arXiv subjects

Gurpreet Singh

Publications and source records attributed to Gurpreet Singh.

At least 55 records · Page 3Linked to original sources

arXiv2021

Positive solutions for quasilinear elliptic inequalities and systems with nonlocal terms

We investigate the existence and nonexistence of positive solutions for the quasilinear elliptic inequality $L_\mathcal{A} u= -{\rm div}[\mathcal{A}(x, u, \nabla u)]\geq (I_α\ast u^p)u^q$ in $Ω$, where $Ω\subset \mathbb{R}^N, N\geq 1,$ is an open set. Here $I_α$ stands for the Riesz potential of order $α\in (0, N)$, $p>0$ and $q\in \mathbb{R}$. For a large class of operators $L_\mathcal{A}$ (which includes the $m$-Laplace and the $m$-mean curvature operator) we obtain optimal ranges of exponents $p,q$ and $α$ for which positive solutions exist. Our methods are then extended to quasilinear elliptic systems of inequalities.

math.AP↗

arXiv2021

Extracting Optimal Solution Manifolds using Constrained Neural Optimization

Constrained Optimization solution algorithms are restricted to point based solutions. In practice, single or multiple objectives must be satisfied, wherein both the objective function and constraints can be non-convex resulting in multiple optimal solutions. Real world scenarios include intersecting surfaces as Implicit Functions, Hyperspectral Unmixing and Pareto Optimal fronts. Local or global convexification is a common workaround when faced with non-convex forms. However, such an approach is often restricted to a strict class of functions, deviation from which results in sub-optimal solution to the original problem. We present neural solutions for extracting optimal sets as approximate manifolds, where unmodified, non-convex objectives and constraints are defined as modeler guided, domain-informed $L_2$ loss function. This promotes interpretability since modelers can confirm the results against known analytical forms in their specific domains. We present synthetic and realistic cases to validate our approach and compare against known solvers for bench-marking in terms of accuracy and computational efficiency.

cs.NE↗

arXiv2021

A study of biharmonic equation involving nonlocal terms and critical Sobolev exponent

In this paper, we investigate the existence of ground state solutions and non-existence of non-trivial weak solution of biharmonic equation with some nonlocal terms and critical Sobolev exponent. Firstly, we prove the non-existence by establishing Pohozaev type of identity. Next, we study the existence of ground state solutions by using the minimization method on the associated Nehari manifold.

math.AP↗

arXiv2020

Coronae of an active fast rotator FR Cnc

We present the first detailed X-ray study and simultaneous optical observations of the active fast rotating star FR Cnc. The X-ray spectra are found to be explained by two-temperature plasma model with temperature of cool and hot components of 0.34 and 1.1 keV, respectively. The X-ray light curve in the 0.5 -- 2.0 keV energy band is found to be rotationally modulated with the degree of rotational modulation of ~17%. We have also found that the X-ray light curve is anti-correlated with the optical light and colour curves in the sense that maximum X-ray light corresponds to minimum optical light and cooler region on the surface of FR Cnc. The X-ray luminosity of FR Cnc is found to be almost consistent in the last 30 years with an average value of 4.85x10^{29} erg/s in the 0.5 -- 2.0 keV energy band.

astro-ph.SR↗

arXiv2020

Quasilinear elliptic inequalities with Hardy potential and nonlocal terms

We study the quasilinear elliptic inequality $$ -Δ_m u - \fracμ{|x|^m}u^{m-1} \geq (I_α*u^p)u^q \quad\mbox{ in }\mathbb{R}^N\setminus \overline B_1, N\geq 1, $$ where $p>0$, $q, μ\in \mathbb{R}$, $m>1$ and $I_α$ is the Riesz potential of order $α\in (0,N)$. We obtain necessary and sufficient conditions for the existence of positive solutions.

math.AP↗

arXiv2020

Prevention is Better than Cure: Handling Basis Collapse and Transparency in Dense Networks

Dense nets are an integral part of any classification and regression problem. Recently, these networks have found a new application as solvers for known representations in various domains. However, one crucial issue with dense nets is it's feature interpretation and lack of reproducibility over multiple training runs. In this work, we identify a basis collapse issue as a primary cause and propose a modified loss function that circumvents this problem. We also provide a few general guidelines relating the choice of activations to loss surface roughness and appropriate scaling for designing low-weight dense nets. We demonstrate through carefully chosen numerical experiments that the basis collapse issue leads to the design of massively redundant networks. Our approach results in substantially concise nets, having $100 \times$ fewer parameters, while achieving a much lower $(10\times)$ MSE loss at scale than reported in prior works. Further, we show that the width of a dense net is acutely dependent on the feature complexity. This is in contrast to the dimension dependent width choice reported in prior theoretical works. To the best of our knowledge, this is the first time these issues and contradictions have been reported and experimentally verified. With our design guidelines we render transparency in terms of a low-weight network design. We share our codes for full reproducibility available at https://github.com/smjtgupta/Dense_Net_Regress.

cs.LG↗

arXiv2020

Weighted Choquard equation perturbed with weighted nonlocal term

We investigate the following problem $$ -{\rm div}(v(x)|\nabla u|^{m-2}\nabla u)+V(x)|u|^{m-2}u= \Big(|x|^{-θ}*\frac{|u|^{b}}{|x|^α}\Big)\frac{|u|^{b-2}}{|x|^α}u+λ\Big(|x|^{-γ}*\frac{|u|^{c}}{|x|^β}\Big)\frac{|u|^{c-2}}{|x|^β}u \quad\mbox{ in }\R^{N}, $$ where $b, c, α, β>0$, $θ,γ\in (0,N)$, $N\geq 3$, $2\leq m< \infty$ and $λ\in \R$. Here, we are concerned with the existence of groundstate solutions and least energy sign-changing solutions and that will be done by using the minimization techniques on the associated Nehari manifold and the Nehari nodal set respectively.

math.AP↗

arXiv2020

TIME: A Transparent, Interpretable, Model-Adaptive and Explainable Neural Network for Dynamic Physical Processes

Partial Differential Equations are infinite dimensional encoded representations of physical processes. However, imbibing multiple observation data towards a coupled representation presents significant challenges. We present a fully convolutional architecture that captures the invariant structure of the domain to reconstruct the observable system. The proposed architecture is significantly low-weight compared to other networks for such problems. Our intent is to learn coupled dynamic processes interpreted as deviations from true kernels representing isolated processes for model-adaptivity. Experimental analysis shows that our architecture is robust and transparent in capturing process kernels and system anomalies. We also show that high weights representation is not only redundant but also impacts network interpretability. Our design is guided by domain knowledge, with isolated process representations serving as ground truths for verification. These allow us to identify redundant kernels and their manifestations in activation maps to guide better designs that are both interpretable and explainable unlike traditional deep-nets.

cs.LG↗

arXiv2019

Error indicator for the incompressible Darcy flow problems using Enhanced Velocity Mixed Finite Element Method

In the flow and transport numerical simulation, mesh adaptivity strategy is important in reducing the usage of CPU time and memory. The refinement based on the pressure error estimator is commonly-used approach without considering the flux error which plays important role in coupling flow and transport systems. We derive a posteriori error estimators for Enhanced Velocity Mixed Finite Element Method (EVMFEM) in the incompressible Darcy flow. We show numerically difference of the explicit residual based error estimator and implicit error estimators, where Arbogast and Chen post-processing procedure from [1] for pressure was used to improve estimators. A residual-based error estimator provides a better indicator for pressure error. Proposed estimators are good indicators in finding of the large error element. Numerical tests confirm theoretical results. We show the advantage of pressure postprocessing on the detecting of velocity error. To the authors' best knowledge, a posteriori error analysis of EVMFEM has been scarcely investigated from the theoretical and numerical point of view. Reference. 1. Arbogast, T., & Chen, Z. (1995). On the implementation of mixed methods as nonconforming methods for second-order elliptic problems. Mathematics of Computation, 64(211), 943-972.

math.NA↗

arXiv2019

Recovery of the Interface Velocity for the Incompressible Flow in Enhanced Velocity Mixed Finite Element Method

The velocity, coupling term in the flow and transport problems, is important in the accurate numerical simulation or in the posteriori error analysis for adaptive mesh refinement. We consider Enhanced Velocity Mixed Finite Element Method for the incompressible Darcy flow. In this paper, our aim to study the improvement of velocity at interface to achieve the better approximation of velocity between subdomains. We propose the reconstruction of velocity at interface by using the post-processed pressure. Numerical results at the interface show improvement on convergence rate.

math.NA↗

arXiv2018

A Domain Decomposition Approach for Local Mesh Refinement in Space and Time

We present an adaptive space-time mesh refinement approach based a domain decomposition approach (Singh and Wheeler, 2018) that allows different time-step sizes and mesh refinements in different subdomains. Our numerical experiments indicate that non-linear solvers fail to converge, to the desired tolerance, due to large non-linear residuals in a smaller subdomain. We exploit this feature to identify subdomains where smaller time-step sizes are necessary while using large time-step sizes in the rest of the reservoir domain. The three key components of our approach are: (1) a space-time, enhanced velocity, domain decomposition approach that allows different mesh refinements and time-step sizes in different subdomains while preserving local mass conservation, (2) a residual based error estimator to identify or mark regions (or subdomains) that pose non-linear convergence issues, and (3) a fully coupled monolithic solver is also presented that solves the coarse and fine subdomain problems, both in space and time, simultaneously. This solution scheme is fully implicit and is therefore unconditionally stable. The proposed space-time domain decomposition approach, with smaller time-step sizes in a subdomain and large time-step sizes everywhere else, circumvents the non-linear convergence issue without adding computational costs. Additionally, a space-time monolithic solver renders a massively parallel, time concurrent framework for solving flow and transport problems in subsurface porous media. Since the proposed approach is similar to the widely used finite difference scheme, it can be easily integrated in any existing legacy reservoir simulator.

math.NA↗

arXiv2018

A Space Time Domain Decomposition Approach using Enhanced Velocity Mixed Finite Element Method

A space-time domain decomposition approach is presented as a natural extension of the enhanced velocity mixed finite element (EVMFE) [Wheeler et. al] for spatial domain decomposition. The proposed approach allows for different space-time discretizations on non-overlapping, subdomains by enforcing a mass continuity argument at the non-matching interface to preserve the local mass conservation property inherent to the mixed finite element methods. To this effect, we consider three different model formulations: (1) a linear single phase flow problem, (2) a non-linear slightly compressible flow and tracer transport, and (3) a non-linear slightly compressible, multiphase flow and transport. We also present a numerical solution algorithm for the proposed domain decomposition approach where a monolithic (fully coupled in space and time) system is constructed that does not require subdomain iterations. This space-time EVMFE method accurately resolves advection-diffusion transport features, in a heterogeneous medium, while circumventing non-linear solver convergence issues associated with large time-step sizes for non-linear problems. Numerical results are presented for the aforementioned, three, model formulations to demonstrate the applicability of this approach to a general class of problems in flow and transport in porous media.

math.NA↗

arXiv2018

Global and blow-up radial solutions for quasilinear elliptic systems arising in the study of viscous, heat conducting fluids

We study positive radial solutions of quasilinear elliptic systems with a gradient term in the form $$ \left\{ \begin{aligned} Δ_{p} u&=v^{m}|\nabla u|^α&&\quad\mbox{ in }Ω,\\ Δ_{p} v&=v^β|\nabla u|^{q} &&\quad\mbox{ in }Ω, \end{aligned} \right. $$ where $Ω\subset\R^N$ $(N\geq 2)$ is either a ball or the whole space, $1 0$, $α\geq 0$, $0\leq β\leq m$ and $(p-1-α)(p-1-β)-qm\neq 0$. We first classify all the positive radial solutions in case $Ω$ is a ball, according to their behavior at the boundary. Then we obtain that the system has non-constant global solutions if and only if $0\leq α<p-1$ and $mq< (p-1-α)(p-1-β)$. Finally, we describe the precise behavior at infinity for such positive global radial solutions by using properties of three component cooperative and irreducible dynamical systems.

math.AP↗

arXiv2018

Multiscale Methods for Model Order Reduction of Non Linear Multiphase Flow Problems

Numerical simulations for flow and transport in subsurface porous media often prove computationally prohibitive due to property data availability at multiple spatial scales that can vary by orders of magnitude. A number of model order reduction approaches are available in the existing literature that alleviate this issue by approximating the solution at a coarse scale. We attempt to present a comparison between two such model order reduction techniques, namely: (1) adaptive numerical homogenization and (2) generalized multiscale basis functions. We rely upon a non-linear, multi-phase, black-oil model formulation, commonly encountered in the oil and gas industry, as the basis for comparing the aforementioned two approaches. An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow and transport problems. To the author's knowledge this is the first time these approaches have been described for a practical non-linear, multiphase flow problem of interest. A numerical benchmark is setup using fine scale property information from the 10$^{th}$ SPE comparative project dataset for the purpose of comparing accuracies of these two schemes. An adaptive criterion is employed in by both the schemes for local enrichment that allows us to preserve solution accuracy compared to the fine scale benchmark problem. The numerical results indicate that both schemes are able to adequately capture the fine scale features of the model problem at hand.

math.NA↗

arXiv2017

Nonlocal Pertubations of Fractional Choquard Equation

We study the equation \begin{equation} (-Δ)^{s}u+V(x)u= (I_α*|u|^{p})|u|^{p-2}u+λ(I_β*|u|^{q})|u|^{q-2}u \quad\mbox{ in } \R^{N}, \end{equation} where $I_γ(x)=|x|^{-γ}$ for any $γ\in (0,N)$, $p, q >0$, $α,β\in (0,N)$, $N\geq 3$ and $ λ\in R$. First, the existence of a groundstate solutions using minimization method on the associated Nehari manifold is obtained. Next, the existence of least energy sign-changing solutions is investigated by considering the Nehari nodal set.

math.AP↗

arXiv2017

On a class of mixed Choquard-Schrödinger-Poisson system

We study the system $$ \left\{ -Δu+u+K(x) ϕ|u|^{q-2}u&=(I_α*|u|^p)|u|^{p-2}u &&\mbox{ in }{\mathbb R}^N, -Δϕ&=K(x)|u|^q&&\mbox{ in }{\mathbb R}^N, \right. $$ where $N\geq 3$, $α\in (0,N)$, $p,q>1$ and $K\geq 0$. Using a Pohozaev type identity we first derive conditions in terms of $p,q,N,α$ and $K$ for which no solutions exist. Next, we discuss the existence of a ground state solution by using a variational approach.

math.AP↗

arXiv2016

Weak Solutions for Singular Quasilinear Elliptic Systems

We investigate the quasilinear elliptic system $-Δ_{m} u&=u^{-p}v^{-q}$, $u>0 \quad\mbox{ in } Ω$, $-Δ_{m} v&=u^{r}v^{-s}$, $v>0 \quad\mbox{ in }Ω$, $u=v=0 \quad\mbox{ on } \partialΩ$, where $Ω\subset{\mathbb R}^{N}(N\geq 1)$ is a bounded and smooth domain, $1 0$. Under certain conditions imposed on the exponents we obtain the existence and uniqueness of a weak solution $(u, v)$ with $u, v \in W_{0}^{1, m}(Ω)\cap C(Ω)$. We also investigate the $W_{0}^{1, τ}(Ω)$ regularity of solution and determine the optimal range of $τ\geq m$ for such regularity.

math.AP↗

arXiv2015

Regularity of Weak Solutions for Singular Elliptic Problems Driven by m-Laplace Operator

We obtain optimal regularity in the Sobolev space $W_0^{1,τ}(Ω)$ for the unique solution of $$ -Δ_m u=K(x)u^{-p} \mbox{in} Ω, \quad u=0\mbox{on}\partial Ω. $$ Here $Ω\subset{\mathbb R}^N$ is a smooth and bounded domain, $m>1$, $p\geq 0$ and $K\in C(Ω)$ is a positive function that behaves like ${\rm dist}(x,\partialΩ)^{-q}$ for some $q\geq 0$ with $p+q< 2-\frac{1-p}{m}$. We obtain that the unique weak solution to the above problem belongs to $W_0^{1,τ}(Ω)$ for $$ m\leq τ<\frac{m+p-1}{p+q-1}\quad \mbox{if}p+q>1, $$ and $$ m\leq τ<\infty\quad \mbox{if}p+q=1. $$ The above range of $τ$ is optimal.

math.AP↗