arXiv subjects

Kundan Kumar

Publications and source records attributed to Kundan Kumar.

At least 55 records · Page 3Linked to original sources

arXiv2019

Nonlinear State Estimation using Gaussian Integral

In this letter, a new filtering technique to solve a nonlinear state estimation problem has been developed. It is well known that for a nonlinear system, the prior and posterior probability density functions (pdf) are non-Gaussian in nature. However, in this work, they are assumed as Gaussian and subsequently mean, and covariance of them are calculated. In the proposed method, nonlinear functions of process dynamics and measurement are expressed in a polynomial form with the help of Taylor series expansion. In order to calculate the prior and the posterior mean and covariance, the functions are integrated over the Gaussian pdf with the help of Gaussian integral. The performance of the proposed method is tested in two nonlinear state estimation problems. The simulation results show that the proposed filter provides more accurate result than other existing deterministic sample point filters such as cubature Kalman filter, unscented Kalman filter, etc.

eess.SP↗

arXiv2019

The gradient flow structures of thermo-poro-visco-elastic processes in porous media

In this paper, the inherent gradient flow structures of thermo-poro-visco-elastic processes in porous media are examined for the first time. In the first part, a modelling framework is introduced aiming for describing such processes as generalized gradient flows requiring choices of physical states, corresponding energies, dissipation potentials and external work rates. It is demonstrated that various existing models can be in fact written within this framework. Ultimately, the particular structure allows for a unified well-posedness analysis performed for different classes of linear and non-linear models. In the second part, the gradient flow structures are utilized for constructing efficient discrete approximation schemes for thermo-poro-visco-elasticity -- in particular robust, physical splitting schemes. Applying alternating minimization to naturally arising minimization formulations of (semi-)discrete models is proposed. For such, the energy decrease per iteration is quantified by applying abstract convergence theory only utilizing convexity and Lipschitz continuity properties of the problem -- a fairly simple but flexible machinery. By this approach, e.g., the widely used undrained and fixed-stress splits for the linear Biot equations are derived and analyzed. By application of the framework to more advanced models, novel splitting schemes with guaranteed theoretical convergence rates are naturally derived. Moreover, based on the minimization character of the (semi-)discrete equations, relaxation of splitting schemes by line search is proposed; numerical results show a potentially great impact on the acceleration of splitting schemes for both linear and nonlinear problems.

math.NA↗

arXiv2019

Blood Vessel Detection using Modified Multiscale MF-FDOG Filters for Diabetic Retinopathy

Blindness in diabetic patients caused by retinopathy (characterized by an increase in the diameter and new branches of the blood vessels inside the retina) is a grave concern. Many efforts have been made for the early detection of the disease using various image processing techniques on retinal images. However, most of the methods are plagued with the false detection of the blood vessel pixels. Given that, here, we propose a modified matched filter with the first derivative of Gaussian. The method uses the top-hat transform and contrast limited histogram equalization. Further, we segment the modified multiscale matched filter response by using a binary threshold obtained from the first derivative of Gaussian. The method was assessed on a publicly available database (DRIVE database). As anticipated, the proposed method provides a higher accuracy compared to the literature. Moreover, a lesser false detection from the existing matched filters and its variants have been observed.

eess.IV↗

arXiv2019

A Mixed Approach to the Poisson Problem with Line Sources

In this work we consider the primal mixed variational formulation of the Poisson equation with a line source. The analysis and approximation of this problem is non-standard as the line source causes the solutions to be singular. We start by showing that this problem admits a solution in appropriately weighted Sobolev spaces. Next, we show that given some assumptions on the problem parameters, the solution admits a splitting into higher and lower regularity terms. The lower regularity terms are here explicitly known and capture the solution singularities. The higher regularity terms, meanwhile, are defined as the solution of its own mixed Poisson equation. With the solution splitting in hand, we then define a singularity removal based mixed finite element method in which only the higher regularity are approximated numerically. This method yields a significant improvement in the convergence rate when compared to approximating the full solution. In particular, we show that the singularity removal based method yields optimal convergence rates for lowest order Raviart-Thomas and discontinuous Lagrange elements.

math.AP↗

arXiv2019

Implicit linearization method for non-standard two-phase flow in porous media

In this paper, we consider a non-local (in time) two-phase flow model. The non-locality is introduced through the wettability alteration induced dynamic capillary pressure function. We present a monotone fixed-point iterative linearization scheme for the resulting non-standard model. The scheme treats the dynamic capillary pressure functions semi-implicitly and introduces an $L$-scheme type \cite{List2016, Radu2015} stabilization term in the pressure as well as the transport equations. We prove the convergence of the proposed scheme theoretically under physically acceptable assumptions and verify the theoretical analysis with numerical simulations. The scheme is implemented and tested for a variety of reservoir heterogeneity in addition to the dynamic change of the capillary pressure function. The proposed scheme satisfies the predefined stopping criterion within a few numbers of iterations. We also compared the performance of the proposed scheme against the iterative IMplicit Pressure Explicit Saturation scheme

physics.comp-ph↗

arXiv2019

An upscaled model for permeable biofilm in a thin channel and tube

In this paper, we derive upscaled equations for modelling biofilm growth in porous media. The resulting macro-scale mathematical models consider permeable multi-species biofilm including water flow, transport, detachment and reactions. The biofilm is composed of extracellular polymeric substances (EPS), water, active bacteria and dead bacteria. The free flow is described by the Stokes and continuity equations and the water flux inside the biofilm by the Brinkman and continuity equations. The nutrients are transported in the water phase by convection and diffusion. This pore-scale model includes variations of the biofilm composition and size due to reproduction of bacteria, production of EPS, death of bacteria and shear forces. The model includes a water-biofilm interface between the free flow and the biofilm. Homogenization techniques are applied to obtain upscaled models in a thin channel and a tube, by investigating the limit as the ratio of the aperture to the length $\varepsilon$ of both geometries approaches to zero. As $\varepsilon$ gets smaller, we obtain that the percentage of biofilm coverage area over time predicted by the pore-scale model approaches the one obtained using the effective equations, which shows a correspondence between both models. The two derived porosity-permeability relations are compared to two empirical relations from the literature. The resulting numerical computations are presented to compare the outcome of the effective (upscaled) models for the two mentioned geometries.

physics.flu-dyn↗

arXiv2019

A Singularity Removal Method for Coupled 1D-3D Flow Models

In reservoir simulations, the radius of a well is inevitably going to be small compared to the horizontal length scale of the reservoir. For this reason, wells are typically modelled as lower-dimensional sources. In this work, we consider a coupled 1D-3D flow model, in which the well is modelled as a line source in the reservoir domain and endowed with its own 1D flow equation. The flow between well and reservoir can then be modelled in a fully coupled manner by applying a linear filtration law. The line source induces a logarithmic type singularity in the reservoir pressure that is difficult to resolve numerically. We present here a singularity removal method for the model equations, resulting in a reformulated coupled 1D-3D flow model in which all variables are smooth. The singularity removal is based on a solution splitting of the reservoir pressure, where it is decomposed into two terms: an explicitly given, lower regularity term capturing the solution singularity and some smooth background pressure. The singularities can then be removed from the system by subtracting them from the governing equations. Finally, the coupled 1D-3D flow equations can be reformulated so they are given in terms of the well pressure and the background reservoir pressure. As these variables are both smooth (i.e. non-singular), the reformulated model has the advantage that it can be approximated using any standard numerical method. The reformulation itself resembles a Peaceman well correction performed at the continuous level.

math.NA↗

arXiv2019

Analysis of dynamic ruptures generating seismic waves in a self-gravitating planet: an iterative coupling scheme and well-posedness

We study the solution of the system of equations describing the dynamical evolution of spontaneous ruptures generated in a prestressed elastic-gravitational deforming body and governed by rate and state friction laws. We propose an iterative coupling scheme based on a weak formulation with nonlinear interior boundary conditions, both for continuous time and with implicit discretization (backward Euler) in time. We regularize the problem by introducing viscosity. This guarantees the convergence of the scheme for solutions of the regularized problems in both cases. We also make precise the conditions on the relevant coefficients for convergence to hold.

physics.comp-ph↗

arXiv2019

Automated retinal vessel segmentation based on morphological preprocessing and 2D-Gabor wavelets

Automated segmentation of vascular map in retinal images endeavors a potential benefit in diagnostic procedure of different ocular diseases. In this paper, we suggest a new unsupervised retinal blood vessel segmentation approach using top-hat transformation, contrast-limited adaptive histogram equalization (CLAHE), and 2-D Gabor wavelet filters. Initially, retinal image is preprocessed using top-hat morphological transformation followed by CLAHE to enhance only the blood vessel pixels in the presence of exudates, optic disc, and fovea. Then, multiscale 2-D Gabor wavelet filters are applied on preprocessed image for better representation of thick and thin blood vessels located at different orientations. The efficacy of the presented algorithm is assessed on publicly available DRIVE database with manually labeled images. On DRIVE database, we achieve an average accuracy of 94.32% with a small standard deviation of 0.004. In comparison with major algorithms, our algorithm produces better performance concerning the accuracy, sensitivity, and kappa agreement.

eess.IV↗

arXiv2019

Iterative solvers for Biot model under small and large deformation

We consider L-scheme and Newton based solvers for Biot model under small or large deformation. The mechanical deformation follows the Saint Venant-Kirchoff constitutive law. Further, the fluid compressibility is assumed to be nonlinear. A Lagrangian frame of reference is used to keep track of the deformation. We perform an implicit discretization in time (backward Euler) and propose two linearization schemes for solving the nonlinear problems appearing within each time step: Newton's method and L-scheme. The linearizations are used monolithically or in combination with a splitting algorithm. The resulting schemes can be applied for any spatial discretization. The convergences of all schemes are shown analytically for cases under small deformation. Illustrative numerical examples are presented to confirm the applicability of the schemes, in particular, for large deformation.

math.NA↗

arXiv2019

A pore-scale study of transport of inertial particles by water in porous media

We study the transport of inertial particles in water flow in porous media. Our interest lies in understanding the accumulation of particles including the possibility of clogging. We propose that accumulation can be a result of hydrodynamic effects: the tortuous paths of the porous medium generate regions of dominating strain/vorticity, which favour the accumulation/dispersion of the inertial particles. Numerical simulations show that essentially two accumulation regimes are identified: for low and for high flow velocities. When particles accumulate in high-velocity regions, at the entrance of a pore throat, a clog is formed. The formation of a clog significantly modifies the flow, as the partial blockage of the pore causes a local redistribution of pressure. This redistribution can divert the upstream water flow into neighbouring pores. Moreover, we show that accumulation in high velocity regions occurs in heterogeneous media, but not in homogeneous media, where we refer to homogeneity with respect to the distribution of the pore throat diameters.

physics.flu-dyn↗

arXiv2018

On the optimization of the fixed-stress splitting for Biot's equations

In this work we are interested in effectively solving the quasi-static, linear Biot model for poromechanics. We consider the fixed-stress splitting scheme, which is a popular method for iteratively solving Biot's equations. It is well-known that the convergence of the method is strongly dependent on the applied stabilization/tuning parameter. In this work, we propose a new approach to optimize this parameter. We show theoretically that it depends also on the fluid flow properties and not only on the mechanics properties and the coupling coefficient. The type of analysis presented in this paper is not restricted to a particular spatial discretization. We only require it to be inf-sup stable. The convergence proof applies also to low-compressible or incompressible fluids and low-permeable porous media. Illustrative numerical examples, including random initial data, random boundary conditions or random source terms and a well-known benchmark problem, i.e. Mandel's problem are performed. The results are in good agreement with the theoretical findings. Furthermore, we show numerically that there is a connection between the inf-sup stability of discretizations and the performance of the fixed-stress splitting scheme.

math.NA↗

arXiv2018

Splitting method for elliptic equations with line sources

In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain $Ω$ when the right-hand side is a (1D) line source $Λ$. The analysis and approximation of such problems is known to be non-standard as the line source causes the solution to be singular. Our main result is a splitting theorem for the solution; we show that the solution admits a split into an explicit, low regularity term capturing the singularity, and a high-regularity correction term $w$ being the solution of a suitable elliptic equation. The splitting theorem states the mathematical structure of the solution; in particular, we find that the solution has anisotropic regularity. More precisely, the solution fails to belong to $H^1$ in the neighbourhood of $Λ$, but exhibits piecewise $H^2$-regularity parallel to $Λ$. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via its correction function $w$. This approach has several benefits. Firstly, it recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to $L^2$, a problem for which the discretizations and solvers are readily available. Secondly, it makes the numerical approximation independent of the discretization of $Λ$; thirdly, it improves the approximation properties of the numerical method. We consider here the Galerkin finite element method, and show that the singularity subtraction then recovers optimal convergence rates on uniform meshes, i.e., without needing to refine the mesh around each line segment. The numerical method presented in this paper is therefore well-suited for applications involving a large number of line segments. We illustrate this by treating a dataset (consisting of $\sim 3000$ line segments) describing the vascular system of the brain.

math.NA↗

arXiv2018

Numerical simulation of biofilm formation in a microchannel

The focus of this paper is the numerical solution of a pore-scale model for the growth of a permeable biofilm. The model includes water flux inside the biofilm, different biofilm components, and shear stress on the biofilm-water interface. To solve the resulting highly coupled system of model equations, we propose a splitting algorithm. The Arbitrary Lagrangian Eulerian (ALE) method is used to track the biofilm-water interface. Numerical simulations are performed using physical parameters from the existing literature. Our computations show the effect of biofilm permeability on the nutrient transport and on its growth.

physics.flu-dyn↗

arXiv2018

A pore-scale model for permeable biofilm: numerical simulations and laboratory experiments

In this paper we derive a pore-scale model for permeable biofilm formation in a two-dimensional pore. The pore is divided in two phases: water and biofilm. The biofilm is assumed to consist of four components: water, extracellular polymeric substances (EPS), active bacteria, and dead bacteria. The flow of water is modeled by the Stokes equation whereas a diffusion-convection equation is involved for the transport of nutrients. At the water/biofilm interface, nutrient transport and shear forces due to the water flux are considered. In the biofilm, the Brinkman equation for the water flow, transport of nutrients due to diffusion and convection, displacement of the biofilm components due to reproduction/dead of bacteria, and production of EPS are considered. A segregated finite element algorithm is used to solve the mathematical equations. Numerical simulations are performed based on experimentally determined parameters. The stress coefficient is fitted to the experimental data. To identify the critical model parameters, a sensitivity analysis is performed. The Sobol sensitivity indices of the input parameters are computed based on uniform perturbation by $\pm 10 \%$ of the nominal parameter values. The sensitivity analysis confirms that the variability or uncertainty in none of the parameters should be neglected.

physics.flu-dyn↗

arXiv2018

Upscaling of unsaturated flow in fractured porous media

In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called $\varepsilon > 0$. Then we take the limit $\varepsilon \to 0$ and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to $\varepsilon$, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

math.NA↗

arXiv2018

Large-Scale Neuromorphic Spiking Array Processors: A quest to mimic the brain

Neuromorphic engineering (NE) encompasses a diverse range of approaches to information processing that are inspired by neurobiological systems, and this feature distinguishes neuromorphic systems from conventional computing systems. The brain has evolved over billions of years to solve difficult engineering problems by using efficient, parallel, low-power computation. The goal of NE is to design systems capable of brain-like computation. Numerous large-scale neuromorphic projects have emerged recently. This interdisciplinary field was listed among the top 10 technology breakthroughs of 2014 by the MIT Technology Review and among the top 10 emerging technologies of 2015 by the World Economic Forum. NE has two-way goals: one, a scientific goal to understand the computational properties of biological neural systems by using models implemented in integrated circuits (ICs); second, an engineering goal to exploit the known properties of biological systems to design and implement efficient devices for engineering applications. Building hardware neural emulators can be extremely useful for simulating large-scale neural models to explain how intelligent behavior arises in the brain. The principle advantages of neuromorphic emulators are that they are highly energy efficient, parallel and distributed, and require a small silicon area. Thus, compared to conventional CPUs, these neuromorphic emulators are beneficial in many engineering applications such as for the porting of deep learning algorithms for various recognitions tasks. In this review article, we describe some of the most significant neuromorphic spiking emulators, compare the different architectures and approaches used by them, illustrate their advantages and drawbacks, and highlight the capabilities that each can deliver to neural modelers.

cs.NE↗

arXiv2018

Anderson accelerated fixed-stress splitting schemes for consolidation of unsaturated porous media

In this paper, we study the robust linearization of nonlinear poromechanics of unsaturated materials. The model of interest couples the Richards equation with linear elasticity equations, employing the equivalent pore pressure. In practice a monolithic solver is not always available, defining the requirement for a linearization scheme to allow the use of separate simulators, which is not met by the classical Newton method. We propose three different linearization schemes incorporating the fixed-stress splitting scheme, coupled with an L-scheme, Modified Picard and Newton linearization of the flow. All schemes allow the efficient and robust decoupling of mechanics and flow equations. In particular, the simplest scheme, the Fixed-Stress-L-scheme, employs solely constant diagonal stabilization, has low cost per iteration, and is very robust. Under mild, physical assumptions, it is theoretically shown to be a contraction. Due to possible break-down or slow convergence of all considered splitting schemes, Anderson acceleration is applied as post-processing. Based on a special case, we justify theoretically the general ability of the Anderson acceleration to effectively accelerate convergence and stabilize the underlying scheme, allowing even non-contractive fixed-point iterations to converge. To our knowledge, this is the first theoretical indication of this kind. Theoretical findings are confirmed by numerical results. In particular, Anderson acceleration has been demonstrated to be very effective for the considered Picard-type methods. Finally, the Fixed-Stress-Newton scheme combined with Anderson acceleration provides a robust linearization scheme, meeting the above criteria.

math.NA↗