arXiv subjects

K. Mitra

Publications and source records attributed to K. Mitra.

10 recordsLinked to original sources

arXiv2025

Robust and fast iterative method for the elliptic Monge-Amp\`ere equation

This paper introduces a fast and robust iterative scheme for the elliptic Monge-Amp\`ere equation with Dirichlet boundary conditions. The Monge-Amp\`ere equation is a nonlinear and degenerate equation, with applications in optimal transport, geometric optics, and differential geometry. The proposed method linearises the equation and uses a fixed-point iteration (L-scheme), solving a Poisson problem in each step with a weighted residual as the right-hand side. This algorithm is robust against discretisation, nonlinearities, and degeneracies. For a weight greater than the largest eigenvalue of the Hessian, contraction in $H^2$ and $L^\infty$ is proven for both classical and generalised solutions, respectively. The method's performance can be enhanced by using preconditioners or Green's functions. Test cases demonstrate that the scheme outperforms Newton's method in speed and stability.

math.NA↗

arXiv2024

Inverse methods for freeform optical design

We present a systematic derivation of three mathematical models of increasing complexity for optical design, based on Hamilton's characteristic functions and conservation of luminous flux, and briefly explain the connection with the mathematical theory of optimal transport. We outline several iterative least-squares solvers for our models and demonstrate their performance for a few challenging problems.

physics.optics↗

arXiv2024

Robust time-discretisation and linearisation schemes for singular and degenerate evolution systems modelling biofilm growth

We propose and analyse numerical schemes for a system of quasilinear, degenerate evolution equations modelling biofilm growth as well as other processes such as flow through porous media and the spreading of wildfires. The first equation in the system is parabolic and exhibits degenerate and singular diffusion, while the second is either uniformly parabolic or an ordinary differential equation. First, we introduce a semi-implicit time discretisation that has the benefit of decoupling the equations. We prove the positivity, boundedness, and convergence of the time-discrete solutions to the time-continuous solution. Then, we introduce an iterative linearisation scheme to solve the resulting nonlinear time-discrete problems. Under weak assumptions on the time-step size, we prove that the scheme converges irrespective of the space discretisation and mesh. Moreover, if the problem is non-degenerate, the convergence becomes faster as the time-step size decreases. Finally, employing the finite element method for the spatial discretisation, we study the behaviour of the scheme, and compare its performance to other commonly used schemes. These tests confirm that the proposed scheme is robust and fast.

math.NA↗

arXiv2021

A posteriori error estimates for the Richards equation

The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection-reaction-diffusion equation that exhibits both parabolic-hyperbolic and parabolic-elliptic kinds of degeneracies. In this study, we provide reliable, fully computable, and locally space-time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated $H^1(H^{-1})$, $L^2(L^2)$, and the $L^2(H^1)$ errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space-time efficiency error bounds are then obtained in a standard $H^1(H^{-1})\cap L^2(H^1)$ norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as flux nonconformity, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.

math.NA↗

arXiv2020

Travelling wave solutions for gravity fingering in porous media flows

We study an imbibition problem for porous media. When a wetted layer is above a dry medium, gravity leads to the propagation of the water downwards into the medium. In experiments, the occurrence of fingers was observed, a phenomenon that can be described with models that include hysteresis. In the present paper, we describe a single finger in a moving frame and set up a free boundary problem to describe the shape and the motion of one finger that propagates with a constant speed. We show the existence of solutions to the travelling wave problem and investigate the system numerically.

math.AP↗

arXiv2020

Capillary Hysteresis and Gravity Segregation in Two Phase Flow Through Porous Media

We study the gravity-driven flow of two fluid phases in a one-dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves, and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function, and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.

math.AP↗

arXiv2020

Existence and properties of solutions of extended play-type hysteresis model

This paper analyses the well-posedness and properties of the extended play-type model which was proposed in [van Duijn & Mitra (2018)] to incorporate hysteresis in unsaturated flow through porous media. The model, when regularised, reduces to a nonlinear degenerate parabolic equation coupled with an ordinary differential equation. This has an interesting mathematical structure which, to our knowledge, still remains unexplored. The existence of solutions for the non-degenerate version of the model is shown using the Rothe's method. An equivalent to maximum principle is proven for the solutions. Existence of solutions for the degenerate case is then shown assuming that the initial condition is bounded away from the degenerate points. Finally, it is shown that if the solution for the unregularised case exists, then it is contained within physically consistent bounds.

math.AP↗

arXiv2020

A Blockchain-based Approach for Assessing Compliance with SLA-guaranteed IoT Services

Within cloud-based internet of things (IoT) applications, typically cloud providers employ Service Level Agreements (SLAs) to ensure the quality of their provisioned services. Similar to any other contractual method, an SLA is not immune to breaches. Ideally, an SLA stipulates consequences (e.g. penalties) imposed on cloud providers when they fail to conform to SLA terms. The current practice assumes trust in service providers to acknowledge SLA breach incidents and executing associated consequences. Recently, the Blockchain paradigm has introduced compelling capabilities that may enable us to address SLA enforcement more elegantly. This paper proposes and implements a blockchain-based approach for assessing SLA compliance and enforcing consequences. It employs a diagnostic accuracy method for validating the dependability of the proposed solution. The paper also benchmarks Hyperledger Fabric to investigate its feasibility as an underlying blockchain infrastructure concerning latency and transaction success/fail rates.

cs.DC↗

arXiv2007

Strong-field effects in the Rabi oscillations of the superconducting phase qubit

Rabi oscillations have been observed in many superconducting devices, and represent prototypical logic operations for quantum bits (qubits) in a quantum computer. We use a three-level multiphoton analysis to understand the behavior of the superconducting phase qubit (current-biased Josephson junction) at high microwave drive power. Analytical and numerical results for the ac Stark shift, single-photon Rabi frequency, and two-photon Rabi frequency are compared to measurements made on a dc SQUID phase qubit with Nb/AlOx/Nb tunnel junctions. Good agreement is found between theory and experiment.

cond-mat.supr-con↗

arXiv2006

Fast High-Fidelity Measurements of the Ground and Excited States of a dc-SQUID Phase Qubit

We have investigated the fidelity and speed of single-shot current-pulse measurements of the three lowest energy states of the dc SQUID phase qubit. We apply a short (2ns) current pulse to one junction of a Nb/AlOx/Nb SQUID that is in the zero voltage state at 25 mK and measure if the system switches to the finite voltage state. By plotting the switching rate versus pulse size we can determine average occupancy of the levels down to 0.01%, quantify small levels of leakage, and find the optimum pulse condition for single-shot measurements. Our best error rate is 3% with a measurement fidelity of 94%. By monitoring the escape rate during the pulse, the pulse current in the junction can be found to better than 10 nA on a 0.1 ns time scale. Theoretical analysis of the system reveals switching curves that are in good agreement with the data, as well as predictions that the ultimate single-shot error rate for this technique can reach 0.4% and the fidelity 99.2%.

cond-mat.supr-con↗