Showing papers in "SIAM Journal on Numerical Analysis in 1992"
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TL;DR: In this article, a new version of the Perona and Malik theory for edge detection and image restoration is proposed, which keeps all the improvements of the original model and avoids its drawbacks.
Abstract: A new version of the Perona and Malik theory for edge detection and image restoration is proposed. This new version keeps all the improvements of the original model and avoids its drawbacks: it is proved to be stable in presence of noise, with existence and uniqueness results. Numerical experiments on natural images are presented.
2,565 citations
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1,088 citations
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TL;DR: In this article, a new approach based on Hamilton-Jacobi-Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions.
Abstract: The problem of recovering a Lambertian surface from a single two-dimensional image may be written as a first-order nonlinear equation which presents the disadvantage of having several continuous and even smooth solutions. A new approach based on Hamilton–Jacobi–Bellman equations and viscosity solutions theories enables one to study non-uniqueness phenomenon and thus to characterize the surface among the various solutions.A consistent and monotone scheme approximating the surface is constructed thanks to the dynamic programming principle, and numerical results are presented.
727 citations
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TL;DR: A theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established.
Abstract: In this note a theoretical analysis of some Krylov subspace approximations to the matrix exponential operation $\exp (A)v$ is presented, and a priori and a posteriors error estimates are established. Several such approximations are considered. The main idea of these techniquesis to approximately project the exponential operator onto a small Krylov subspace and to carry out the resulting small exponential matrix computation accurately. This general approach, which has been used with success in several applications, provides a systematic way of defining high-order explicit-type schemes for solving systems of ordinary differential equations or time-dependent partial differential equations.
700 citations
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TL;DR: Beylkin and Rokhlin this paper presented exact and explicit representations of the differential operators in orthonormal bases of compactly supported wavelets as well as the representations of Hilbert transform and fractional derivatives.
Abstract: This paper describes exact and explicit representations of the differential operators, ${{d^n } / {dx^n }}$, $n = 1,2, \cdots $, in orthonormal bases of compactly supported wavelets as well as the representations of the Hilbert transform and fractional derivatives. The method of computing these representations is directly applicable to multidimensional convolution operators.Also, sparse representations of shift operators in orthonormal bases of compactly supported wavelets are discussed and a fast algorithm requiring $O(N\log N)$ operations for computing the wavelet coefficients of all N circulant shifts of a vector of the length $N = 2^n $ is constructed. As an example of an application of this algorithm, it is shown that the storage requirements of the fast algorithm for applying the standard form of a pseudodifferential operator to a vector (see [G. Beylkin, R. R. Coifman, and V. Rokhlin, Comm. Pure. Appl. Math., 44 (1991), pp. 141–183]) may be reduced from $O(N)$ to $O(\log ^2 N)$ significant entries.
614 citations
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TL;DR: In this paper, the authors studied numerical methods for the one-dimensional heat equation with a singular forcing term, where the delta function was replaced by a discrete approximation, and the resulting equation was solved by a Crank-Nicolson method on a uniform grid.
Abstract: Numerical methods are studied for the one-dimensional heat equation with a singular forcing term, $u_t = u_{xx} + c(t)\delta (x - \alpha (t)).$ The delta function $\delta (x)$ is replaced by a discrete approximation $d_h (x)$ and the resulting equation is solved by a Crank–Nicolson method on a uniform grid. The accuracy of this method is analyzed for various choices of $d_h $. The case where $c(t)$ is specified and also the case where c is determined implicitly by a constraint on the solution at the point a are studied. These problems serve as a model for the immersed boundary method of Peskin for incompressible flow problems in irregular regions. Some insight is gained into the accuracy that can be achieved and the importance of choosing appropriate discrete delta functions.
261 citations
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TL;DR: In this paper, a semi-discretized form of the Navier-Stokes equations in a two-or three-dimensional bounded domain is studied and error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method.
Abstract: In this paper projection methods (or fractional step methods) are studied in the semi-discretized form for the Navier–Stokes equations in a two- or three-dimensional bounded domain. Error estimates for the velocity and the pressure of the classical projection scheme are established via the energy method. A modified projection scheme which leads to improved error estimates is also proposed.
257 citations
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TL;DR: A numerical scheme for the approximation of a parameter-dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value as mentioned in this paper.
Abstract: A numerical scheme for the approximation of a parameter-dependent problem is said to exhibit locking if the accuracy of the approximations deteriorates as the parameter tends to a limiting value. A robust numerical scheme for the problem is one that is essentially uniformly convergent for all values of the parameter. Precise mathematical definitions for these terms are developed, their quantitative characterization is given, and some general theorems involving locking and robustness are proven. A model problem involving heat transfer is analyzed in detail using this mathematical framework, and various related computational results are described. Applications to some different problems involving locking are presented.
245 citations
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TL;DR: This paper shows that the use of wavelets to discretize an elliptic problem with Dirichlet or Neumann boundary conditions has two advantages: an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type (in contrast with classical methods).
Abstract: This paper shows that the use of wavelets to discretize an elliptic problem with Dirichlet or Neumann boundary conditions has two advantages: an explicit diagonal preconditioning makes the condition number of the corresponding matrix become bounded by a constant and the order of approximation is locally of spectral type (in contrast with classical methods); using a conjugate gradient method, one thus obtains fast numerical algorithms of resolution. A comparison is also drawn between wavelet and classical methods.
206 citations
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TL;DR: In this paper, a class of second-order numerical schemes for the compressible Euler equations is described, and their stability (i.e., $L^1 $ stability) is proved.
Abstract: A class of second-order numerical schemes for the compressible Euler equations is described, and their $L^1 $ stability (i.e., $\rho \geqq 0$, $T \geqq 0$) is proved. Following Van Leer’s approach, the solution ($\rho $, u,$\sqrt T $ here) is represented as piecewise linear functions. The necessity of a slope limitation appears naturally in the derivation of the schemes, but it can be less strict than the slope reconstructions usually used. These schemes are written in terms of explicit flux splitting formula and are naturally multidimensional in space; the upwinding is obtained through a very generalized notion of characteristics: the kinetic one.
190 citations
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TL;DR: In this article, the quality of a piecewise linear interpolation surface (PUS) over triangles depends on the specific triangulation of the data points, and a new interpretation of existing error bounds for interpolating general smooth functions by PUS is studied.
Abstract: Given a set of data points in $R^2 $ and corresponding data values, it is clear that the quality of a Piecewise Linear Interpolation Surface (PUS) over triangles depends on the specific triangulation of the data points. In this paper, the question of what are good triangles (and triangulations) for linear interpolation is studied further. First, the model problem of constructing optimal triangulations for interpolating quadratic functions by PLIS is considered. Next, a new interpretation of existing error bounds for interpolating general smooth functions by PUS is studied. The conclusion is that triangles should be long in directions where the magnitude of the second directional derivative of F is small and thin in directions where the magnitude of the second directional derivative of F is large.
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TL;DR: In this article, a control volume method for planar div-curl systems is proposed, which is independent of potential and least squares formulations, and works directly with the divcurl system.
Abstract: A control volume method is proposed for planar div-curl systems. The method is independent of potential and least squares formulations, and works directly with the div-curl system. The novelty of the technique lies in its use of a single local vector field component and two control volumes rather than the other way around. A discrete vector field theory comes quite naturally from this idea and is developed. Error estimates are proved for the method, and other ramifications investigated.
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TL;DR: In this paper, the authors generalized Synge's maximum angle condition for triangular elements to tetrahedral elements and proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains $O(h)$ in the $W_p^1 (\Omega )$-norm for sufficiently smooth functions and $p \in [1, ∞ ]$.
Abstract: Synge’s maximum angle condition for triangular elements is generalized to tetrahedral elements. For the generalized condition, it is proved that tetrahedra may degenerate in a certain way and the error of the standard linear interpolation remains $O(h)$ in the $W_p^1 (\Omega )$-norm for sufficiently smooth functions and $p \in [1,\infty ]$.
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TL;DR: In this paper, the use of finite elements to discretize the time dependent Maxwell equations on a bounded domain in 3D space is analyzed and energy norm error estimates are provided when general finite element methods are used to discrete the equations in space.
Abstract: The use of finite elements to discretize the time dependent Maxwell equations on a bounded domain in three-dimensional space is analyzed Energy norm error estimates are provided when general finite element methods are used to discretize the equations in space In addition, it is shown that if some curl conforming elements due to Nedelec are used, error estimates may also be proved in the $L^2 $ norm
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TL;DR: In this article, a new technique is proposed to solve NSPD (nonsymmetric or indefinite) problems that are "compact" perturbations of some SPD (symmetric positive definite) problems.
Abstract: A new technique is proposed to solve NSPD (nonsymmetric or indefinite) problems that are “compact” perturbations of some SPD (symmetric positive definite) problems. In the new algorithm, a direct method is first used to solve the original equation restricted on a coarser space (that has a considerably smaller dimension), then an SPD equation for the residue is solved by using one or a few iterations of a given iterative algorithm. It is shown that for any convergent iterative method for the SPD problem, the new algorithm always converges with essentially the same rate if the coarse space is properly chosen. In applications, for multiplicative domain decomposition methods, the algorithm consists of solving the original NSPD problem on the coarse mesh and solving SPD equations on all subdomains; for multigrid methods, except when the correction on the coarsest mesh is first performed for the original NSPD equation, all smoothings are carried out for SPD equations on all other levels. It is shown that most o...
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TL;DR: A method is given for computing bounds for the solution set of a system of linear equations with interval coefficients with interval coefficient bounds.
Abstract: A method is given for computing bounds for the solution set of a system of linear equations with interval coefficients.
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TL;DR: In this paper, the optimal linear, positive schemes for constant-coefficient advection in two or three dimensions are presented, which are generalizations of first-order upwinding in one dimension.
Abstract: In this paper the optimal linear, positive schemes for constant-coefficient advection in two or three dimensions are presented. These are the generalizations of first-order upwinding in one dimension. By comparison with a dimension-by-dimension treatment the optimum schemes have much lower numerical diffusion, and permit larger timesteps.
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TL;DR: In this paper, the authors analyzed the MAC discretization of fluid flow for the stationary Stokes equations and proved that the discrete approximations do in fact converge to the exact solutions of the flow equations.
Abstract: The MAC (Marker and Cell) discretization of fluid flow is analysed for the stationary Stokes equations. It is proved that the discrete approximations do in fact converge to the exact solutions of the flow equations. Estimates using mesh dependent norms analogous to the standard ${\bf H}^1 $ and $L^2 $ norms are given for the velocity and pressure, respectively.
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TL;DR: In this article, the problem of direct numerical integration of differential Riccati equations (DREs) and some related issues are considered, and a useful matrix interpretation is given for many integration schemes, such as backward differentiation formulas, when applied to the DRE.
Abstract: . In this paper the problem of direct numerical integration of differential Riccati equations (DREs) and some related issues are considered. The DRE is an expression of a particular change of variables for a linear system of ordinary differential equations. The error that an approximate solution of the DRE induces on the original variables of the system is considered, and it is related to geometrical properties of the system itself. Sharp bounds on the global error for the computed solution are also given in terms of local errors and geometrical properties of the original system. Nonstiff and stiff DREs of unsymmetric and symmetric type are considered. A useful matrix interpretation is given for many integration schemes (such as the backward differentiation formulas, BDF), when applied to the DRE. This allows the matrix structure of the problem to be exploited. In particular, for stiff DREs, the resulting strategy allows for a saving of three orders of magnitude with respect to the standard reform...
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TL;DR: In this paper, a modification of the bending technique was proposed by Mayne and Polak, requiring evaluation of the constraints function at an auxiliary point at each iteration, without losing superlinear convergence.
Abstract: When solving inequality constrained optimization problems via Sequential Quadratic Programming (SQP), it is potentially advantageous to generate iterates that all satisfy the constraints: all quadratic programs encountered are then feasible and there is no need for a surrogate merit function. (Feasibility of the successive iterates is in fact required in many contexts such as in real-time applications or when the objective function is not defined outside the feasible set.) It has recently been shown that this is, indeed, possible, by means of a suitable perturbation of the original SQP iteration, without losing superlinear convergence. In this context, the well-known Maratos effect is compounded by the possible infeasibility of the full step of one even close to a solution. These difficulties have been accommodated by making use of a suitable modification of a “bending” technique proposed by Mayne and Polak, requiring evaluation of the constraints function at an auxiliary point at each iteration.In Part I...
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TL;DR: The upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method are given explicitly for a certain concrete estimator for linear elements as mentioned in this paper.
Abstract: This paper adresses the problem of determining upper and lower bounds for the effectivity index on the a posteriori estimate of the error in the finite element method. These bounds are given explicitly for a certain concrete estimator for linear elements and unstructured triangular meshes. They depend strongly on the geometry of the triangles and (relatively weakly) on the smoothness of the solution. An example shows that the bounds are not over pessimistic. In Babuska, Plank, and Rodriguez (4) detailed numerical experimentation is given.
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TL;DR: In this article, for homogeneous strongly elliptic operators with constant coefficients on polygonal domains, different methods are used based on the expressions of the coefficients obtained in the first two parts; the dual singular function method is also generalized.
Abstract: In the two first parts of this work [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 27e52], [RAIRO Model. Math. Anal. Numer., 24 (1990), pp. 343–367] formulas giving the coefficients arising in the singular expansion of the solutions of elliptic boundary value problems on nonsmooth domains are investigated. Now, for the case of homogeneous strongly elliptic operators with constant coefficients on polygonal domains, the solution of such problems by the finite element method is considered. In order to approximate the solution or the coefficients, different methods are used based on the expressions of the coefficients that were obtained in the first two parts; the dual singular function method is also generalized.
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TL;DR: The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law and it is proved for Lip{sup +}-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency.
Abstract: The convergence rate is discussed of approximate solutions for the nonlinear scalar conservation law. The linear convergence theory is extended into a weak regime. The extension is based on the usual two ingredients of stability and consistency. On the one hand, the counterexamples show that one must strengthen the linearized L(sup 2)-stability requirement. It is assumed that the approximate solutions are Lip(sup +)-stable in the sense that they satisfy a one-sided Lipschitz condition, in agreement with Oleinik's E-condition for the entropy solution. On the other hand, the lack of smoothness requires to weaken the consistency requirement, which is measured in the Lip'-(semi)norm. It is proved for Lip(sup +)-stable approximate solutions, that their Lip'convergence rate to the entropy solution is of the same order as their Lip'-consistency. The Lip'-convergence rate is then converted into stronger L(sup p) convergence rate estimates.
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TL;DR: In this article, the Laplace transform is introduced to analyze the method of orthogonal spline collocation for semidiscretization of a class of linear partial integrodifferential equations arising in viscoelasticity problems.
Abstract: The Laplace transform is introduced to analyze the method of orthogonal spline collocation for the semidiscretization of a class of linear partial integrodifferential equations arising in viscoelasticity problems, for example. The stability and convergence of the space discretization are examined.
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TL;DR: In this article, the stability of discrete approximations for the Dirichlet problem for fully nonlinear, uniformly elliptic partial differential equations in situations where classical solutions are not known to exist is established.
Abstract: This paper exhibits and proves the stability of discrete approximations for the solution of the Dirichlet problem for fully nonlinear, uniformly elliptic partial differential equations in situations where classical solutions are not known to exist. The resulting solutions are characterized in the viscosity sense of Crandall and Lions [Traps. Amer. Math. Soc., 177 (1983) pp. 1–42] and our stability analysis depends on estimates for nonlinear difference equations of positive type which are consequences of earlier work on linear difference equations with random coefficients.
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TL;DR: In this paper, Lagrange multipliers are used for finite element approximations of the Stokes and Navier-Stokes equations with in-homogeneous essential boundary conditions.
Abstract: Finite element approximations of the Stokes and Navier–Stokes equations with in-homogeneous essential boundary conditions are considered. Boundary conditions are enforced weakly by introducing Lagrange multipliers. Optimal error estimates, including some for the stress vector on the boundary, are derived under minimal regularity assumptions on the data. Particular attention is paid to the analysis of a practical choice of finite element spaces for which the Lagrange multiplier calculation uncouples from that for the velocity and pressure. The results are also applicable to general second-order elliptic systems.
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TL;DR: In this paper, the authors studied the Cauchy and initial boundary value problems for a linear advection equation with a nonlinear source term, where the source term is chosen to have two equilibrium states, one unstable and the other stable as solutions of the underlying characteristic equation.
Abstract: The Cauchy and initial boundary value problems are studied for a linear advection equation with a nonlinear source term. The source term is chosen to have two equilibrium states, one unstable and the other stable as solutions of the underlying characteristic equation. The true solutions exhibit travelling waves which propagate from one equilibrium to another. The speed of propagation is dependent on the rate of decay of the initial data at infinity A class of monotone explicit finite-difference schemes are proposed and analysed; the schemes are upwind in space for the advection term with some freedom of choice for the evaluation of the nonlinear source term. Convergence of the schemes is demonstrated and the existence of numerical waves, mimicking the travelling waves in the underlying equation, is proved. The convergence of the numerical wave-speeds to the true wave-speeds is also established. The behaviour of the scheme is studied when the monotonicity criteria are violated due to stiff source terms, and oscillations and divergence are shown to occur. The behaviour is contrasted with a split-step scheme where the solution remains monotone and bounded but where incorrect speeds of propagation are observed as the stiffness of the problem increases.
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TL;DR: In this paper, an optimal order multigrid method for the lowest-order Raviart-Thomas mixed triangular finite element was developed, and the convergence analysis was based on the equivalence between RAVIART-THM mixed methods and certain nonconforming methods.
Abstract: An optimal order multigrid method for the lowest-order Raviart–Thomas mixed triangular finite element is developed. The algorithm and the convergence analysis are based on the equivalence between Raviart–Thomas mixed methods and certain nonconforming methods. Both the Dirichlet and singular Neumann boundary value problems for second-order elliptic equations are discussed.
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TL;DR: In this article, the numerical aspects of variational problems which fail to be convex are studied and an estimate of the approximate deformation as it approximates a measure and some numerical results are also given.
Abstract: In this paper, some numerical aspects of variational problems which fail to be convex are studied. It is well known that for such a problem, in general, the infimum of the energy (the functional that has to be minimized) fails to be attained. Instead, minimizing sequences develop oscillations which allow them to decrease the energy.It is shown that there exists a minimizes for an approximation of the problem and the oscillations in the minimizing sequence are analyzed. It is also shown that these minimizing sequences choose their gradients in the vicinity of the wells with a probability which tends to be constant. An estimate of the approximate deformation as it approximates a measure and some numerical results are also given.
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TL;DR: Estimates for the condition numbers of the corresponding discretization matrices show that a conjugate gradient method applied to the hierarchicalDiscretization (the so-called hierarchical multilevel method) will yield suboptimal convergence rates in comparison with standard multigrid schemes.
Abstract: The paper deals with hierarchical bases in spaces of conforming $C^1 $ elements in connection with the approximate solution of the biharmonic equation \[ \Delta ^2 u = f\quad {\text{in }}\Omega ,\qquad u = \frac{{\partial u}}{{\partial n}} = 0\quad {\text{on }}\partial \Omega \]on a plane polygonal domain $\Omega $. Two different composite finite elements are studied: piecewise quadratic Powell–Sabin elements and piecewise cubic elements of Clough–Tocher type.The main result are estimates for the condition numbers of the corresponding discretization matrices that show that a conjugate gradient method applied to the hierarchical discretization (the so-called hierarchical multilevel method) will yield suboptimal convergence rates in comparison with standard multigrid schemes.