Gradient Of Solution Research Articles

Active Contour Directed by the Poisson Gradient Vector Field and Edge Tracking

This paper develops a new active contour (AC) model capable of multiple complex objects segmentation in the presence of heavy noise. The model segments images in the framework of two types of partial differential equations (PDEs): the Euler–Lagrange and Poisson PDEs. The former is used to build an evolution algorithm, while the Poisson solution gradient vector field (PGVF) directs the evolution toward the boundaries of all image objects. The AC halts on boundaries and PGVF separatrices, splits on the latter, and leaves at least one segment (called label) on every boundary. Each label tracks its boundary until the corresponding object is enveloped. The advantages of the new method are validated on a number of skin lesions, road, and aircraft images of varying sizes and in the presence of Gaussian noise. The obtained results are compared against results by contemporary and established active contours and neural networks.

Maximal Differentiability for a General Class of Quasilinear Elliptic Equations with Right-hand Side Measures

Abstract We prove maximal differentiability for the gradient of solutions to a certain type of nonlinear elliptic equations with a measure on the right-hand side. Our results generalize the limiting case of Calderón–Zygmund theory for an elliptic equation with $p$-growth to the case of Orlicz growth.

Concentration Gradients in Material Sciences: Methods to Design and Biomedical Applications

AbstractConcentration gradients are at the basis of many natural phenomena and hold great importance in human and plant biology. This basic phenomenon has inspired many material scientists to confer novel and original properties to materials with numerous biomedical and pharmaceutical applications such as drug discovery, controlled release of drugs, tissue engineering, orthopedic implants and time–temperature indicators, amongst others. While the generation of gradients in solution is well described, their incorporation in solid materials remains challenging. This study reviews concepts and methods reported to generate gradients in respective solutions and (at the surface or in) solid materials like polymer scaffolds and matrices. This article also discusses how innovative materials possessing such gradients may be applied in biomedicine and pharmacy.

Effect of Magnetic Field on Thermosolutal Instability of Rotating Ferromagnetic Fluid Under Varying Gravity Field

This paper deals with the theoretical investigation of the effect of a magnetic field, rotation and magnetization on a ferromagnetic fluid under varying gravity field. To find the exact solution for a ferromagnetic fluid layer contained between two free boundaries, we have used a linear stability analysis and normal mode analysis method. For the case of stationary convection, a stable solute gradient has a stabilizing effect, while rotation has a stabilizing effect if λ><i>0</i> and destabilizing effect if λ<<i>0</i>. Further, the magnetic field is discovered to have both a stabilizing and destabilizing effect for both λ><i>0</i> and λ<<i>0</i>. It is likewise discovered that magnetization has a stabilizing effect for both λ><i>0</i> and λ<<i>0</i> in the absence of the stable solute gradient. Graphs have been plotted by giving numerical values of various parameters. In the absence of rotation, magnetic field and stable solute gradient, the principle of exchange of stabilities is found to hold true for certain conditions.

Lorentz estimates for quasilinear elliptic double obstacle problems involving a Schrödinger term

Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to p‐Laplace double obstacle problems involving the Schrödinger term: with bound constraints ψ1 ≤ u ≤ ψ2 in nonsmooth domains. This problem has its own interest in mathematics, engineering, physics, and other branches of science. Our approach makes a novel connection between the study of Calderón‐Zygmund theory for nonlinear Schrödinger type equations and variational inequalities for double obstacle problems.

A flexible gradient method for unstructured‐grid solvers

AbstractIn this short note, we introduce a flexible explicit gradient method for efficient and robust unstructured‐grid computations. As shown, the method serves as a memory‐efficient alternative to the multiple least‐squares gradients needed by conflicting requirements for inviscid, viscous, and turbulence‐model source terms (e.g., vorticity), in practical computational‐fluid‐dynamics solvers. The flexible method computes the gradient of a numerical solution in a cell by averaging the face gradients defined by the alpha‐damped formula with a parameter based on a single set of least‐squares gradients. We show that a small gives less‐accurate gradients similar to unweighted least‐squares gradients, and a large produces accurate gradients similar to weighted least‐squares gradients. Therefore, it allows us to obtain two types of gradients by changing the value of without storing more than one set of least‐squares coefficients. Moreover, the method yields fourth‐order accurate gradients with on a regular grid. The method is demonstrated for two‐dimensional inviscid and viscous flows on unstructured grids.

Global regularity for degenerate/singular parabolic equations involving measure data

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure.

Spurious interface and boundary behaviour beyond physical solutions in lattice Boltzmann schemes

The scalar field and the non-equilibrium solutions of the linear advection-diffusion d2Q9 Lattice Boltzmann (LBM) two-relaxation-times (TRT) scheme are constructed analytically. The scheme copes with an infinite number of suitable, second-order accurate, equilibrium weights. Here, the simplest, translation-invariant geometry with an implicitly located, straight or diagonal, grid-aligned interface (boundary) is addressed. We show that these two interface (boundary) orientations are accommodated with the help of two distinctive, anisotropic, discrete-exponential algebraic solution components, referred to as the A-layer and the B-layer. Being unpredicted by the perturbative analysis, such as the Chapman-Enskog, asymptotic or truncation, their solution is derived symbolically from the TRT recurrence equations, subject to the local mass conservation solvability and effective closure conditions. When the interface (boundary) is “diagonal”, the A-layer perturbs the simplest physical solutions, like the piece-wise linear, polynomial or exponential scalar field, rendering the macroscopic solution weight-dependent and delaying its convergence to the first order; the A-layer base depends upon the weights, free relaxation parameter Λ and physical numbers. In contrast, the B-layer, invisible to the scalar field, typically accommodates the non-equilibrium discrepancy between the normal and diagonal directions on the “straight” interface (boundary); the B-layer base is fixed by Λ alone. The A-layer and B-layer may coexist and degrade the physical solution gradient and its convergence. Only the D2Q5 model is free from all these effects in the straight and diagonal orientations, while the diagonally-rotated D2Q5 model is unsuitable because of the “checkerboard” effect. These spurious corrections are not the Knudsen layers, but they present the LBM response for any-order bulk mismatch with the implicit or explicit interface (boundary) treatment; the A-layer and B-layer bring them in evidence and provide excellent benchmarks for their attenuation through interface-conjugate or adaptive refinement techniques. Our approach extends to any lattice, linear collision, source term, heterogeneity and LBM problem class.

A Level-Set Adjoint-State Method for Transmission Traveltime Tomography in Irregular Domains

We propose an efficient PDE based approach for solving the first-arrival traveltime tomography problem in irregular domains. We consider the mismatch functional based on the $L_1$ or $L_2$ fidelity term, respectively, and we compute both functional gradients with respect to the sound speed by the adjoint-state method. The novelty of the proposed method consists of two aspects. First, since the tomography problem is formulated in an irregular domain, we develop new efficient level-set based fast sweeping methods for solving the eikonal and adjoint-state equations in the irregular domain. Second, since the computed adjoint state is associated with a source singularity originated from the gradient of a point source eikonal solution, we propose a novel decomposition approach to remove this singularity, which just requires solving the same adjoint-state equation again but with a different boundary condition. Finally, we show the performance of the proposed algorithm on several numerical examples including a synthetic data set from ultrasound computerized tomography.

Pointwise gradient bounds for a class of very singular quasilinear elliptic equations

A pointwise gradient bound for weak solutions to Dirichlet problem for quasilinear elliptic equations \begin{document}$ -\mathrm{div}(\mathbb{A}(x,\nabla u)) = \mu $\end{document} is established via Wolff type potentials. It is worthwhile to note that the model case of \begin{document}$ \mathbb{A} $\end{document} here is the non-degenerate \begin{document}$ p $\end{document} -Laplacian operator. The central objective is to extend the pointwise regularity results in [Q.-H. Nguyen, N. C. Phuc, Pointwise gradient estimates for a class of singular quasilinear equations with measure data, J. Funct. Anal. 278(5) (2020), 108391] to the very singular case \begin{document}$ 1 , where the data \begin{document}$ \mu $\end{document} on right-hand side is assumed belonging to some classes that close to \begin{document}$ L^1 $\end{document} . Moreover, a global pointwise estimate for gradient of weak solutions to such problem is also obtained under the additional assumption that \begin{document}$ \Omega $\end{document} is sufficiently flat in the Reifenberg sense.

Study of double-diffusive convection in a rotating couple-stress ferromagnetic fluid in the presence of varying gravitational field and horizontal magnetic field saturating in a porous medium

The results of double-diffusive convection on the liquid coatings of ferromagnetic liquids is studied by researchers by various theoretical and hydrodynamics concepts of hydrodynamics. However this study is done in presence of varying gravitational field. The study of such couple-stress ferromagnetic fluids in presence of fluctuating gravitational field is still not explored. As we know that gravitational field varies on different points on the earth; for this reason the study of couple-stress ferromagnetic liquids in presence of variable gravity fields is highly required. In this paper we studied the impact of medium permeability, stable solute gradient, couple-stress, rotation, magnetic field, and magnetization of a couple-stress ferromagnetic fluid in presence of variable gravitational field. For this study a coating of couple-stress ferromagnetic fluid saturating in a porous medium when fluid layer is heated and soluted. We examined the results of the magnetic field on couple-stress ferromagnetic fluids, its thermosolutal instability during rotation. A linearized concept and normal mode technique are used to acquire the dispersion relation. For the case of stationary convection, medium permeability, couple-stress, and magnetic field have both stabilizing and destabilizing results under certain conditions. A stable solute gradient has a stabilizing result on the system. Also, the rotation has a stabilizing result for λ>0 and destabilizing result for λ<0 the system. It is likewise observed that in the absence of a stable solute gradient, magnetization has a stabilizing result on the system for both λ>0 and λ<0. The critical Rayleigh number for the onset of instability is decided numerically and graphically also. The principle of exchange of stabilities is observed to keep genuine in the absence of rotation, magnetic field, and stable solute gradient under certain conditions.

Implicit-Discrete-Maximum-Principle-Based Production Optimization in Reservoir Development

Cost-effective development of proved oil resources is the goal of oil and gas field development workers. One of the ways to achieve this is to make the reservoir development in an optimal state based on the existing production conditions, which is also one of the core contents of intelligent oilfield. This paper proposes a new production optimization control method, which aims to maximize the net present value of reservoir development and production. By solving the mathematical model of development and production, the input and output control parameters of the reservoir are optimized in real time to obtain the optimal production plan. The conventional maximum principle method requires two time series solution calculations. In this paper, the coefficient matrix of the full implicit simulator is used to directly obtain the required adjoint equations. The efficient combination of simulation calculation and gradient solution of discrete maximum principle greatly improves the solution efficiency of the model and saves the calculation time. The reservoir examples are analyzed through theoretical research. The results show that the optimal production scheme is in line with the actual situation of the oilfield, which provides theoretical and technical support for the intelligent oilfield system.

Convergence of the kinetic hydrostatic reconstruction scheme for the Saint Venant system with topography

We prove the convergence of the hydrostatic reconstruction scheme with kinetic numerical flux for the Saint Venant system with Lipschitz continuous topography. We use a recently derived fully discrete sharp entropy inequality with dissipation, that enables us to establish an estimate in the inverse of the square root of the space increment ∆x of the L 2 norm of the gradient of approximate solutions. By Diperna's method we conclude the strong convergence towards bounded weak entropy solutions.

A hybrid-Trefftz finite element platform for solid and porous elastodynamics

Hybrid-Trefftz finite elements are well suited for modeling the response of materials under highly transient loading. Their approximation bases are built using functions that satisfy exactly the differential equations governing the problem. This option embeds relevant physical information into the approximation basis and removes the well-known sensitivity of the conventional finite elements to high solution gradients and short wavelength excitations. Despite such advantages, no public software using hybrid-Trefftz finite elements to model wave propagation through solid and porous media exists to date. This paper covers the formulation and implementation of hybrid-Trefftz finite elements for single-phase, biphasic and triphasic media, subjected to dynamic loads. The formulation is cast in a unified framework, valid for the three types of materials alike, and independent of the nature (harmonic, periodic or transient) of the applied load. Displacement, traction, elastic and absorbing boundary conditions are accommodated. The implementation is made in three novel, open-source and user-friendly computational modules which are freely distributed online.

Development and Validation of a UPLC-DAD Method for the Simultaneous Quantification of Eight Antihypertensive Drugs in the Pharmaceutical Matrix.

Hypertension is a critical health problem; it is a prevalent risk factor for cardiovascular disease. Many treatments to combat hypertension are available, however many patients are resistant to the standard therapeutic approaches. The association of two or more substances in a fixed-dose combination is effective and tolerated as a substitute for the standard therapeutic approach. The new ultra performance liquide chromatography method was developed and validated to assay a combination of eight antihypertensive drugs including a diuretic: hydrochlorothiazide, dihydropyridine calcium channel blocker: Amlodipine and angiotensin II type 1 receptor blockers (sartans): valsartan, candesartan, eprosartan, olmesartan, losartan, and irbesartan in the pharmaceutical matrix. Chromatographic separation was performed on an Acquity® UPLC C18 1.7 µm 2.1 × 100 mm column, with a gradient of buffer solution and acetonitrile, in the proportion of (80:20 v/v). Good resolution was obtained, and an optimal analysis time of less than 5 min was achieved. The method was validated according to the International Conference on Harmonization guidelines following the classical approach and accuracy profile, and it is shown to be suitable for intended applications. The method was successfully used for quality control laboratories and the determination of these drugs combinations in pharmaceutical dosage forms.

The parabolic p-Laplacian with fractional differentiability

Abstract We study the parabolic $p$-Laplacian system in a bounded domain. We deduce optimal convergence rates for the space–time discretization based on an implicit Euler scheme in time. Our estimates are expressed in terms of Nikolskiǐ spaces and therefore cover situations when the (gradient of the) solution has only fractional derivatives in space and time. The main novelty is that, different to all previous results, we do not assume any coupling condition between the space and time resolutions $h$ and $\tau $. For this we show that the $L^2$-projection is compatible with the quasi-norm. The theoretical error analysis is complemented by numerical experiments.

A new perspective on flux and slope limiting in discontinuous Galerkin methods for hyperbolic conservation laws

In this work, we discuss and develop multidimensional limiting techniques for discontinuous Galerkin (DG) discretizations of scalar hyperbolic problems. To ensure that each cell average satisfies a local discrete maximum principle (DMP), we impose inequality constraints on the local Lax–Friedrichs fluxes of a piecewise-linear (P1) approximation. Since the piecewise-constant (P0) version corresponds to a property-preserving low-order finite volume method, the validity of DMP conditions can always be enforced using slope and/or flux limiters. We show that the (currently rather uncommon) use of direct flux limiting makes it possible to construct more accurate DMP-satisfying approximations in which a weak form of slope limiting is used to prevent unbounded growth of solution gradients. After presenting two flux limiters that ensure the validity of local DMPs for cell averages, we discuss the design of slope limiters based on different kinds of inequality constraints. In particular, we derive new limiting procedures based on flux constraints and constraints for directional derivatives. The latter approach makes it possible to preserve directional monotonicity in applications to problems that require different treatment of different space directions. At the flux limiting stage, the anisotropy of the problem at hand can be taken into account by using a customized definition of local bounds for the DMP constraints. At the slope limiting stage, we adjust the magnitude of individual directional derivatives using low-order reconstructions from cell averages to define the bounds. In this way, we avoid unnecessary limiting of well-resolved derivatives at smooth peaks and in internal/boundary layers. The properties of selected algorithms are explored in numerical studies for DG-P1 discretizations of two-dimensional test problems. In the context of hp-adaptive DG methods, the new limiting procedures can be used in P1 subcells of macroelements marked as ‘troubled’ by a smoothness indicator.

Lorentz gradient estimates for a class of elliptic p-Laplacian equations with a Schrödinger term

We prove in this paper the global Lorentz estimate in term of fractional-maximal function for gradient of weak solutions to a class of p-Laplace elliptic equations containing a non-negative Schrödinger potential which belongs to reverse Hölder classes. In particular, this class of p-Laplace operator includes both degenerate and non-degenerate cases. The interesting idea is to use an efficient approach based on the level-set inequality related to the distribution function in harmonic analysis.

Global continuity and BMO estimates for non-Newtonian fluids with perfect slip boundary conditions

We study the generalized stationary Stokes system in a bounded domain in the plane equipped with perfect slip boundary conditions. We show natural stability results in oscillatory spaces, i.e., Hölder spaces and Campanato spaces, including the border-line spaces of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In particular, we show that, under appropriate assumptions, gradients of solutions are globally continuous. Since the stress tensor is assumed to be governed by a general Orlicz function, our theory includes various cases of (possibly degenerate) shear thickening and shear thinning fluids; including the model case of power law fluids. The global estimates seem to be new even in the case of the linear Stokes system. We include counterexamples that demonstrate that our assumptions on the right-hand side and on the boundary regularity are optimal.

Storm event impacts on in-stream nitrate concentration and discharge dynamics: A comparison of high resolution in-situ measured data with model simulations

The relationship between nitrogen and discharge (N-Q) in a stream can be captured with high frequency nitrate nitrogen (NO3−-N) samplers. In Austria, the Raab catchment (998 km2) has high frequency NO3−-N data measured with a spectrometer probe. This study evaluated if the widely-used and typically calibrated eco-hydrological model Soil and Water Assessment Tool (SWAT) can reproduce the hysteresis loop direction and the dilution or accretion effects of NO3−-N dynamics during storm events in this agricultural catchment. The daily aggregated NO3−-N measurements were compared with the daily SWAT simulated discharge and NO3−-N concentrations of 14 storm events by computing hysteresis indices - loop direction and area (h index), loop direction (HInew) and solute gradient (∆C). Overall, the SWAT model was able to replicate the predominant anticlockwise hysteresis and dilution effect of NO3−-N in the Raab catchment. The loop direction was simulated correctly in 9 and 10 events, for the h and HInew indices, respectively. The hysteresis direction inferred from both indices did not always concur due to the differences in the calculation methods. The dilution or accretion effect was simulated correctly in 9 of the events. However, the SWAT model only correctly simulated the N-Q relationships for all three hysteresis criteria in 5 of the 14 events. Due to the aggregation of measured data to the daily time step, information pertaining to the hysteresis shape was sometimes lost, particularly if the storm event was <4 days in duration. Structural limitations of the SWAT as well as specific relevant basin parameters (parameters that have one value for the entire catchment) may restrict simulating N-Q dynamics. An enhanced calibrated and validated model would possibly improve the results, since the events during the better calibrated period more often reproduced the measured hysteresis indices.