arXiv subjects

Navneet Pratap Singh

Publications and source records attributed to Navneet Pratap Singh.

9 recordsLinked to original sources

arXiv2025

Stability Analysis of Inexact Solves in Model Reduction of Non-parametric Second-order Dynamical systems

Here, we focus on Model Order Reduction (MOR) of non-parametric second-order dynamical systems. In these MOR algorithms, sequences of large and sparse linear systems arise during the model reduction process. Solving such linear systems is the main computational bottleneck in efficient scaling of these MOR algorithms for reducing extremely large dynamical systems. Preconditioned iterative methods are often used for solving such linear systems. These iterative methods introduce errors because they solve the linear systems up to a certain tolerance. Hence, our focus is to analyze the stability of these MOR algorithms when using inexact linear solves. Adaptive Iterative Rational Global Arnoldi (AIRGA) is a popular MOR algorithm belonging to this category. We prove that, under four mild conditions, the AIRGA algorithm is backward stable with respect to the errors introduced by these inexact linear solves. Our results easily extend to other MOR algorithms belonging to this category. Our first condition enforces the use of a Ritz-Galerkin based linear solver, where the residual of a linear system is made orthogonal to the corresponding Krylov subspace. Our second condition requires satisfying few extra orthogonalities. We show how to modify the underlying linear solver to achieve these extra orthogonalities. We further demonstrate that using a recycling variant of the underlying linear solver helps us achieve these orthogonalities cheaply and with no code changes. Our third condition involves existence and invertibility of a matrix mostly dependent upon the input dynamical system, with the norm of this matrix bounded by one. Our fourth and final condition involves being able to compute a perturbation from the derived expression and bounding its norm by one as well. The last two conditions are easily satisfied by all our models.

math.NA↗

arXiv2025

Simulating and Learning Quantum Evolution: A CTQW-ML Framework

We present an approach to simulate the Schrödinger equation through continuous time quantum walks. The CTQW-based simulation applies unitary evolution driven by a quantum walk to generate probability amplitude distributions at various time steps. Additionally, we implemented a supervised neural network model to evaluate the effectiveness of data-driven techniques. The model learns to predict the squared modulus of the wavefunction given spatial and temporal coordinates. A comparative analysis demonstrates that the ML model can reproduce the qualitative structure and temporal progression of the quantum system with high accuracy. This study provides the synergy between quantum walk-based simulation and machine learning for solving quantum dynamical equations.

quant-ph↗

arXiv2025

Hybrid Quantum Walks Dynamics and Construction of a Quantum Cryptographic Primitive

In this research article, we design a quantum hash function model from hybrid quantum walks on finite path graph. The hybrid evolution operator consisting of integrated framework of continuous time quantum walks and lackadaisical quantum walks as per choice of bits in binary input message, acts on initial quantum state and generate quantum hash values from probability distribution of final quantum state. The quantum hash values generated through the proposed method is showing strong cryptographic properties in sensitivity analysis, collision analysis, statistical features analysis, birthday attack and uniform analysis. Our proposed framework is showing successful results utilizing mathematically well defined structure of hybrid quantum walks based on quantum mechanics phenomena.

quant-ph↗

arXiv2025

Quantum communication scheme to teleport arbitrary quantum state via discrete time quantum walks

In this article, we propose a quantum communication protocol via 2-step discrete time quantum walks with two coins on a graph of 10 vertices containing both cycles and paths. Quantum walks are known for their ability to integrate quantum mechanics dynamics like superposition and entanglement during the procedure. We calculate the total final quantum state of the system as well as the recovery operators to rescue the initial quantum state back at the receiver's location. Our work provides a fundamental block for a quantum teleportation scheme on a specific series of cycle and path graphs.

quant-ph↗

arXiv2024

Quantum fractional revival governed by adjacency matrix Hamiltonian in unitary Cayley graphs

In this article, we give characterization for existence of quantum fractional revival in unitary Cayley graph utilizing adjacency matrix Hamiltonian. Unitary Cayley graph $X=( Z_n, S)$ is a special graph as connection set $S \subseteq Z_n$ is the collection of coprimes to $n$. Unitary Cayley graph is an integral graph and its adjacency matrix is a circulant one. We prove that quantum fractional revival in unitary Cayley graphs exists only when the number of vertices is even. Number-theoretic and spectral characterizations are given for unitary Cayley graph admitting quantum fractional revival. Quantum fractional revival is analogous to quantum entanglement. It is one of qubit state transfer phenomena useful in communication of quantum information.

quant-ph↗

arXiv2021

Cube Sampled K-Prototype Clustering for Featured Data

Clustering large amount of data is becoming increasingly important in the current times. Due to the large sizes of data, clustering algorithm often take too much time. Sampling this data before clustering is commonly used to reduce this time. In this work, we propose a probabilistic sampling technique called cube sampling along with K-Prototype clustering. Cube sampling is used because of its accurate sample selection. K-Prototype is most frequently used clustering algorithm when the data is numerical as well as categorical (very common in today's time). The novelty of this work is in obtaining the crucial inclusion probabilities for cube sampling using Principal Component Analysis (PCA). Experiments on multiple datasets from the UCI repository demonstrate that cube sampled K-Prototype algorithm gives the best clustering accuracy among similarly sampled other popular clustering algorithms (K-Means, Hierarchical Clustering (HC), Spectral Clustering (SC)). When compared with unsampled K-Prototype, K-Means, HC and SC, it still has the best accuracy with the added advantage of reduced computational complexity (due to reduced data size).

cs.LG↗

arXiv2020

Reusing Preconditioners in Projection based Model Order Reduction Algorithms

Dynamical systems are pervasive in almost all engineering and scientific applications. Simulating such systems is computationally very intensive. Hence, Model Order Reduction (MOR) is used to reduce them to a lower dimension. Most of the MOR algorithms require solving large sparse sequences of linear systems. Since using direct methods for solving such systems does not scale well in time with respect to the increase in the input dimension, efficient preconditioned iterative methods are commonly used. In one of our previous works, we have shown substantial improvements by reusing preconditioners for the parametric MOR (Singh et al. 2019). Here, we had proposed techniques for both, the non-parametric and the parametric cases, but had applied them only to the latter. We have four main contributions here. First, we demonstrate that preconditioners can be reused more effectively in the non-parametric case as compared to the parametric one because of the lack of parameters in the former. Second, we show that reusing preconditioners is an art and it needs to be fine-tuned for the underlying MOR algorithm. Third, we describe the pitfalls in the algorithmic implementation of reusing preconditioners. Fourth, and final, we demonstrate this theory on a real life industrial problem (of size 1.2 million), where savings of upto 64% in the total computation time is obtained by reusing preconditioners. In absolute terms, this leads to a saving of 5 days.

math.NA↗

arXiv2018

Preconditioned Linear Solves for Parametric Model Order Reduction

The main computational cost of algorithms for computing reduced-order models of parametric dynamical systems is in solving sequences of very large and sparse linear systems. We focus on efficiently solving these linear systems, arising while reducing second-order linear dynamical systems, by iterative methods with appropriate preconditioners. We propose that the choice of underlying iterative solver is problem dependent. We propose the use of block variant of the underlying iterative method because often all right-hand-side are available together. Since, Sparse Approximate Inverse (SPAI) preconditioner is a general preconditioner that can be naturally parallelized, we propose its use. Our most novel contribution is a technique to cheaply update the SPAI preconditioner, while solving the parametrically changing linear systems. We support our proposed theory by numerical experiments where we first show benefit of 80% in time by using a block iterative method, and a benefit of 70% in time by using SPAI updates.

math.NA↗

arXiv2017

Preconditioned Iterative Solves in Model Reduction of Second Order Linear Dynamical Systems

Recently a new algorithm for model reduction of second order linear dynamical systems with proportional damping, the Adaptive Iterative Rational Global Arnoldi (AIRGA) algorithm, has been proposed. The main computational cost of the AIRGA algorithm is in solving a sequence of linear systems. These linear systems do change only slightly from one iteration step to the next. Here we focus on efficiently solving these systems by iterative methods and the choice of an appropriate preconditioner. We propose the use of relevant iterative algorithm and the Sparse Approximate Inverse (SPAI) preconditioner. A technique to cheaply update the SPAI preconditioner in each iteration step of the model order reduction process is given. Moreover, it is shown that under certain conditions the AIRGA algorithm is stable with respect to the error introduced by iterative methods. Our theory is illustrated by experiments. It is demonstrated that SPAI preconditioned Conjugate Gradient (CG) works well for model reduction of a one dimensional beam model with AIRGA algorithm. Moreover, the computation time of preconditioner with update is on an average 2/3 rd of the computation time of preconditioner without update. With average timings running into hours for very large systems, such savings are substantial.

math.NA↗