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International audience; Neural field models are nonlinear integro-differential equations for the evolution of neuronal activity, and they are a prototypical large-scale, coarse-grained neuronal model in continuum cortices. Neural fields are often simulated heuristically and, in spite of their popularity in mathematical neuroscience, their numerical analysis is not yet fully established. We introduce generic projection methods for neural fields, and derive a-priori error bounds for these schemes. We extend an existing framework for stationary integral equations to the time-dependent case, which is relevant for neuroscience applications. We find that the convergence rate of a projection scheme for a neural field is determined to a great extent by the convergence rate of the projection operator. This abstract analysis, which unifies the treatment of collocation and Galerkin schemes, is carried out in operator form, without resorting to quadrature rules for the integral term, which are introduced only at a later stage, and whose choice is enslaved by the choice of the projector. Using an elementary timestepper as an example, we demonstrate that the error in a time stepper has two separate contributions: one from the projector, and one from the time discretisation. We give examples of concrete projection methods: two collocation schemes (piecewise-linear and spectral collocation) and two Galerkin schemes (finite elements and spectral Galerkin); for each of them we derive error bounds from the general theory, introduce several discrete variants, provide implementation details, and present reproducible convergence tests.

In this paper we propose a product space reformulation to transform monotone inclusions described by finitely many operators on a Hilbert space into equivalent two-operator problems. Our approach relies on Pierra’s classical reformulation with a different decomposition, which results in a reduction of the dimension of the outcoming product Hilbert space. We discuss the case of not necessarily convex feasibility and best approximation problems. By applying existing splitting methods to the proposed reformulation we obtain new parallel variants of them with a reduction in the number of variables. The convergence of the new algorithms is straightforwardly derived with no further assumptions. The computational advantage is illustrated through some numerical experiments.

In this paper, we investigate the projected dynamical system associated with the generalized variational inequality with hesitant fuzzy relation. We establish the equivalence between the generalized variational inequality with hesitant fuzzy relation and the fuzzy fixed point problem. And we analyze the existence theorem and iterative algorithm of solutions to such problem. Furthermore, using the projection method, we propose a projection neural network for solving the generalized variational inequality with hesitant fuzzy relation and discuss the stability of the proposed projected dynamical system.

This paper studies fully discrete approximations to the evolutionary Navier{ Stokes equations by means of inf-sup stable H1-conforming mixed nite elements with a grad-div type stabilization and the Euler incremental projection method in time. We get error bounds where the constants do not depend on negative powers of the viscosity. We get the optimal rate of convergence in time of the projection method. For the spatial error we get a bound O(hk) for the L2 error of the velocity, k being the degree of the polynomials in the velocity approximation. We prove numerically that this bound is sharp for this method., MINECO grant MTM2016-78995-P (AEI), Junta de Castilla y León grant VA024P17, Junta de Castilla y León grant VA105G18, MINECO grant MTM2015-65608-P

Block Krylov subspace methods (KSMs) comprise building blocks in many state-of-the-art solvers for large-scale matrix equations as they arise, for example, from the discretization of partial differential equations. While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well-explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs.

An analysis of existing and newly derived fast-projection methods for the numerical integration of incompressible Navier&ndash, Stokes equations is proposed. Fast-projection methods are based on the explicit time integration of the semi-discretized Navier&ndash, Stokes equations with a Runge&ndash, Kutta (RK) method, in which only one Pressure Poisson Equation is solved at each time step. The methods are based on a class of interpolation formulas for the pseudo-pressure computed inside the stages of the RK procedure to enforce the divergence-free constraint on the velocity field. The procedure is independent of the particular multi-stage method, and numerical tests are performed on some of the most commonly employed RK schemes. The proposed methodology includes, as special cases, some fast-projection schemes already presented in the literature. An order-of-accuracy analysis of the family of interpolations here presented reveals that the method generally has second-order accuracy, though it is able to attain third-order accuracy only for specific interpolation schemes. Applications to wall-bounded 2D (driven cavity) and 3D (turbulent channel flow) cases are presented to assess the performances of the schemes in more realistic configurations.

Variational inequality theory is an effective tool for engineering, economics, transport and mathematical optimization. Some of the approaches used to resolve variational inequalities usually involve iterative techniques. In this article, we introduce a new modified viscosity-type extragradient method to solve monotone variational inequalities problems in real Hilbert space. The result of the strong convergence of the method is well established without the information of the operator’s Lipschitz constant. There are proper mathematical studies relating our newly designed method to the currently state of the art on several practical test problems.

We propose a semi-Lagrangian method for the numerical solution of the incompressible Navier–Stokes equations. The method is based on the Chorin–Temam fractional step projection method, combined with a fully semi-Lagrangian scheme to approximate both advective and diffusive terms in the momentum equation. A standard finite element method is used instead to solve the Poisson equation for the pressure. The proposed method allows to employ large time steps, while avoiding the solution of large linear systems to compute the velocity components, which would be required by a semi-implicit approach. We report numerical results obtained in two dimensions using triangular meshes on classical benchmarks, showing good agreement with reference solutions in spite of the very large time step employed.

A modified Chorin–Teman (Euler non-incremental) projection method and a modified Euler incremental projection method for non inf-sup stable mixed finite elements are analyzed. The analysis of the classical Euler non-incremental and Euler incremental methods are obtained as a particular case. We first prove that the modified Euler non-incremental scheme has an inherent stabilization that allows the use of non inf-sup stable mixed finite elements without any kind of extra added stabilization. We show that it is also true in the case of the classical Chorin–Temam method. For the second scheme, we study a stabilization that allows the use of equal-order pairs of finite elements. The relation of the methods with the so-called pressure stabilized Petrov Galerkin method (PSPG) is established. The influence of the chosen initial approximations in the computed approximations to the pressure is analyzed. Numerical tests confirm the theoretical results, Research sup-ported by Spanish MINECO under grants MTM2013-42538-P (MINECO, ES) and MTM2016-78995-P (AEI/FEDER UE)

In the numerical treatment of large-scale Sylvester and Lyapunov equations, projection methods require solving a reduced problem to check convergence. As the approximation space expands, this solution takes an increasing portion of the overall computational effort. When data are symmetric, we show that the Frobenius norm of the residual matrix can be computed at significantly lower cost than with available methods, without explicitly solving the reduced problem. For certain classes of problems, the new residual norm expression combined with a memory-reducing device make classical Krylov strategies competitive with respect to more recent projection methods. Numerical experiments illustrate the effectiveness of the new implementation for standard and extended Krylov subspace methods.

Since the appearance of the Barzilai–Borwein (BB) step sizes strategy for unconstrained optimization problems, it received more and more attention of the researchers. It was applied in various fields of the nonlinear optimization problems and recently was also extended to optimization problems with bound constraints. In this paper, we further extend the BB step sizes to more general variational inequality (VI) problems, i.e., we adopt them in projection methods. Under the condition that the underlying mapping of the VI problem is strongly monotone and Lipschitz continuous and the modulus of strong monotonicity and the Lipschitz constant satisfy some further conditions, we establish the global convergence of the projection methods with BB step sizes. A series of numerical examples are presented, which demonstrate that the proposed methods are convergent under mild conditions, and are more efficient than some classical projection-like methods.

In the present paper, delta functions (DFs) are proposed as a new set of basis functions. Their properties and relations to well-known triangular functions (TFs) are described. The simplicity and useful properties of newly proposed sets led us to use them with more accuracy and less computational burden.Furthermore, DFs are applied to propose an efficient method for approximating the solution of integral equations systems. Convergence analysis and the rate of convergence have been considered as well. Some numerical examples are provided to illustrate the computational efficiency and accuracy of the method.

A numerical method for simulating periodic travelling-wave solutions of some nonlinear dispersive wave equations is proposed. The construction of the scheme is based on an efficient computation of the elements that characterize these solutions: the initial profile and the velocity of the wave.

The solution of linear systems with a parameter is an important problem in engineering applications, including structural dynamics, acoustics, and electronic circuit simulations and related model order reduction methods such as Pade via Lanczos. In this paper, we present a method for solving parameterized symmetric linear systems with multiple right-hand sides, based on the Lanczos method. We show that for this class of applications, a simple deflation method can be used. ispartof: SIAM Journal on Matrix Analysis and Applications vol:31 issue:4 pages:1642-1662 status: published

This note describes how the incomplete markets model with aggregate uncertainty in Den Haan et al. [Comparison of solutions to the incomplete markets model with aggregate uncertainty. Journal of Economic Dynamics and Control, this issue] is solved using standard quadrature and projection methods. This is made possible by linking the aggregate state variables to a parameterized density that describes the cross-sectional distribution. A simulation procedure is used to find the best shape of the density within the class of approximating densities considered. This note compares several simulation procedures in which there is—as in the model—no cross-sectional sampling variation.

The main goal of this paper is to present recovery type a posteriori error estimators and superconvergence for the nonconforming finite element eigenvalue approximation of self-adjoint elliptic equations by projection methods. Based on the superconvergence results of nonconforming finite element for the eigenfunction we derive superconvergence and recovery type a posteriori error estimates of the eigenvalue. The results are based on some regularity assumption for the elliptic problem and are applicable to the lowest order nonconforming finite element approximations of self-adjoint elliptic eigenvalue problems with quasi-regular partitions. Therefore, the results of this paper can be employed to provide useful a posteriori error estimators in practical computing under unstructured meshes.

We deal with the time-dependent Navier–Stokes equations (NSE) with Dirichlet boundary conditions on the whole domain or, on a part of the domain and open boundary conditions on the other part. It is shown numerically that combining the penalty-projection method with spatial discretization by the Marker And Cell scheme (MAC) yields reasonably good results for solving the above-mentioned problem. The scheme which has been introduced combines the backward difference formula of second-order (BDF2, namely Gear’s scheme) for the temporal approximation, the second-order Richardson extrapolation for the nonlinear term, and the penalty-projection to split the velocity and pressure unknowns. Similarly to the results obtained for other projection methods, we estimate the errors for the velocity and pressure in adequate norms via the energy method.

In this paper, we propose a new projection method for solving variational inequality problems, which can be viewed as an improvement of the method of Han and Lo [D.R. Han, Hong K. Lo, Two new self-adaptive projection methods for variational inequality problems, Computers & Mathematics with Applications 43 (2002) 1529–1537], by adopting a new step-size rule. The method is as simple as Han and Lo’s methods [D.R. Han, Hong K. Lo, Two new self-adaptive projection methods for variational inequality problems, Computers & Mathematics with Applications 43 (2002) 1529–1537] and other extra-gradient-type methods, which uses only function evolutions and projections onto the feasible set. We prove that under the condition that the underlying function is co-coercive, the sequence generated by the method converges to a solution of the variational inequality problem globally. Some preliminary computational results are reported, which illustrated that the new method is more efficient than Han and Lo’s method [D.R. Han, Hong K. Lo, Two new self-adaptive projection methods for variational inequality problems, Computers & Mathematics with Applications 43 (2002) 1529–1537].

International audience; This paper presents two algorithms for solving the general dynamic equation for aerosols. Coagulation and condensation are solved simultaneously without using splitting. The first method is based on a collocation method for size discretization and a Rosenbrock solver for time integration. In the second method, time discretization is first performed. This leads to a Volterra equation of first kind, which is solved with cubic splines. Both methods are compared to a reference solution for constant coagulation and linear condensation.

In this paper we characterise the weighting subspaces associated with two approximation techniques for solving ordinary differential equations: the Tau Method [E.L. Ortiz, The Tau Method, SIAM J. Numer. Anal. 6 (1969) 480–92] and the orthogonal collocation method. We show that approximations constructed by means of these two methods are always expressible in terms of a prescribed orthogonal polynomials basis, by projection on a suitably chosen finite dimensional weighting subspace. We show, in particular, that collocation is a special Tau Method with a twisted basis.

In this paper, we suggest and analyze a new inexact proximal point method for solving general variational inequalities, which can be considered as an implicit predictor–corrector method. An easily measurable error term is proposed with further relaxed error bound and an optimal step length is obtained by maximizing the profit-function and is dependent on the previous points. Our results include several known and new techniques for solving variational inequalities and related optimization problems. Results obtained in this paper can be viewed as an important improvement and refinement of the previously known results. Preliminary numerical experiments are included to illustrate the advantage and efficiency of the proposed method.

In this paper, we consider and analyze some new projection-proximal methods for solving general variational inequalities. The modified methods converge for pseudomonotone operators which is a weaker condition than monotonicity. The proposed methods include several new and known methods as special cases. Our results can be considered as a novel and important extension of the previously known results. Since the general variational inequalities include the quasi-variational inequalities and implicit complementarity problems as special cases, results proved in this paper continue to hold for these problems.

Kronecker structured representations are used to cope with the state space explosion problem in Markovian modeling and analysis. Currently, an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of Markov chains (MCs) underlying such representations. This paper proposes a block successive overrelaxation (BSOR) preconditioner for hierarchical Markovian models (HMMs; throughout the paper, the HMM acronym stands for hierarchical Markovian models and should not be confused with the HMM that is sometimes used for hidden Markov models.) that are composed of multiple low-level models and a high-level model that defines the interaction among low-level models. The Kronecker structure of an HMM yields nested block partitionings in its underlying continuous-time MC which may be used in the BSOR preconditioner. The computation of the BSOR preconditioned residual in each iteration of a preconditioned projection method becomes the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning. The proposed BSOR preconditioner solves these systems using sparse LU or real Schur factors of diagonal blocks. The fill-in of sparse LU factorized diagonal blocks is reduced using the column approximate minimum degree (COLAMD) ordering. A set of numerical experiments is presented to show the merits of the proposed BSOR preconditioner.

The purpose of this paper is to investigate the problem of iterative computation of approximately balanced reduced order systems. The resulting approach is completely automatic once an error tolerance is specified and also yields an error bound. This is to be contrasted with existing projection methods, namely Pade via Lanczos (PVL) and rational Krylov, which do not satisfy these properties. Our approach is based on the computation and approximation of the cross gramian of the system. The cross gramian is the solution of a Sylvester equation and therefore some effort is dedicated to the study of this equation leading to some new insights. Our method produces a low rank approximation to this gramian in factored form and thus directly provides a reduced order model and a reduced basis for the original system. It is well suited to large scale problems because there are no matrix factorizations of the large (sparse) system matrix. Only matrix–vector products are required.

This paper deals with the treatment of incompressibility in solid mechanics in finite-strain elastoplasticity. A finite-strain model proposed by Miehe, Apel and Lambrecht, which is based on a logarithmic strain measure and its work-conjugate stress tensor is chosen. Its main interest is that it allows for the adoption of standard constitutive models established in a small-strain framework. This model is extended to take into account the plastic incompressibility constraint intrinsically. In that purpose, an extension of this model to a three-field mixed finite element formulation is proposed, involving displacements, a strain variable and pressure as nodal variables with respect to standard finite element. Numerical examples of finite-strain problems are presented to assess the performance of the formulation. To conclude, an industrial case for which the classical under-integrated elements fail is considered. HighlightsA three-field mixed finite element formulation in finite-strain is proposed.It is based on an additive split of the logarithmic strain into elastic and plastic parts.The developed formulation is easy, effective and generic.Our approach was validated on both academic cases and on a complex industrial case.

Consider the linear system Ax=b , where A∈ C N×N is a singular matrix. In the present work we propose a general framework within which Krylov subspace methods for Drazin-inverse solution of this system can be derived in a convenient way. The Krylov subspace methods known to us to date treat only the cases in which A is hermitian and its index ind (A) is unity necessarily. In the present work A is not required to be hermitian. It can have any type of spectrum and ind (A) is arbitrary. We show that, as is the case with nonsingular systems, the Krylov subspace methods developed here terminate in a finite number of steps that is at most N− ind (A) . For one of the methods derived here we also provide an analysis by which we are able to bound the errors, the relevant bounds decreasing with increasing dimension of the Krylov subspaces involved. The results of this paper are applicable to consistent systems as well as to inconsistent ones. An interesting feature of the approach to singular systems presented in this work is that it is formulated as a generalization of the standard Krylov subspace approach to nonsingular systems. Indeed, our approach here reduces to that relevant for nonsingular systems upon setting ind (A)=0 everywhere.

Cataloged from PDF version of article. In this research we use the projection method (reported by Judd) to find numerical solutions to the Euler equations of a stochastic dynamic growth model. The mode1 that we solve is Brock’s asset pricing model for a variety of parameterizations of the production functions. Using simulated data from the model, conjectures (which are not analytically tractable) can be verified. We show that the market portfolio is mean-variance efficient in this dynamic context. We also show a result that is not available from the static CAPM theory: the efficient frontier shifts up and down over the business cycle.

International audience; The generalized Stokes problem (GSP) is broadly recognized as a keystone in implicit or semi-implicit discretizations of the Navier--Stokes equations (NSE), either incompressible or of low Mach number type, i.e., for which there exists a constraint on the vector field. The GSP is also known as the steady Darcy--Brinkman model for flows in porous media or in Hele--Shaw configurations. Up to now, only pressure- (preconditioned) Uzawa methods claimed to solve the three-dimensional (3D) GSP exactly (i.e., without introducing an error due to some time stepping). In the present article, we present another exact 3D solver that is developed in the particular context of the Helmholtz--Hodge projection in $(H^1)^d$. Instead of working iteratively with the pressure operator (i.e., the Schur complement), we are interested here in using an alternative scalar field, which performs the projection and is called the gauge. It turns out that iterative computation of the gauge presents efficient properties in terms of operator conditioning. To prove the relevance of the gauge method, we demonstrate several mathematical properties of the operator acting on the gauge, especially when preconditioned in an appropriate way. In particular, we prove that the gauge operator can be solved efficiently with any gradient method. The convergence properties of the continuous preconditioned operator are quantitatively estimated in the context of a semiperiodic two-dimensional (2D) domain. We then study the discrete gauge operator implemented in the framework of Chebyshev 2D-3D pseudospectral methods. It exhibits the same favorable properties as those found for the continuous operator. Finally, we perform numerical experiments that corroborate our analyses.

Projection methods have emerged as competitive techniques for solving large scale matrix Lyapunov equations. We explore the numerical solution of this class of linear matrix equations when a Minimal Residual (MR) condition is used during the projection step. We derive both a new direct method, and a preconditioned operator-oriented iterative solver based on CGLS, for solving the projected reduced least squares problem. Numerical experiments with benchmark problems show the effectiveness of an MR approach over a Galerkin procedure using the same approximation space.

We consider Volterra integral equations having a finite dimensional feature space. This provides us flexibility to construct an orthonormal basis with small support that can be preserved by the Volterra integral operator. Under the projection method, such a basis yields a sparse discretization matrix for the Volterra integral operator. When the feature space is refinable, we introduce a construction of such an orthonormal basis from existing references. Finally, we present applications to numerical differentiation for which we obtain a quasi-linear lossless compression of the discretization matrix.

This study provides a step further in the computation of the transition path of a continuous time endogenous growth model discussed by Privileggi (Nonlinear dynamics in economics, finance and social sciences: essays in honour of John Barkley Rosser Jr., Springer, Berlin, Heidelberg, pp. 251---278, 2010)--based on the setting first introduced by Tsur and Zemel (J Econ Dyn Control 31:3459---3477, 2007)--in which knowledge evolves according to the Weitzman (Q J Econ 113:331---360, 1998) recombinant process. A projection method, based on the least squares of the residual function corresponding to the ODE defining the optimal policy of the `detrended' model, allows for the numeric approximation of such policy for a positive Lebesgue measure range of values of the efficiency parameter characterizing the probability function of the recombinant process. Although the projection method's performance rapidly degenerates as one departs from a benchmark value for the efficiency parameter, we are able to numerically compute time-path trajectories which are sufficiently regular to allow for sensitivity analysis under changes in parameters' values.

A direct method is described for the approximation of nonsimple turning points, corresponding to cusp points, of nonlinear operator equations depending on two parameters. The procedure is based on the application of a special projection method to the computation of simple turning points of a suitable augmented system. Numerical examples illustrate the features of the proposed algorithm.

International audience; Relying on the prox-regularity notion, we introduce and establish the convergence of two-step projection methods for a system of nonconvex variational inequality problems. We obtain as a particular case some known results in this field.

The paper presents the first implementation of a primitive variable spectral method for calculating viscous flows inside a sphere. A variational formulation of the Navier-Stokes equations is adopted using a fractional-step time discretization with the classical second-order backward difference scheme combined with explicit extrapolation of the nonlinear term. The resulting scalar and vector elliptic equations are solved by means of the direct spectral solvers developed recently by the authors. The spectral matrices for radial operators are characterized by a minimal sparsity - diagonal stiffness and tridiagonal mass matrix. Closed-form expressions of their nonzero elements are provided here for the first time, showing that the condition number of the relevant matrices grows as the second power of the truncation order. A new spectral elliptic solver for the velocity unknown in spherical coordinates is also described that includes implicitly the Coriolis force in a rotating frame, but requires a minimal coupling between the modal velocity components in the Fourier space. The numerical tests confirm that the proposed method achieves spectral accuracy and ensures infinite differentiability to all orders of the numerical solution, by construction. These results indicate that the new primitive variable spectral solver is an effective alternative to the spectral method recently proposed by Kida and Nakayama, where the velocity field is represented in terms of poloidal and toroidal functions.

In many fields of numerical analysis there appear transformations of the form T k v = Σ v + k i = v α k i,v T i . When v varies, a sequence of transformations is obtained. This approach covers, for example, the E− and Θ-algorithms, the recursion formulae for B-splines, Bernstein polynomials and orthogonal polynomials, Pade approximants, the divided difference scheme and projection methods. In this paper it will be proved that such transformations can be written as a ratio of determinants and can be recursively computed by a triangular recursion scheme. The reciprocal of these results also holds. Furthermore, we will show that T v k can be represented in terms of a complex contour integral. Throughout the paper we will study several examples in some detail, and it will turn out that the application of our general theory leads to interesting new results in the special cases. Among others, we will derive a new determinantal representation formula for B-splines, a recurrence relation for generalized Bernstein polynomials, a generalization of the E-algorithm and we will prove that the Θ-algorithm can be represented as a quotient of determinants.

In this paper we combine dynamic programming methods with projection methods for solving stochastic growth models. As an application of these methods, we solve Brock's asset pricing model with a variety of parameterizations. We focused on finding parameterizations that result in an equity premium that is high relative to the variation in consumption. We show (both analytically and numerically) that the equity premium can be higher in a production based asset pricing model than it is in the consumption based asset pricing model, even when the real output level is the same in both models. © 2006 Elsevier B.V. All rights reserved.

A new algorithm is developed to solve models with heterogeneous agents and aggregate uncertainty that avoids some disadvantages of the prevailing algorithm that strongly relies on simulation techniques and is easier to implement than existing algorithms. A key aspect of the algorithm is a new procedure that parameterizes the cross-sectional distribution, which makes it possible to avoid Monte Carlo integration. The paper also develops a new simulation procedure that not only avoids cross-sectional sampling variation but is also more than ten times faster than the standard procedure of simulating an economy with a large but finite number of agents. This procedure can help to improve the efficiency of the most popular algorithm in which simulation procedures play a key role., Cet article développe une nouvelle méthode de résolution des modèles macrodynamiques à agents hétérogènes. Cette méthode propose de directement paramétriser l'évolution de la distribution des richesses suite à des chocs macroéconomiques ou des politiques économiques plutôt que de la simuler. Cette stratégie se révèle non seulement plus rapide mais aussi plus précise que les techniques traditionnelles de simulation de Monte Carlo en évitant tout biais d'échantillonnage. Nous montrons que cette avancée se révèle cruciale pour évaluer les effets redistributifs des fluctuations et des politiques économiques.

Multiphase flow through porous media is a highly nonlinear process that can be solved numerically with the aid of finite elements (FE) in space and finite differences (FD) in time. For an accurate solution much refined FE grids are generally required with the major computational effort consisting of the resolution to the nonlinearity frequently obtained with the classical Picard linearization approach. The efficiency of the repeated solution to the linear systems within each individual time step represents the key to improve the performance of a multiphase flow simulator. The present paper discusses the performance of the projection solvers (GMRES with restart, TFQMR, and BiCGSTAB) for two global schemes based on a different nodal ordering of the unknowns (ORD1 and ORD2) and a scheme (SPLIT) based on the straightforward inversion of the lumped mass matrix which allows for the preliminary elimination and substitution of the unknown saturations. It is shown that SPLIT is between two and three time faster than ORD1 and ORD2, irrespective of the solver used. Copyright © 2005 John Wiley & Sons, Ltd.

It is well known that projection schemes for certain linear ill-posed problems A퓍 = y can be regularized by a proper choice of the discretization level only, where no additional regularization is needed. The previous study of this self-regularization phenomenon was restricted to the case of so-called moderately ill-posed problems, i.e., when the singular values σ푘(A), 푘 = 1,2,..., of the operator A tend to zero with polynomial rate. The main accomplishment of the present paper is a new strategy for a discretization level choice that provides optimal order accuracy also for severely ill-posed problems, i.e., when σ푘(A) tend to zero exponentially. The proposed strategy does not require a priori information regarding the solution smoothness and the exact rate of σ푘(A).

We consider the problem of estimating a functional of a density of the type $\int \phi (f, \cdot)$. Starting from efficient estimators of linear and quadratic functionals of f and using a Taylor expansion of $\phi$, we build estimators that achieve the $n^{-1/2}$ rate whenever f is smooth enough. Moreover, we show that these estimators are efficient. Concerning the estimation of quadratic functionals (more precisely, of integrated squared density) Bickel and Ritov have already built efficient estimators. We propose here an alternative construction based on projections, which seems more natural.

Date of Conference: 18-21 August 1996 Conference Name: 3rd International Workshop on Applied Parallel Computing, PARA 1996 The linear feasibility problem arises in several areas of applied mathematics and medical science, in several forms of image reconstruction problems. The surrogate constraint algorithm of Yang and Murty for the linear feasibility problem is implemented and analyzed. The sequential approach considers projections one at a time. In the parallel approach, several projections are made simultaneously and their convex combination is taken to be used at the next iteration. The sequential method is compared with the parallel method for varied numbers of processors. Two improvement schemes for the parallel method are proposed and tested.

The paper presents a selection of modern results concerning the numerical analysis of one-dimensional Cauchy singular integral equations, in particular the stability of operator sequences associated with different projection methods. The aim of the paper is to show the main ideas and approaches, such as the concept of transforming the question of the stability of an operator sequence into an invertibility problem in a certain Banach algebra or the concept of certain scales of weighted Besov spaces to prove convergence rates of the sequence of the approximate solutions. Moreover, computational aspects, in particular the construction of fast algorithms, are discussed.

Existence and uniqueness of solutions, as well as their explicit representations, are obtained for singular integral equations with weighted Carleman shift which cannot be reduced to binomial boundary value problems.

This paper examines a variation on Newton's method in which the Euclidian norm of the residue vector is reduced at each step by a projection method which depends on the Jacobian matrix. Rather than inverting the full Jacobian at each step, this method solves a projection problem on a (possibly) lower dimensional subspace. We examine the suitability of the method for underdetermined or overdetermined problems as well. Convergence criteria are developed for certain subspace selection methods and numerical examples are given.