Discretization Mesh Size Research Articles

Determination of the right-hand side in elliptic equations

ABSTRACT The problem of determining a term in the right-hand side of elliptic equations from an observation on a part of the boundary is investigated. The inverse problem is formulated as an operator equation and then stabilized by Tikhonov regularization method. The regularized problem is discretized based on Hinze's variational discretization concept and the regularization parameter is chosen guaranteeing that when noise level and the discretization mesh size tend to zero, the solution of the discretized regularized problem converges to the f ∗ -minimum norm solution of the continuous inverse problem. Some numerical examples are presented for illustrating the performance of the proposed method.

Multidimensional sum-up rounding for integer programming in optimal experimental design

We present a numerical method for approximating the solution of convex integer programs stemming from optimal experimental design. The statistical setup consists of a Bayesian framework for linear inverse problems for which the direct relationship is described by a discretized integral equation. Specifically, we aim to find the optimal sensor placement from a set of candidate locations where data are collected with measurement error. The convex objective function is a measure of the uncertainty, described here by the trace or log-determinant of the posterior covariance matrix, for the discretized linear inverse problem solution. The resulting convex integer program is relaxed, producing a lower bound. An upper bound is obtained by extending the sum-up rounding approach to multiple dimensions. For this extension, we analyze its accuracy as a function of the discretization mesh size for a rectangular domain. We show asymptotic optimality of the integer solution defining the upper bound for different experimental design criteria (A- and D-optimal), by proving the convergence to zero of the gap between the upper and lower bounds as the mesh size goes to zero. The technique is illustrated on a two-dimensional gravity surveying problem for both A-optimal and D-optimal sensor placement where our designs yield better results compared with a thresholding rounding approach.

Stability and regularization for determining sets of discrete Laplacian

We study the determining sets for discrete harmonic functions on the square lattices. The stability and regularization of the reconstruction of harmonic functions from its values on a part of a domain are discussed. For some specific configurations, we use the logarithmic convexity estimates to obtain error bounds and propose an optimal choice of the mesh size of discretization.

Discretization of interior point methods for state constrained elliptic optimal control problems: optimal error estimates and parameter adjustment

An adjustment scheme for the relaxation parameter of interior point approaches to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method is based on error estimates of an associated finite element discretization of the relaxed problems and optimally selects the relaxation parameter in dependence on the mesh size of discretization. The finite element analysis for the relaxed problems is carried out and a numerical example is presented which confirms our analytical findings.

Moreau–Yosida Regularization in State Constrained Elliptic Control Problems: Error Estimates and Parameter Adjustment

An adjustment scheme for the regularization parameter of a Moreau-Yosida-based regularization, or relaxation, approach to the numerical solution of pointwise state constrained elliptic optimal control problems is introduced. The method utilizes error estimates of an associated finite element discretization of the regularized problems for the optimal selection of the regularization parameter in dependence on the mesh size of discretization and error estimates for the approximation error due to regularization. The theoretical results are verified numerically.

Wavelet-based Spatiotemporal Multiscaling in Diffusion Problems with Chemically Reactive Boundary

Chemically reacting flows over catalytic and non-catalytic surfaces are one of the elementary operations in chemical processing plants. The underlying physical phenomena span time- and length-scales over several orders of magnitude, for which a robust and flexible modeling framework must efficiently account for. With this purpose as the eventual goal, we propose a wavelet-based multiscale numerical framework and demonstrate it on coupling of two prototype methods for the problem of species generated on a chemically reactive boundary and diffusing through the bulk. The two methods consider different time and length scales. The first method in this coupling, termed fine, models the chemical reactions on the reactive boundary stochastically by the kinetic Monte Carlo method (KMC) and the diffusion in the medium deterministically using relatively small time increments and small spatial discretization mesh size. The second method, termed coarse, models both the reaction and the diffusion deterministically and uses drastically larger time increments and spatial discretization size than the fine model. The two methods are coupled by forming a spatiotemporal Compound Wavelet Matrix (CWM) that combines information about the time and spatial scales contained in them.

Stability in convex semi-infinite programming and rates of convergence of optimal solutions of discretized finite subproblems

We consider convex semiinfinite programming (SIP) problems with an arbitrary fixed index set T. The article analyzes the relationship between the upper and lower semicontinuity (lsc) of the optimal value function and the optimal set mapping, and the so-called Hadamard well-posedness property (allowing for more than one optimal solution). We consider the family of all functions involved in some fixed optimization problem as one element of a space of data equipped with some topology, and arbitrary perturbations are premitted as long as the perturbed problem continues to be convex semiinfinite. Since no structure is required for T, our results apply to the ordinary convex programming case. We also provide conditions, not involving any second order optimality one, guaranteeing that the distance between optimal solutions of the discretized subproblems and the optimal set of the original problem decreases by a rate which is linear with respect to the discretization mesh-size.

Adaptive Preconditioned Variational Methods For Solving Self-Adjoint Partial Differential Equations

The implementation of the Preconditioned Conjugate Gradient method for the solution of large linear systems arising from the discretization of differential operators, requires the pre-determination of only one iteration parameter. The numerical determination of the optimal value of this constant parameter, involves the spectral bounds of some matrices and can be obtained in O ( N 2 ) sine function evaluations, where 1/ N is the discretization mesh size. It is shown that this parameter can be chosen in a stable manner in O (1) operations per iteration, if it is allowed to vary with the iteration index from information derived from the gradient parameters.

Performance and analysis of saddle point preconditioners for the discrete steady-state Navier-Stokes equations

We examine the convergence characteristics of iterative methods based on a new preconditioning operator for solving the linear systems arising from discretization and linearization of the steady-state Navier-Stokes equations. With a combination of analytic and empirical results, we study the effects of fundamental parameters on convergence. We demonstrate that the preconditioned problem has an eigenvalue distribution consisting of a tightly clustered set together with a small number of outliers. The structure of these distributions is independent of the discretization mesh size, but the cardinality of the set of outliers increases slowly as the viscosity becomes smaller. These characteristics are directly correlated with the convergence properties of iterative solvers.

Discretization in semi-infinite programming: the rate of convergence

The discretization approach for solving semi-infinite optimization problems is considered. We are interested in the convergence rate of the error between the solution of the semi-infinite problem and the solution of the discretized program depending on the discretization mesh-size. It will be shown how this rate depends on whether the minimizer is strict of order one or two and on whether the discretization includes boundary points of the index set in a specific way. This is done for ordinary and for generalized semi-infinite problems.

Preconditioning for the Steady-State Navier--Stokes Equations with Low Viscosity

We introduce a preconditioner for the linearized Navier--Stokes equations that is effective when either the discretization mesh size or the viscosity approaches zero. For constant coefficient problems with periodic boundary conditions, we show that the preconditioning yields a system with a single eigenvalue equal to 1, so that performance is independent of both viscosity and mesh size. For other boundary conditions, we demonstrate empirically that convergence depends only mildly on these parameters and we give a partial analysis of this phenomenon. We also show that some expensive subsidiary computations required by the new method can be replaced by inexpensive approximate versions of these tasks based on iteration, with virtually no degradation of performance.

Efficient Iterative Solution of the Three-Dimensional Helmholtz Equation

We examine preconditioners for the discrete indefinite Helmholtz equation on a three-dimensional box-shaped domain with Sommerfeld-like boundary conditions. The preconditioners are of two types. The first is derived by discretization of a related continuous operator that differs from the original only in its boundary conditions. The second is derived by a block Toeplitz approximation to the descretized problem. The resulting preconditioning matrices allow the use of fast transform methods and differ from the discrete Helmholtz operator by an operator of low rank. We present experimental results demonstrating that when these methods are combined with Krylov subspace iteration, convergence rates depend only mildly on both the wave number and discretization mesh size. In addition, the methods display high efficiencies in an implementation on an IBM SP-2 parallel computer.

Perturbation of Eigenvalues of Preconditioned Navier--Stokes Operators

We study the sensitivity of algebraic eigenvalue problems associated with matrices arising from linearization and discretization of the steady-state Navier--Stokes equations. In particular, for several choices of preconditioners applied to the system of discrete equations we derive upper bounds on perturbations of eigenvalues as functions of the viscosity and discretization mesh size. The bounds suggest that the sensitivity of the eigenvalues is at worst linear in the inverse of the viscosity and quadratic in the inverse of the mesh size and that scaling can be used to decrease the sensitivity in some cases. Experimental results supplement these results and confirm the relatively mild dependence on viscosity. They also indicate a dependence on the mesh size of magnitude smaller than the analysis suggests.

A preconditioned conjugate gradient Uzawa‐type method for the solution of the stokes problem by mixed Q1–P0 stabilized finite elements

AbstractWe study the behaviour of a conjugate gradient Uzawa‐type method for a stabilized finite element approximation of the Stokes problem. Many variants of the Uzawa algorithm have been described for different finite elements satisfying the well‐known Inf‐Sup condition of Babuška and Brezzi, but it is surprising that developments for unstable ‘low‐order’ discretizations with stabilization procedures are still missing. Our paper is presented in this context for the popular (so‐called) Q1–P0 element.First we show that a simple stabilization technique for this element permits us to retain the property of a convergence factor bounded independently of the discretization mesh size. The second contribution of this work deals with the construction of a less costly preconditioner taking full advantages of the block diagonal structure of the stabilization matrix. Its efficiency is supported by 2D and. 3D numerical results.

Preconditioning by Fast Direct Methods for Nonself-Adjoint Nonseparable Elliptic Equations

We consider the use of fast direct methods as preconditioners for iterative methods for computing the numerical solution of nonself-adjoint elliptic boundary value problems. We derive bounds on convergence rates that are independent of discretization mesh size. For two-dimensional problems on rectangular domains, discretized on an $n \times n$ grid, these bounds lead to asymptotic operation counts of $O(n^2 \log n\log \varepsilon ^{ - 1} )$ to achieve relative error $\varepsilon $ and $O(n^2 (\log n)^2 )$ to reach truncation error.

The preconditioned variational methods for solving large linear systems

The implementation of the Preconditioned Conjugate Gradient method for the solution of large linear systems arising from the discretization of differential operators, requires the predetermination of only one iteration parameter. The numerical determination of the optimal value of this constant parameter, involve the spectral bounds of some matrices and can be obtained in O(N 2) sine function evaluations, where 1/N is the discretization mesh size. It is shown that this parameter can be chosen in a stable manner in O(1) operations per iteration, if it is allowed to vary with the iteration index from information derived from the gradient parameters.