Bounded Variation Research Articles

Numerical solutions to Hyperbolic Maxwell quasi-variational inequalities in Bean–Kim model for type-II superconductivity

This paper is devoted to the finite element analysis for the Bean–Kim model governed by the full 3D Maxwell equations. Describing type-II superconductivity at the macroscopic level, this model leads to a challenging coupled system consisting of the Faraday equation and a hyperbolic quasi-variational inequality (QVI) of the second kind with L1-type nonlinearity, that arises explicitly from the magnetic field dependency in the critical current. With the involved Maxwell coupling in the 3D H(curl)-setting, the hyperbolic QVI character poses the primary challenge in the numerical investigation. Two mixed finite element methods based on implicit Euler and leapfrog time-stepping are proposed. On the one hand, the implicit Euler method results in a nonstandard system of curl-curl elliptic QVI with a first-order curl-type nonlinearity. Though the well-posedness of this system is guaranteed, its numerical realization is not straightforward and requires the use of a two-stage iteration process of high computational complexity. On the other hand, by approximating the electric and magnetic fields at two different time step levels, the leapfrog method turns out to be more suitable as it naturally eliminates the notorious QVI structure. More importantly, utilizing suited subdifferential and optimization techniques, we are able to prove an efficiently computable explicit formula for its exact solution in terms of the electric field, which makes its numerical computation substantially more favorable than the Euler method. As further advantages, the leapfrog method applies to broad scenarios involving low regular data of bounded variation (BV) in time for both the applied current source and the temperature distribution. Through nonstandard technical arguments tailored to the BV data, our analysis proves the conditional stability and, eventually, the uniform convergence of the proposed leapfrog method. This paper is closed by 3D numerical tests showcasing the reasonable and efficient performance of the proposed numerical solution.

Convergence of the numerical approximations and well-posedness: Nonlocal conservation laws with rough flux

We study a class of nonlinear nonlocal conservation laws with discontinuous flux, modeling crowd dynamics and traffic flow. The discontinuous coefficient of the flux function is assumed to be of bounded variation (BV) and bounded away from zero, and hence the spatial discontinuities of the flux function can be infinitely many with possible accumulation points. Strong compactness of the Godunov and Lax-Friedrichs type approximations is proved, providing the existence of entropy solutions. A proof of the uniqueness of the adapted entropy solutions is provided, establishing the convergence of the entire sequence of finite volume approximations to the adapted entropy solution. As per the current literature, this is the first well-posedness result for the aforesaid class and connects the theory of nonlocal conservation laws (with discontinuous flux), with its local counterpart in a generic setup. Some numerical examples are presented to display the performance of the schemes and explore the limiting behavior of these nonlocal conservation laws to their local counterparts.

BV Solutions to Evolution Inclusion with a Time and Space Dependent Maximal Monotone Operator

This paper deals with the research of solutions of bounded variation (BV) to evolution inclusion coupled with a time and state dependent maximal monotone operator. Different problems are studied: existence of solutions, unicity of the solution, existence of periodic and bounded variation right continuous (BVRC) solutions. Second-order evolution inclusions and fractional (Caputo and Riemann–Liouville) differential inclusions are also considered. A result of the Skorohod problem driven by a time- and space-dependent operator under rough signal and a Volterra integral perturbation in the BRC setting is given. The paper finishes with some results for fractional differential inclusions under rough signals and Young integrals. Many of the given results are novel.

A variational level set model based on additive decomposition for segmenting noisy images with intensity inhomogeneity

The variational level set model is widely used in image segmentation. However, it suffers from the limited performance when the images are contaminated by noise and intensity inhomogeneity. To solve this problem, a variational level set model based on additive decomposition is proposed in this paper. First, we decompose the image into three components: structure, intensity inhomogeneity and noise, which are regularized by different metrics. Specially, the structure component is regularized as a piecewise constant function in bounded variation (BV) space; the intensity inhomogeneity is modeled as a smooth function that is regularized by H1 space, and the residual containing noise is measured by L2 norm. Furthermore, a new indirect regularization term for level set function is designed to enhance the accuracy of the segmentation outcomes. And then, alternate direction iteration algorithm combining with gradient descent and ADMM to solve the proposed model. Experiments on both synthetic and real images validate the proposed model. Compared with five state-of-the-art models, the experimental results demonstrate that the proposed model exhibits the significant improvement in both accuracy and efficiency, outperforming all other methods. Quantitative evaluations show the average segmentation coefficients for Jaccard, Dice and Accuracy against ground truth are 0.93, 0.95 and 0.96, respectively, and the average running time is 0.80 seconds, further confirming its superior performance.

Exponential time‐decay for a one‐dimensional wave equation with coefficients of bounded variation

AbstractWe consider the initial‐value problem for a one‐dimensional wave equation with coefficients that are positive, constant outside of an interval, and have bounded variation (BV). Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges. The key ingredient of the proof is a high‐frequency resolvent estimate for an associated Helmholtz operator with a BV potential.

On the rate of convergence of a numerical scheme for Fractional conservation laws with noise

Abstract We consider a semidiscrete finite volume scheme for a degenerate fractional conservation law driven by a cylindrical Wiener process. Making use of the bounded variation (BV) estimates, and a clever adaptation of classical Kružkov theory, we provide estimates on the rate of convergence for approximate solutions to degenerate fractional problems. The main difficulty stems from the degenerate fractional operator and requires a significant departure from the existing strategy to establish Kato’s type of inequality. Indeed, recasting the mathematical framework recently developed in Bhauryal et al. (2021, J. Differential Equations, 284, 433–521), we establish such Kato’s type of inequality for a finite volume scheme. Finally, as an application of this theory, we demonstrate numerical convergence rates.

Magnetic resonance imaging enhancement using prior knowledge and a denoising scheme that combines total variation and histogram matching techniques

IntroductionBrain perfusion-weighted images obtained through dynamic contrast studies play a critical and clinical role in diagnosis and treatment decisions. However, due to the patient's limited exposure to radiation, computed magnetic resonance imaging (MRI) suffers from low contrast-to-noise ratios (CNRs). Denoising MRI images is a critical task in many e-health applications for disease detection. The challenge in this research field is to define novel algorithms and strategies capable of improving accuracy and performance in terms of image vision quality and computational cost to process data. Using MRI statistical information, the authors present a method for improving image quality by combining a total variation-based denoising algorithm with histogram matching (HM) techniques.MethodsThe total variation is the Rudin–Osher–Fatemi total variation (TV-ROF) minimization approach, TV-L2, using the isotropic TV setting for the bounded variation (BV) component. The dual-stage approach is tested against two implementations of the TV-L2: the split Bregman (SB) algorithm and a fixed-point (FP) iterations scheme. In terms of HM, the study explores approximate matching and the exact histogram matching from Coltuc.ResultsAs measured by the structural similarity index (SIMM), the results indicate that in the more realistic study scenarios, the FP with an HM pairing is one of the best options, with an improvement of up to 12.2% over the one without an HM.DiscussionThe findings can be used to evaluate and investigate more advanced machine learning-based approaches for developing novel denoising algorithms that infer information from ad hoc MRI histograms. The proposed methods are adapted to medical image denoising since they account for the preference of the medical expert: a single parameter can be used to balance the preservation of (expert-dependent) relevant details against the degree of noise reduction.

Variational discretization of one-dimensional elliptic optimal control problems with BV functions based on the mixed formulation

<p style='text-indent:20px;'>We consider optimal control of an elliptic two-point boundary value problem governed by functions of bounded variation (BV). The cost functional is composed of a tracking term for the state and the BV-seminorm of the control. We use the mixed formulation for the state equation together with the variational discretization approach, where we use the classical lowest order Raviart-Thomas finite elements for the state equation. Consequently the variational discrete control is a piecewise constant function over the finite element grid. We prove error estimates for the variational discretization approach in combination with the mixed formulation of the state equation and confirm our analytical findings with numerical experiments.</p>

Convergence, bounded variation properties and Kruzhkov solution of a fully discrete Lagrangian–Eulerian scheme via weak asymptotic analysis for 1D hyperbolic problems

AbstractWe design and implement an effective fully discrete Lagrangian–Eulerian scheme for a class of scalar, local and nonlocal models, and systems of hyperbolic problems in 1D. We propose statements, via a weak asymptotic analysis, which include existence, uniqueness, regularity, and numerical approximations of entropy‐weak solutions computed with the scheme for the corresponding nonlinear initial value problem for the local scalar case. We study both convergence and weak bounded variation (BV) properties of the scheme to the entropy solution (for the local and scalar case) in the sense of Kruzhkov. The approach is based on the improved concept of no‐flow curves, as introduced by the authors, and we highlight the strengths of the method: (i) the scheme for systems of hyperbolic problems does not require computation of the eigenvalues (exact or approximate) either to the numerical flux function or the weak CFL stability condition (wCFL) and (ii) we prove the properties: positivity‐preserving, total variation nonincreasing, and maximum principle subject to the wCFL. We present numerical experiments to evaluate the shock capturing capabilities of the scheme in resolving the main features for hyperbolic problems: shock waves, rarefaction waves, contact discontinuities, positivity‐preserving properties, and nonlinear wave formations and interactions.

On BV functions and essentially bounded divergence-measure fields in metric spaces

By employing the differential structure recently developed by N. Gigli, we first give a notion of functions of bounded variation (BV) in terms of suitable vector fields on a complete and separable metric measure space (\mathbb{X},d,\mu) equipped with a non-negative Radon measure \mu finite on bounded sets. Then, we extend the concept of divergence-measure vector fields \mathcal{DM}^p(\mathbb{X}) for any p\in[1,\infty] and, by simply requiring in addition that the metric space is locally compact, we determine an appropriate class of domains for which it is possible to obtain a Gauss–Green formula in terms of the normal trace of a \mathcal{DM}^\infty(\mathbb{X}) vector field. This differential machinery is also the natural framework to specialize our analysis for \mathsf{RCD}(K,\infty) spaces, where we exploit the underlying geometry to determine the Leibniz rules for \mathcal{DM}^\infty(\mathbb{X}) and ultimately to extend our discussion on the Gauss–Green formulas.

Besov class via heat semigroup on Dirichlet spaces III: BV functions and sub-Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Émery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $$L^1$$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $$L^1$$ . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

Rough traces of BV functions in metric measure spaces

Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation (BV) to the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such functions. We then compare our analysis with the study done in a recent work by Lahti and Shanmugalingam, where traces of BV functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.

A fractional degenerate parabolic-hyperbolic Cauchy problem with noise

We consider the Cauchy problem for a stochastic scalar parabolic-hyperbolic equation in any space dimension with nonlocal, nonlinear, and possibly degenerate diffusion terms. The equations are nonlocal because they involve fractional diffusion operators. We adapt the notion of stochastic entropy solution and provide a new technical framework to prove the uniqueness. The existence proof relies on the vanishing viscosity method. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities and derive error estimate for the stochastic vanishing viscosity method. In addition, we develop uniqueness method “à la Kružkov” for more general equations where the noise coefficient may depend explicitly on the spatial variable.

Stability of Conical Shocks in the Three-Dimensional Steady Supersonic Isothermal Flows Past Lipschitz Perturbed Cones

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are governed by the steady isothermal Euler equations for potential flow with axisymmetry so that the equations contain a singular geometric source term. We first formulate the shock stability problem as an initial-boundary value problem with the leading conical shock-front as a free boundary, and then establish the existence and asymptotic behavior of global entropy solutions of bounded variation (BV) of the problem. To achieve this, we first develop a modified Glimm scheme to construct approximate solutions via self-similar solutions as building blocks in order to incorporate with the geometric source term. Then we introduce the Glimm-type functional, based on the local interaction estimates between weak waves, the strong leading conical shock, and self-similar solutions, as well as the estimates of the center changes of the self-similar solutions. To make sure the decreasing of the Glimm-type functional, we choose appropriate weights by careful asymptotic analysis of the reflection coefficients in the interaction estimates, when the Mach number of the incoming flow is sufficiently large. Finally, we establish the existence of global entropy solutions involving a strong leading conical shock-front, besides weak waves, under the conditions that the Mach number of the incoming flow is sufficiently large and the weighted total variation of the slopes of the generating curve of the Lipschitz perturbed cone is sufficiently small. Furthermore, the entropy solution is shown to approach asymptotically the self-similar solution that is determined by the incoming flow and the asymptotic tangent of the cone boundary at infinity.

Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

With a view toward fractal spaces, by using a Korevaar–Schoen space approach, we introduce the class of bounded variation (BV) functions in the general framework of strongly local Dirichlet spaces with a heat kernel satisfying sub-Gaussian estimates. Under a weak Bakry–Emery curvature type condition, which is new in this setting, this BV class is identified with a heat semigroup based Besov class. As a consequence of this identification, properties of BV functions and associated BV measures are studied in detail. In particular, we prove co-area formulas, global $$L^1$$ Sobolev embeddings and isoperimetric inequalities. It is shown that for nested fractals or their direct products the BV class we define is dense in $$L^1$$ . The examples of the unbounded Vicsek set, unbounded Sierpinski gasket and unbounded Sierpinski carpet are discussed.

The Cauchy problem for fractional conservation laws driven by Lévy noise

In this article, we explore some of the main mathematical problems connected to multidimensional fractional conservation laws driven by Lévy processes. Making use of an adapted entropy formulation, a result of existence and uniqueness of a solution is established. Moreover, using bounded variation (BV) estimates for vanishing viscosity approximations, we derive an explicit continuous dependence estimate on the nonlinearities of the entropy solutions under the assumption that the Lévy noise depends only on the solution. This result is used to show the error estimate for the stochastic vanishing viscosity method. Furthermore, we establish a result on vanishing non-local regularization of scalar stochastic conservation laws.

A perfect reconstruction property for PDE-constrained total-variation minimization with application in Quantitative Susceptibility Mapping

We study the recovery of piecewise constant functions of finite bounded variation (BV) from their image under a linear partial differential operator with unknown boundary conditions. It is shown that minimizing the total variation (TV) semi-norm subject to the associated PDE-constraints yields perfect reconstruction up to a global constant under a mild geometric assumption on the jump set of the function to reconstruct. The proof bases on establishing a structural result about the jump set associated with BV-solutions of the homogeneous PDE. Furthermore, we show that the geometric assumption is satisfied up to a negligible set of orthonormal transformations. The results are then applied to Quantitative Susceptibility Mapping (QSM) which can be formulated as solving a two-dimensional wave equation with unknown boundary conditions. This yields in particular that total variation regularization is able to reconstruct piecewise constant susceptibility distributions, explaining the high-quality results obtained with TV-based techniques for QSM.

Functions of bounded variation and the [formula omitted]-modulus in [formula omitted

Moduli of path families are widely used to study Sobolev functions. Similarly, the recently introduced approximation (AM-) modulus is helpful in the theory of functions of bounded variation (BV) in Rn (Martio, 2016). We continue this direction of research. Let ΓE be the family of all paths which meet E⊂Rn. We introduce the outer measure E↦AM(ΓE) and compare it with other (n−1)-dimensional measures. In particular, we show that AM(ΓE)=2Hn−1(E) whenever E lies on a countably (n−1)-rectifiable set. Further, we study functions which have bounded variation on AM-a.e. path and we relate these functions to the classical BV functions which have only bounded essential variation on AM-a.e. path. We also characterize sets E of finite perimeter in terms of the AM-modulus of two path families naturally associated with E.

Solutions to the Einstein-Scalar-Field System in Spherical Symmetry with Large Bounded Variation Norms

It is well-known that small, regular, spherically symmetric characteristic initial data to the Einstein-scalar-field system which are decaying towards (future null) infinity give rise to solutions which are foward-in-time global (in the sense of future causal geodesic completeness). We construct a class of spherically symmetric solutions which are global but the initial norms are consistent with initial data not decaying towards infinity. This gives the following consequences: (1) We prove that there exist foward-in-time global solutions with arbitrarily large (and in fact infinite) initial bounded variation (BV) norms and initial Bondi masses. (2) While general solutions with non-decaying data do not approach Minkowski spacetime, we show using the results of Luk--Oh that if a sufficiently strong asymptotic flatness condition is imposed on the initial data, then the solutions we construct (with large BV norms) approach Minkowski spacetime with a sharp inverse polynomial rate. (3) Our construction can be easily extended so that data are posed at past null infinity and we obtain solutions with large BV norms which are causally geodesically complete both to the past and to the future. Finally, we discuss applications of our method to construct global solutions for other nonlinear wave equations with infinite critical norms.

Which special functions of bounded deformation have bounded variation?

Functions of bounded deformation (BD) arise naturally in the study of fracture and damage in a geometrically linear context. They are related to functions of bounded variation (BV), but are less well understood. We discuss here the relation to BV under additional regularity assumptions, which may require the regular part of the strain to have higher integrability or the jump set to have finite area or the Cantor part to vanish. On the positive side, we prove that BD functions that are piecewise affine on a Caccioppoli partition are in GSBV, and we prove that SBDp functions are approximately continuous -almost everywhere away from the jump set. On the negative side, we construct a function that is BD but not in BV and has distributional strain consisting only of a jump part, and one that has a distributional strain consisting of only a Cantor part.