Pointwise Discretization Research Articles

Modeling Variational Inpainting Methods With Splines

Mathematical methods of image inpainting often involve the discretization of a given continuous model. Typically, this is done by a pointwise discretization. We present a method that avoids this by modeling known variational approaches using a finite dimensional spline space. The way we build our algorithm we are able to prove that the basis of the spline space is stable. Further, due to the compact supports and structure of the basis the model involves a sparse system matrix allowing a fast and memory efficient implementation. Besides the analysis of the resulting model, we present a numerical implementation based on the alternating method of multipliers. We compare the results numerically with classical TV inpainting and give examples of applications such as text removal, restoration and noise removal.

An ALE formulation for compressible flows based on multi-moment finite volume method

ABSTRACTA direct Arbitrary Lagrangian Eulerian (ALE) formulation using multi-moment finite volume method for viscous compressible flows on unstructured moving grids has been developed and presented in this paper. High-order polynomials are reconstructed over a compact mesh stencil by making use of both volume integrated average value (VIA) and point values (PV) at the vertices of mesh cells which change in time. By formulating the governing equation in ALE integral form and differential form, the computational variables, VIA and PV, are updated, respectively, by a finite volume scheme and a point-wise discretization using multi-moments (VIA and PV). A simple and efficient formulation is derived for moving mesh which satisfies the geometrical conservation law. In the computations involving moving-body, the PVs of velocity at the vertices of cells adjacent to the body surface are coincided with the motion of the body, which eliminates the numerical approximation to find the cell boundary values on the body surface in conventional finite volume method. Numerical tests are presented to verify the present method as a high-order ALE formulation for compressible flows on both stationary and moving grids.

Density of Spherically Embedded Stiefel and Grassmann Codes

The density of a code is the fraction of the coding space covered by packing balls centered around the codewords. A high density indicates that a code performs well when used as a uniform point-wise discretization of an ambient space. This paper investigates the density of codes in the complex Stiefel and Grassmann manifolds equipped with the chordal distance arising from an Euclidean embedding, including the unitary group as a special case. The choice of distance enables the treatment of the manifolds as subspaces of Euclidean hyperspheres. In this geometry, the densest packings are not necessarily equivalent to maximum–minimum-distance codes. Computing a code’s density follows from computing: 1) the normalized volume of a metric ball and 2) the kissing radius, the radius of the largest balls one can pack around the codewords without overlapping. First, the normalized volume of a metric ball is evaluated by asymptotic approximations. The volume of a small ball can be well-approximated by the volume of a locally equivalent tangential ball. In order to properly normalize this approximation, the precise volumes of the manifolds induced by their spherical embedding are computed. For larger balls, a hyperspherical cap approximation is used, which is justified by a volume comparison theorem showing that the normalized volume of a ball in the Stiefel or Grassmann manifold is asymptotically equal to the normalized volume of a ball in its embedding sphere as the dimension grows to infinity. Then, bounds on the kissing radius are derived alongside corresponding bounds on the density. Unlike spherical codes or codes in flat spaces, the kissing radius of Grassmann or Stiefel codes cannot be exactly determined from its minimum distance. It is nonetheless possible to derive bounds on density as functions of the minimum distance. Stiefel and Grassmann codes have larger density than their image spherical codes when dimensions tend to infinity. Finally, the bounds on density lead to refinements of the standard Hamming bounds for Stiefel and Grassmann codes.

Worst-Case Pointwise Discretization Error Bounds for Systems with Geometrically Induced Singular Flux Solutions Using Interval Boundary Element Method

This paper describes the interval boundary element method treatment of the pointwise discretization error for systems with geometrically generated flux singularities. Worst-case interval bounds are provided for the local discretization error for all elements except for the element with a singular flux solution, for which the flux intensity factor is enclosed. An example is presented showing the behavior of the interval bounds on the local discretization error for systems with geometrically induced singularities.

Worst case bounds on the point-wise discretization error in boundary element method for the elasticity problem

In this work, point-wise discretization error is bounded via interval approach for the elasticity problem using interval boundary element formulation. The formulation allows for computation of the worst case bounds on the boundary values for the elasticity problem. From these bounds the worst case bounds on the true solution at any point in the domain of the system can be computed. Examples are presented to demonstrate the effectiveness of the treatment of local discretization error in elasticity problem via interval methods.

Friction term discretization and limitation to preserve stability and conservation in the 1D shallow‐water model: Application to unsteady irrigation and river flow

AbstractFriction is one of the relevant forces included in the momentum equation of the 1D shallow‐water model. This work shows that a pointwise discretization of the friction term unbalances this term with the rest of the terms in the equation in steady state. On the other hand, an upwind discretization of the friction term ensures the correct discrete balance. Furthermore, a conservative technique based on the limitation of the friction value is proposed in order to avoid unbounded values of the friction term in unsteady cases of advancing front over dry and rough surfaces. This limitation improves the quality of unsteady solutions in wet/dry fronts and guarantees the numerical stability in cases with dominant friction terms. The proposed discretization is validated in some test cases with analytical solution or with measured data and used in some practical cases. Copyright © 2007 John Wiley & Sons, Ltd.

POINTWISE AND UPWIND DISCRETIZATIONS OF SOURCE TERMS IN OPEN-CHANNEL FLOOD ROUTING

Upwind algorithms are becoming progressively popular for river flood routing due to their capability of resolving trans-critical flow regimes. For consistency, these algorithms suggest natural upwind discretization of the source term, which may be essential for natural channels with irregular geometry. Yet applications of these upwind algorithms to natural river flows are rare, and in such applications the traditional and simpler pointwise, rather than upwind discretization of the source term is used. Within the framework of a first-order upwind algorithm, this paper presents a comparison of upwind and pointwise discretizations of the source term. Numerical simulations were carried out for a selected irregular channel comprising a pool-riffle sequence in the River Lune, England with observed data. It is shown that the impact of pointwise discretization, compared to the upwind, is appreciable mainly in flow zones with the Froude number closer to or larger than unity. The discrepancy due to pointwise and upwind discretizations of the source term is negligible in flow depth and hence in water surface elevation, but well manifested in mean velocity and derived flow quantities. Also the occurrence of flow reversal and equalisation over the pool-riffle sequence in response to increasing discharges is demonstrated.

Inequality path constraints in optimal control: a finite iteration ε-convergent scheme based on pointwise discretization

This paper presents a new result in the analysis and implementation of path constraints in optimal control problems (OCPs). The scheme uses the well-known concept of discretizing path constraints on a finite number of points, yielding a set of interior-time point constraints replacing the original path constraints. The approach replaces the original OCP by a sequence of OCPs which is shown to converge in a finite number of steps to the solution of the original path constrained problem with ε-accuracy. Numerical results, verifying the theoretical analysis, are presented. The method is shown to be effective and promising for future applications, particularly in control vector parameterization implementations.

Inverse analysis method using spectral decomposition of Green's function

SUMMARY An inverse analysis method using the spectral decomposition of Green's function is proposed. For linear inverse problems of identifying inner sources from surface responses, Green's function, relates the sources to the responses. Since this function behaves as a compact operator, it admits the spectral decomposition and is expressed as a sum of distinct eigen values and eigen functions. A suitable inverse operator that maps the responses to the sources is then determined. The proposed method consists of the following three procedures: (1) numerically computing the spectral decomposition and determining the inverse operator; (2) estimating a response function using a set of measured data; and (3) predicting a source function from the response function using inverse operator. A simple inverse analysis method, which uses a pointwise discretization of Green's function and computes a generalized inverse matrix applying the singular-value decomposition, is regarded as an approximation to compute the inverse operator. The accuracy, however, is much lower than the proposed method, because of the pointwise discretization and the less accurate computation of the spectral decomposition. Illustrative examples are solved to demonstrate the usefulness of the proposed inverse analysis method. Errors due to improper calculation of the inverse operator of Green’s function are shown.

A Shape Decomposition Technique in Electrical Impedance Tomography

Consider a two-dimensional domain containing a medium with unit electrical conductivity and one or more non-conducting objects. The problem considered here is that of identifying shape and position of the objects on the sole basis of measurements on the external boundary of the domain. An iterative technique is presented in which a sequence of solutions of the direct problem is generated by a boundary element method on the basis of assumed positions and shapes of the objects. The key new aspect of the approach is that the boundary of each object is represented in terms of Fourier coefficients rather than a point-wise discretization. These Fourier coefficients generate the fundamental “shapes” mentioned in the title in terms of which the object shape is decomposed. The iterative procedure consists in the successive updating of the Fourier coefficients at every step by means of the Levenberg–Marquardt algorithm. It is shown that the Fourier decomposition—which, essentially, amounts to a form of image compression—enables the algorithm to image the embedded objects with unprecedented accuracy and clarity. In a separate paper, the method has also been extended to three dimensions with equally good results.

A Nyström algorithm for electrostatics of an anisotropic composite

The construction of efficient numerical algorithms for the electrostatics of locally anisotropic composites seems to be a poorly explored area. In this paper coupled first and second kind Fredholm integral equations for the electrostatics of anisotropic inclusions in an anisotropic matrix are derived. For a square array of circular cylinders the integral equations are solved to high accuracy through pointwise discretization. In the process a recent renormalization method is extended for the evaluation of lattice sums [J. Math. Phys. 35, 6036 (1994)] to encompass the calculation of sums on stretched lattices. Conditionally convergent stretched lattice sums are computed from physical considerations.