Space Of Piecewise Constant Functions Research Articles

Staircasing effect for minimizers of the one-dimensional discrete Perona–Malik functional

We consider the one-dimensional Perona–Malik functional, that is the energy associated to the celebrated forward-backward equation introduced by P. Perona and J. Malik in the context of image processing, with the addition of a forcing term. We discretize the functional by restricting its domain to a finite dimensional space of piecewise constant functions, and by replacing the derivative with a difference quotient. We investigate the asymptotic behavior of minima and minimizers as the discretization scale vanishes. In particular, if the forcing term has bounded variation, we show that any sequence of minimizers converges in the sense of varifolds to the graph of the forcing term, but with tangent component which is a combination of the horizontal and vertical directions. If the forcing term is more regular, we also prove that minimizers actually develop a microstructure that looks like a piecewise constant function at a suitable scale, which is intermediate between the macroscopic scale and the scale of the discretization.

An extended finite element method for coupled Darcy–Stokes problems

AbstractIn this article, we propose an extended finite element method to deal with coupled Darcy–Stokes problems with straight or curved interfaces. A mixed approximate form is presented based on Brezzi‐Douglas‐Marini finite element space and piecewise constant function space for Darcy equations, and a mixed approximate form is shown based on piecewise quadratic function space and piecewise constant function space for Stokes equations. By introducing some stabilization terms, we prove that the inf‐sup condition for the discrete coupled problem is valid. Moreover, the optimal convergence is derived. Finally, some numerical examples are presented to demonstrate our theoretical results.

An extended mixed finite element method for elliptic interface problems

In this paper, we propose an extended mixed finite element method for elliptic interface problems. By adding some stabilization terms, we present a mixed approximation form based on Brezzi-Douglas-Marini element space and the piecewise constant function space, and show that the discrete inf-sup constant is independent of how the interface intersects the triangulation. Furthermore, we derive that the optimal convergence holds independent of the location of the interface relative to the mesh. Finally, some numerical examples are presented to verify our theoretical results.

Nonconvex and nonsmooth total variation regularization method for diffuse optical tomography based on RTE * *Submitted Funding: this work is supported by the National Natural Science Foundation of China (Grant No. 11701159).

Diffuse optical tomography with near-infrared light is a promising technique for noninvasive study of the functional characters of human tissues. Mathematically, it is a seriously ill-posed parameter identification problem. For the purpose of better providing both segmentation and piecewise constant approximation of the underlying solution, nonconvex nonsmooth total variation based regularization functional is considered in this paper. We first give a theoretical study on the well-posedness of solutions corresponding to this minimization problem in the Banach space of piecewise constant functions. Moreover, our theoretical results show that the minimizers corresponding to a sequence nonconvex nonsmooth potential functions which converge to the 0–1 functions, can be used to approximate the solution to the weak Mumford–Shah regularization. Then from the numerical side, we propose a double graduated nonconvex Gauss–Newton algorithm to solve this nonconvex nonsmooth regularization. All illustrations and numerical experiments give a flavor of the possibilities offered by the minimizers of the proposed algorithm.

An expanded mixed covolume element method for integro-differential equation of Sobolev type on triangular grids

The expanded mixed covolume Element (EMCVE) method is studied for the two-dimensional integro-differential equation of Sobolev type. We use a piecewise constant function space and the lowest order Raviart-Thomas (mathit{RT}_{0}) space as the trial function spaces of the scalar unknown u and its gradient σ and flux λ, respectively. The semi-discrete and backward Euler fully-discrete EMCVE schemes are constructed, and the optimal a priori error estimates are derived. Moreover, numerical results are given to verify the theoretical analysis.

A Mumford–Shah-type approach to simultaneous reconstruction and segmentation for emission tomography problems with Poisson statistics

Abstract We propose a variational model to simultaneous reconstruction and segmentation in emission tomography. As in the original Mumford–Shah model [27] we use the contour length as penalty term to preserve edge information whereas a different data fidelity term is used to measure the information discrimination between the computed tomography data of the reconstructed object and the observed (or simulated) data. As data fidelity term we use the Kullback–Leibler divergence which originates from the Poisson distribution present in emission tomography. In this paper we focus on piecewise constant reconstructions which is a reasonable assumption in medical imaging. The segmenting contour as well as the corresponding reconstructions are found as minimizers of a Mumford–Shah-type functional over the space of piecewise constant functions. The numerical scheme is implemented by evolving the level-set surface according to the shape derivative of the functional. The method is validated for simulated data with different levels of noise.

Error Estimates of Projection Type Methods for Solving Weakly Singular Integral Equations

We study projection type methods for numerical solution of one class of weakly singular integral Fredholm equations of the second kind including the integral radiative transfer equation. We consider Galerkin type methods and their modifications: the iterated Galerkin method (the Sloan method), the Kantorovich method, and the iterated Kantorovich method. For the approximating space we take the space of piecewise constant functions or the space of piecewise linear functions related to an arbitrary nonuniform grid. Error estimates are obtained.

Variational Mesh Denoising Using Total Variation and Piecewise Constant Function Space.

Mesh surface denoising is a fundamental problem in geometry processing. The main challenge is to remove noise while preserving sharp features (such as edges and corners) and preventing generating false edges. We propose in this paper to combine total variation (TV) and piecewise constant function space for variational mesh denoising. We first give definitions of piecewise constant function spaces and associated operators. A variational mesh denoising method will then be presented by combining TV and piecewise constant function space. It is proved that, the solution of the variational problem (the key part of the method) is in some sense continuously dependent on its parameter, indicating that the solution is robust to small perturbations of this parameter. To solve the variational problem, we propose an efficient iterative algorithm (with an additional algorithmic parameter) based on variable splitting and augmented Lagrangian method, each step of which has closed form solution. Our denoising method is discussed and compared to several typical existing methods in various aspects. Experimental results show that our method outperforms all the compared methods for both CAD and non-CAD meshes at reasonable costs. It can preserve different levels of features well, and prevent generating false edges in most cases, even with the parameters evaluated by our estimation formulae.

Optimization of parameters in SDFEM for different spaces of parameters

This paper is devoted to the numerical solution of the scalar convection–diffusion equation. We present new results of the adaptive technique in finite element method based on minimizing a functional called error indicator. Particularly, we use more different spaces (space of piecewise constant functions, piecewise linear continuous functions, and piecewise linear discontinuous functions) for the parameter τ from the SUPG (SDFEM) method.

Nonconforming Discontinuous Galerkin Methods for Nonlocal Variational Problems

We present a nonconforming discontinuous Galerkin finite element scheme for nonlocal variational problems associated with some linear nonlocal diffusion and nonlocal peridynamic operators subject to volumetric constraints. The nonlocal operators under consideration have nonlocal interaction kernels that exhibit singularities at the origin so that the natural energy spaces do not allow conforming discontinuous finite element functions. The key idea in our method is based on the introduction of modified nonlocal interaction kernels near the origin. It works for multidimensional problems defined on bounded domains partitioned with general meshes and approximations using general basis functions. A convergence theory is established for two types of finite dimensional spaces, with the first type including finite element spaces that lead to unconditional convergence if they contain all continuous piecewise linear functions and the other type being the discontinuous piecewise constant function space that is shown...

Discontinuous Galerkin Immersed Finite Volume Element Method for Anisotropic Flow Models in Porous Medium

By choosing the trial function space to the immersed finite element space and the test function space to be piecewise constant function space, we develop a discontinuous Galerkin immersed finite volume element method to solve numerically a kind of anisotropic diffusion models governed by the elliptic interface problems with discontinuous tensor-conductivity. The existence and uniqueness of the discrete scheme are proved, and an optimal-order energy-norm estimate andL2-norm estimate for the numerical solution are derived.

W1,q stability of the Fortin operator for the MAC scheme

We prove in this paper the continuity of the natural projection operator from $W^{1,q}_{0}(\varOmega )^{d}$ , q?[1,+?), d=2 or d=3, to the MAC discrete space of piecewise constant functions over the dual cells, endowed with the finite volume $W_{0}^{1,q}$ -discrete norm. Since this projection operator is also a Fortin operator (that is an operator which preserves the divergence), this result may be applied to control the pressure in mixed problems where the test function for the velocity must be more regular than $H^{1}_{0}(\varOmega )^{d}$ .

Discretization of the Frobenius–Perron Operator Using a Sparse Haar Tensor Basis: The Sparse Ulam Method

The global macroscopic behavior of a dynamical system is encoded in the eigenfunctions of the associated Frobenius-Perron operator. For systems with low dimensional long term dynamics, efficient techniques exist for a numerical approximation of the most important eigenfunctions; cf. [M. Dellnitz and O. Junge, SIAM J. Numer. Anal., 36 (1999), pp. 491-515]. They are based on a projection of the operator onto a space of piecewise constant functions supported on a neighborhood of the attractor—Ulam's method. In this paper we develop a numerical technique which makes Ulam's approach applicable to systems with higher dimensional long term dynamics. It is based on ideas for the treatment of higher dimensional partial differential equations using sparse grids [C. Zenger, Sparse grids, in Parallel Algorithms for Partial Differential Equations (Kiel, 1990), Vieweg, Braunschweig, 1991, pp. 241-251; H.-J. Bungartz and M. Griebel, Acta Numer., 13 (2004), pp. 147-269]. Here, we use a sparse Haar tensor basis as the underlying approximation space. We develop the technique, establish statements about its complexity and convergence, and present two numerical examples.

Converse Sturm–Hurwitz–Kellogg theorem and related results

We prove that if Vn is a Chebyshev system on the circle and f is a continuous real-valued function with at least n + 1 sign changes then there exists an orientation preserving diffeomorphism of S1 that takes f to a function L2-orthogonal to V. We also prove that if f is a function on the real projective line with at least four sign changes then there exists an orientation preserving diffeomorphism of \({\mathbb{RP}}^1\) that takes f to the Schwarzian derivative of a function on \({\mathbb{RP}}^1\). We show that the space of piecewise constant functions on an interval with values ± 1 and at most n + 1 intervals of constant sign is homeomorphic to n-dimensional sphere.

A Gradient Flow of the One-Dimensional Alt-Caffarelli Functional

We construct a candidate of the gradient flow for a variational functional with a singular term in the one-dimensional case. Applied is the scheme using the theory of Γ-convergence for variational functionals, where both the Dirichlet integral and the singular term are approximated at the same time. An approximated solution is constructed on the space of piecewise constant functions, and the desired gradient flow is built as the limit of it. We establish some properties of the gradient flow as well as obtain PDE including a term of integration by Radon measure.

CBS constants for multilevel splitting of graph-Laplacian and application to preconditioning of discontinuous Galerkin systems

The goal of this work is to derive and justify a multilevel preconditioner of optimal arithmetic complexity for symmetric interior penalty discontinuous Galerkin finite element approximations of second order elliptic problems. Our approach is based on the following simple idea given in [R.D. Lazarov, P.S. Vassilevski, L.T. Zikatanov, Multilevel preconditioning of second order elliptic discontinuous Galerkin problems, Preprint, 2005]. The finite element space V of piece-wise polynomials, discontinuous on the partition T , is projected onto the space of piece-wise constant functions on the same partition that constitutes the largest space in the multilevel method. The discontinuous Galerkin finite element system on this space is associated to the so-called “graph-Laplacian”. In 2-D this is a sparse M-matrix with - 1 as off diagonal entries and nonnegative row sums. Under the assumption that the finest partition is a result of multilevel refinement of a given coarse mesh, we develop the concept of hierarchical splitting of the unknowns. Then using local analysis we derive estimates for the constants in the strengthened Cauchy–Bunyakowski–Schwarz (CBS) inequality, which are uniform with respect to the levels. This measure of the angle between the spaces of the splitting was used by Axelsson and Vassilevski in [Algebraic multilevel preconditioning methods II, SIAM J. Numer. Anal. 27 (1990) 1569–1590] to construct an algebraic multilevel iteration (AMLI) for finite element systems. The main contribution in this paper is a construction of a splitting that produces new estimates for the CBS constant for graph-Laplacian. As a result we have a preconditioner for the system of the discontinuous Galerkin finite element method of optimal arithmetic complexity.

Galerkin Methods in Age and Space for a Population Model with Nonlinear Diffusion

We present Galerkin methods in both the age and space variables for an age-dependent population undergoing nonlinear diffusion. The methods presented are a generalization of the methods presented in {\it A Variable Time Step Method for an Age-dependent Population Model with Nonlinear Diffusion}, where the approximation space in age was taken to be the space of piecewise constant functions. In this paper, we allow the use of discontinuous piecewise polynomial subspaces of $L^2$ as the approximation space in age. As in the piecewise constant case, we move the discretization along characteristic lines. The time variable has been left continuous. The methods are shown to be superconvergent in the age variable.

Full discretization of the porous medium/fast diffusion equation based on its very weak formulation

SKA ‡ Abstract. The very weak formulation of the porous medium/fast diffusion equation yields an evolution problem in a Gelfand triple with the pivot space H 1. This allows to employ methods of the theory of monotone operators in order to study fully discrete approximations combining a Galerkin method (including conforming finite element methods) with the backward Euler scheme. Convergence is shown even for rough initial data and right-hand sides. The theoretical results are illustrated for the piecewise constant finite element approximation of the porous medium equation with the�-distribution as initial value. As a byproduct, L p -stability of theH 1 -orthogonal projection onto the space of piecewise constant functions is shown. ut − �(juj p−2 u) = f; p >1; u= u(x;t); u(�;0) = u0; (x;t) 2 � (0;T); whereR d is a nice bounded domain, −� is the Laplace operator acting on the spatial variables, u0 are given initial data, f is a given right-hand side and we assume some boundary condition on the boundary of . We immediately see that for p = 2 this is the classical heat equation. For p > 2, it is referred to as the porous medium equation. In this case u is the density of the gas at a given point and time. For 1 2, in the one dimensional case, the solution is given by (5.2). We will later use this in numerical experiments. The solution is also known in higher dimensions, see again Vazquez (22, Chapter 4). Throughout the article, we focus on the following generalization of the above