
abla \cdot [a(x,c)\{
abla p - \gamma (x,c)
abla z\} ] = q(x,t), \hfill \\
abla \cdot [b(x,c,
abla p)
abla c] - u(x,c,
abla p) \cdot
abla c = \phi (x)\frac{{\partial c}}{{\partial t}} + g(x,t,c) \hfill \\ \end{gathered} \] for $x \in \Omega ,t \in (0,T)$, and appropriate Neumann boundary and initial conditions are considered. Equations of this type arise in models for the miscible displacement of one incompressible fluid by another in a porous medium. Estimates for both continuous time and fully-discrete time Galerkin methods are presented.
168 citations
TL;DR: In this paper, it was shown that the convergence of a strongly continuous semigroup to a rational approximation of a semigroup on a Banach space is bounded by a constant factor.
Abstract: We show that if $r^n (hA),nh = t$, is an A-acceptable rational approximation of a strongly continuous semigroup $e^{tA} $ on a Banach space, then for t bounded, $\| {r^n (hA)} \| \leqq Cn^{{1/2}} $, with certain improvements under additional hypotheses on r. We also discuss the convergence of $r^n (hA)v$ to $e^{tA} v$ as $h \to 0$ under various assumptions on r and v.
151 citations
TL;DR: In this article, an extension of the Golomb-Weinberger method for estimating a linear operator U from inaccurate data is presented, showing that smoothing of the data followed by U is an optimal procedure.
Abstract: Among all possible methods of estimating a linear operator U from inaccurate data, smoothing of the data followed by U is an optimal procedure. This result which we formulate and prove here in a Hilbert space setting may be viewed as an extension of the Golomb–Weinberger method of estimation. It provides as well a rationale for an optimal choice of a smoothing parameter. Our results also provide a method of finding the exact value of the n-width of subsets of a Hilbert space determined by two ellipsoidal bounds. We give an application of this method to the space of essentially time- and band-limited signals introduced by D. Slepian.
134 citations
TL;DR: It is shown that the BFGS and CG algorithms are related to one another in a particularly close way and a new family of algorithms is proposed.
Abstract: On the basis of analysis and numerical experience, the BFGS (Broyden-Fletcher-Goldfarb-Shanno) algorithm is currently considered to be one of the most effective algorithms for finding a minimum of an unconstrained function, f(x), x an element of R/sup n/. However, when computer storage is at a premium, the usual alternative is to use a conjugate gradient (CG) method. It is shown here that the two algorithms are related to one another in a particularly close way. Based upon these observations, a new family of algorithms is proposed. 2 tables.
125 citations
TL;DR: In this paper, the authors established the basic convergence conditions for the method of nonconforming finite elements applied to a class of generalized elliptic boundary value problems with variable, not necessarily smooth coefficients.
Abstract: The paper establishes the basic convergence conditions for the method of nonconforming finite elements applied to a class of generalized elliptic boundary value problems with variable, not necessarily smooth coefficients. The main result is a new, generalized patch test. Approximability and success in this test is the necessary and sufficient condition for convergence of the nonconforming approximations. It is proved that nonconforming elements of Wilson, Adini, Crouzeit–Raviart, Morley, and de Veubeke pass the generalized patch test and thus yield convergent approximations of the boundary value problems.
TL;DR: In this paper, a general theory for calculating equidistributing meshes was developed for difference methods for boundary-value problems of the form \[ u' = f(u,t),\qquad b(u(0),u(1)) = 0.
Abstract: A general theory is developed for calculating equidistributing meshes $\{ t_i \} $ for difference methods for boundary-value problems of the form \[ u' = f(u,t),\qquad b(u(0),u(1)) = 0. \] It is shown that the original problem and the equidistribution constraints on the mesh $\{ t_i \} $ can be replaced by a transformed boundary-value problem on a uniform mesh. Existence, uniqueness, and convergence of Newton’s method for the discrete solution and the equidistributing mesh are proved. Equidistribution of arc length is given for boundary-layer problems. Five sample problems are solved with different methods of choosing the mesh $\{ t_i \} $.
TL;DR: The main purpose is to find ways of taking advantage of the sparsity structure and the symmetry of the second derivative matrix, in order to make small the number of first derivative vectors that have to be calculated.
Abstract: This paper studies automatic procedures for estimating second derivatives of a real valued function of several variables. The estimates are obtained from differences in first derivative vectors, and it is supposed that the required matrix is sparse and that its sparsity structure is known. Our main purpose is to find ways of taking advantage of the sparsity structure and the symmetry of the second derivative matrix, in order to make small the number of first derivative vectors that have to be calculated. Two new algorithms are proposed, which seem to be very successful in practice and which do not require much computer arithmetic. One is a direct method and the other is a substitution method, these terms being explained in the paper. Some examples show, however, that the given methods may not minimize the number of first derivative vector calculations.
TL;DR: The sensitivity of the solution X to the matrix equation $AX - XB = C$ is primarily dependent on the quantity ${\operatorname{sep}}(A,B)$ introduced by Stewart (1973) in connection with the resolution of invariant subspaces as mentioned in this paper.
Abstract: The sensitivity of the solution X to the matrix equation $AX - XB = C$ is primarily dependent on the quantity ${\operatorname{sep}}(A,B)$ introduced by Stewart (1973) in connection with the resolution of invariant subspaces. In this paper, we discuss some properties of ${\operatorname{sep}}(A,B)$, give some examples to show how very small it can be for seemingly harmless problems, and discuss the feasibility of the iteration $AX^{(k + 1)} = X^{(k)} B + C$ for solving the matrix equation.
TL;DR: It is proved that only a small amount of computational work is needed for the approximation of one eigenvalue and the corresponding eigenfunctions.
Abstract: The eigenvalues and eigenfunctions can be approximated by finite element methods. Then the original problem is replaced by a finite-dimensional problem. In this paper we propose a multi-grid method for solving these finite-dimensional problems. It is proved that only a small amount of computational work is needed for the approximation of one eigenvalue and the corresponding eigenfunctions.
TL;DR: In this paper, the authors consider a stiff nonlinear system and assume that the large eigenvalues of the system are purely imaginary, and conditions are given that the system has smooth solutions in long time intervals.
Abstract: Consider a stiff nonlinear system $y' = f(y)$ of ordinary differential equations. Assume that the large eigenvalues of ${{\partial f} / {\partial y}}$ are purely imaginary. Conditions are given that the system has smooth solutions in long time intervals and methods are discussed to obtain these solutions.
TL;DR: In this paper, a weighted least-squares approximation of a discrete function on the sphere in terms of surface spherical harmonics has been obtained, which has the property that waves are resolved uniformly on a sphere that is, if the discrete function is replaced by a tabulation of its spectral representation.
Abstract: Several topics are discussed which concern the representation of a discrete function on the sphere in terms of surface spherical harmonics Several methods are reviewed, with particular attention given to the recent work of Machenhauer and DaleyThe aliasing of the spherical harmonics is discussed and for a given grid a finite number of spherical harmonics are chosen as the discrete basis A harmonic is included in the basis if and only if it does not alias on the grid The number of basis functions is about half the number of grid points, with the result that the approximation may not fit all the function values Nevertheless, it is shown that a weighted least-squares approximation is obtained This approximation has the property that waves are resolved uniformly on the sphere That is, if the discrete function is replaced by a tabulation of its spectral representation, then the high frequencies which are artificially induced by the closeness of the grid points near the pole are removedThe approximation
TL;DR: An algorithm based on plane rotations which is designed for fast computation, especially for large structured problems, and can be made to handle any rank deficiency in the matrices is made.
Abstract: The generalized linear least squares problem is treated here as a linear least squares problem with linear equality constraints. Advantage is taken of this formulation to produce a numerically stable algorithm based on plane rotations which is designed for fast computation, especially for large structured problems. The algorithm can be made to handle any rank deficiency in the matrices. A rounding error analysis and operation counts are given. The use of nonunitary transformations is considered.
TL;DR: In this paper, the Chow-Yorke algorithm was applied to a class of nonlinear two-point boundary value problems by solving the nonlinear system of equations defined by shooting.
Abstract: It is shown that the Chow–Yorke algorithm (a homotopy-type method) may be applied to a class of nonlinear two-point boundary value problems by solving the nonlinear system of equations defined by shooting. The resultant method is globally convergent with probability one, in the sense that it may fail for starting points in a set of Lebesgue measure zero. Some numerical results are given.
TL;DR: Schubert's method for solving sparse nonlinear equations is an extension of Broyden's method The zero-nonzero structure defined by the sparse Jacobian is preserved by updating the approximate Jacobian row by row as discussed by the authors.
Abstract: Schubert’s method for solving sparse nonlinear equations is an extension of Broyden’s method The zero-nonzero structure defined by the sparse Jacobian is preserved by updating the approximate Jacobian row by row An estimate is presented which permits the extension of the convergence results for Broyden’s method to Schubert’s method The analysis for local and q-superlinear convergence given here includes, as a special case, results in a recent paper by B Lam; this generalization seems theoretically and computationally more satisfying A Kantorovich analysis paralleling one for Broyden’s method is given This leads to a convergence result for linear equations that includes another result by Lam A result by More and Trangenstein is extended to show that a modified Schubert’s method applied to linear equations is globally and q-superlinearly convergent
TL;DR: An algorithm for reducing a real matrix A to block diagonal form by a real similarity transformation that attempts to control the condition of the transformation matrices, so that the reducing subspaces are well conditioned and the basis vectors are numerically independent.
Abstract: This paper describes an algorithm for reducing a real matrix A to block diagonal form by a real similarity transformation. The columns of the transformation corresponding to a block span a reducing subspace of A, and the block is the representation of A in that subspace with respect to the basis. The algorithm attempts to control the condition of the transformation matrices, so that the reducing subspaces are well conditioned and the basis vectors are numerically independent.
TL;DR: In this article, the authors consider the case where the generator of a strongly continuous semigroup is a linear operator and the generator is a rational function of one variable, such that the generator can be expressed as a nonnegative integer.
Abstract: For $t > 0$, let $u(t)$ satisfy \[ \frac{{du}}{{dt}} = Au, \] where A is a linear operator, the generator of a strongly continuous semigroup. Let $v(t)$ satisfy $v(t + h) = r(hA)v(t)$, where $r( \cdot )$ is a rational function of one variable, such that $| {r(z)} | \leqq 1$ for all $\operatorname{Re} z \leqq 0$. Let $u(0) = v(0)$ and be in the domain of $A^{q + 2} $. Then, if \[ \left| {r(z) - e^z } \right| = O\left( {| z |^{q + 1} } \right)\quad {\text{as }} | z | \to 0,\] we have \[ \| {u(t) = v(t)} \| = O\left( {h^q } \right)\] as $h \to 0$, where q is an arbitrary nonnegative integer.
TL;DR: An iterative method is presented and analyzed which is based on using a preconditioned conjugate gradient iteration for approximately solving the linear equations produced at each time step by an e-magnitude gradient.
Abstract: An iterative method is presented and analyzed which is based on using a preconditioned conjugate gradient iteration for approximately solving the linear equations produced at each time step by an e...
TL;DR: This paper develops “dynamic” ADI methods involving a computerized strategy for completely automatic change of the iteration parameter $\Delta t$ in alternating direction implicit methods for linea...
Abstract: This paper develops “dynamic” ADI methods involving a computerized strategy for completely automatic change of the iteration parameter $\Delta t$ in alternating direction implicit methods for linea...
TL;DR: In this paper, a general class of finite difference methods for solving nonlinear two point boundary value problems is considered, which can also be interpreted as collocation methods and a convergence analysis on uniform meshes is given.
Abstract: A general class of finite difference methods for solving nonlinear two point boundary value problems is considered. These methods can also be interpreted as collocation methods. A convergence analysis on uniform meshes is given. This analysis is based upon a theorem of H. B. Kelley and a previous paper by the author. A specific example is given in detail and results of some numerical computations are included.
TL;DR: In this article, the authors developed numerical methods for the integration of systems of differential equations in which some of the dependent variables are more slowly variant torsion torsions than others.
Abstract: The primary purpose of this paper is the development of numerical methods for the integration of systems of differential equations in which some of the dependent variables are more slowly variant t...
TL;DR: In this paper, a Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions is analyzed and an extrapolation in time is used to yield a system of linear algebraic equations to be solved at each level.
Abstract: A Galerkin method for nonlinear parabolic equations with nonlinear boundary conditions is analyzed. The parabolic equations studied are more general than those previously analyzed and new optimal order error estimates are obtained. An extrapolation in time is used to yield a system of linear algebraic equations to be solved at each time level.
TL;DR: In this paper, an algorithm for the approximate solution of a complete singular integral equation (with Cauchy principal value integral) taken over the arc $( - 1,1) is described.
Abstract: An algorithm is described for the approximate solution of a complete singular integral equation (with Cauchy principal value integral) taken over the arc $( - 1,1)$. The coefficients in the dominant part of this equation are not necessarily restricted to be constants so that the approximate solution of a wide class of singular integral equations is possible. No restriction is placed on the index of the equation.
TL;DR: In this paper, a numerical scheme for solving heat conduction problems involving a change of phase is presented, which avoids geometrical front tracking, and requires only simple algebraic operations.
Abstract: A numerical scheme for solving heat conduction problems involving a change of phase is presented. The numerical solution is obtained for heat flow in a two phase medium by using a method which treats each phase alternately, The resulting scheme avoids geometrical front tracking, and requires only simple algebraic operations. A maximum principle for the method and results of numerical experiments are given. An analytical version of the algorithm for a one dimensional one phase Stefan problem is shown to have an $O(\Delta t \cdot \ln {{1}/{\Delta t}})^{{1/2}} $ rate of convergence. Numerical experiments indicate that this estimate is sharp.
TL;DR: In this paper, it is shown how mixed formulations of boundary-value problems with constraints, such as the Stokes problem and the equilibrium formulation in elasticity, may be solved iteratively by preconditioning with a regularized operator.
Abstract: It is shown how mixed formulations of boundary-value problems with constraints, such as the Stokes problem and the equilibrium formulation in elasticity, may be solved iteratively by preconditioning with a regularized operator. Advantages of using this approach in comparison with the frequently-used augmented Lagrangian method are pointed out.