arXiv subjects

Vishal Vaibhav

Publications and source records attributed to Vishal Vaibhav.

At least 19 recordsLinked to original sources

arXiv2025

End-to-End Argument Mining through Autoregressive Argumentative Structure Prediction

Argument Mining (AM) helps in automating the extraction of complex argumentative structures such as Argument Components (ACs) like Premise, Claim etc. and Argumentative Relations (ARs) like Support, Attack etc. in an argumentative text. Due to the inherent complexity of reasoning involved with this task, modelling dependencies between ACs and ARs is challenging. Most of the recent approaches formulate this task through a generative paradigm by flattening the argumentative structures. In contrast to that, this study jointly formulates the key tasks of AM in an end-to-end fashion using Autoregressive Argumentative Structure Prediction (AASP) framework. The proposed AASP framework is based on the autoregressive structure prediction framework that has given good performance for several NLP tasks. AASP framework models the argumentative structures as constrained pre-defined sets of actions with the help of a conditional pre-trained language model. These actions build the argumentative structures step-by-step in an autoregressive manner to capture the flow of argumentative reasoning in an efficient way. Extensive experiments conducted on three standard AM benchmarks demonstrate that AASP achieves state-of-theart (SoTA) results across all AM tasks in two benchmarks and delivers strong results in one benchmark.

cs.CL↗

arXiv2024

Nonreflecting Boundary Condition for the free Schrödinger equation for hyperrectangular computational domains

In this article, we discuss the efficient ways of implementing the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a hyperrectangular computational domain in $\field{R}^d$ with periodic boundary conditions along the $(d-1)$ unbounded directions. In particular, we consider Padé approximant based rational approximation of the exact TBC and a spatially local form of the exact TBC obtained under its high-frequency approximation. For the spatial discretization, we use a Legendre-Galerkin spectral method with a boundary-adapted basis to ensure the bandedness of the resulting linear system. Temporal discretization is then addressed with two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). Finally, several numerical tests are presented to demonstrate the effectiveness of the methods where we study the stability and convergence behaviour empirically.

math.NA↗

arXiv2024

Transparent boundary condition and its high frequency approximation for the Schrödinger equation on a rectangular computational domain

This paper addresses the numerical implementation of the transparent boundary condition (TBC) and its various approximations for the free Schrödinger equation on a rectangular computational domain. In particular, we consider the exact TBC and its spatially local approximation under high frequency assumption along with an appropriate corner condition. For the spatial discretization, we use a Legendre-Galerkin spectral method where Lobatto polynomials serve as the basis. Within variational formalism, we first arrive at the time-continuous dynamical system using spatially discrete form of the initial boundary-value problem incorporating the boundary conditions. This dynamical system is then discretized using various time-stepping methods, namely, the backward-differentiation formula of order 1 and 2 (i.e., BDF1 and BDF2, respectively) and the trapezoidal rule (TR) to obtain a fully discrete system. Next, we extend this approach to the novel Padé based implementation of the TBC presented by Yadav and Vaibhav [arXiv:2403.07787(2024)]. Finally, several numerical tests are presented to demonstrate the effectiveness of the boundary maps (incorporating the corner conditions) where we study the stability and convergence behavior empirically.

math.NA↗

arXiv2024

Transparent boundary condition and its effectively local approximation for the Schrödinger equation on a rectangular computational domain

The transparent boundary condition for the free Schrödinger equation on a rectangular computational domain requires implementation of an operator of the form $\sqrt{\partial_t-i\triangle_Γ}$ where $\triangle_Γ$ is the Laplace-Beltrami operator. It is known that this operator is nonlocal in time as well as space which poses a significant challenge in developing an efficient numerical method of solution. The computational complexity of the existing methods scale with the number of time-steps which can be attributed to the nonlocal nature of the boundary operator. In this work, we report an effectively local approximation for the boundary operator such that the resulting complexity remains independent of number of time-steps. At the heart of this algorithm is a Padé approximant based rational approximation of certain fractional operators that handles corners of the domain adequately. For the spatial discretization, we use a Legendre-Galerkin spectral method with a new boundary adapted basis which ensures that the resulting linear system is banded. A compatible boundary-lifting procedure is also presented which accommodates the segments as well as the corners on the boundary. The proposed novel scheme can be implemented within the framework of any one-step time marching schemes. In particular, we demonstrate these ideas for two one-step methods, namely, the backward-differentiation formula of order 1 (BDF1) and the trapezoidal rule (TR). For the sake of comparison, we also present a convolution quadrature based scheme conforming to the one-step methods which is computationally expensive but serves as a golden standard. Finally, several numerical tests are presented to demonstrate the effectiveness of our novel method as well as to verify the order of convergence empirically.

math.NA↗

arXiv2020

New method of bandlimited extrapolation

The paper deals with numerical solution of the Fredholm integral equation associated with the classical problem of extrapolating bandlimited functions known on $(-1,1)$ to the entire real line. The approach presented can be characterized as the degenerate kernel method using the spherical Bessel functions as basis functions where the Tikhonov regularization is applied at the discrete level in order to deal with the ill-posedness of the problem.

math.NA↗

arXiv2019

A Chebyshev Spectral Method for Nonlinear Fourier Transform: Norming Constants

In this paper, we present a Chebyshev based spectral method for the computation of the Jost solutions corresponding to complex values of the spectral parameter in the Zakharov--Shabat scattering problem. The discrete framework is then used to devise a new algorithm based on a minimum total variation (MTV) principle for the computation of the norming constants which comprise the discrete part of the nonlinear Fourier spectrum. The method relies on the MTV principle to find the points where the expressions for norming constants are numerically well-conditioned.

physics.comp-ph↗

arXiv2019

A Fast Chebyshev Spectral Method for Nonlinear Fourier Transform

In this letter, we present a fast and well-conditioned spectral method based on the Chebyshev polynomials for computing the continuous part of the nonlinear Fourier spectrum. The algorithm achieves a complexity of $O(N_{\text{iter.}}N\log N)$ per spectral node for $N$ samples of the signal at the Chebyshev nodes where $N_{\text{iter.}}$ is the number of iterations of the biconjugate gradient stabilized method.

physics.comp-ph↗

arXiv2019

Fast Nonlinear Fourier Transform using Chebyshev Polynomials

We explore the class of exponential integrators known as exponential time differencing (ETD) method in this letter to design low complexity nonlinear Fourier transform (NFT) algorithms that compute discrete approximations of the scattering coefficients in terms of the Chebyshev polynomials for real values of the spectral parameter. In particular, we discuss ETD Runge-Kutta methods which yield algorithms with complexity $O(N\log^2N)$ (where $N$ is the number of samples of the signal) and an order of convergence that matches the underlying one-step method.

physics.comp-ph↗

arXiv2019

Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of $4$ on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield algorithms with complexity $O(N\log^2N)$ (where $N$ is the number of samples of the signal) which have superior accuracy-complexity trade-off than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield algorithms of complexity $O(N^2)$ that have superior error behavior even for moderately small step-sizes and higher signal strengths.

math.NA↗

arXiv2019

Darboux Transformation: New Identities

This letter reports some new identities for multisoliton potentials that are based on the explicit representation provided by the Darboux matrix. These identities can be used to compute the complex gradient of the energy content of the tail of the profile with respect to the discrete eigenvalues and the norming constants. The associated derivatives are well defined in the framework of the so-called Wirtinger calculus which can aid a complex variable based optimization procedure in order to generate multisolitonic signals with desired effective temporal and spectral width.

nlin.SI↗

arXiv2018

On the nonreflecting boundary operators for the general two dimensional Schrödinger equation

Of the two main objectives we pursue in this paper, the first one consists in the studying operators of the form $(\partial_t-i\triangle_Γ)^α,\,\,α=1/2,-1/2,-1,\ldots,$ where $\triangle_Γ$ is the Laplace-Beltrami operator. These operators arise in the context of nonreflecting boundary conditions in the pseudo-differential approach for the general Schrödinger equation. The definition of such operators is discussed in various settings and a formulation in terms of fractional operators is provided. The second objective consists in deriving corner conditions for a rectangular domain in order to make such domains amenable to the pseudo-differential approach. Stability and uniqueness of the solution is investigated for each of these novel boundary conditions.

physics.comp-ph↗

arXiv2018

Numerical Methods for Fast Nonlinear Fourier Transformation, Part I: Exponential Runge-Kutta and Linear Multistep Methods

The main objective of this series of papers is to explore the entire landscape of numerical methods for fast nonlinear Fourier transformation (NFT) within the class of integrators known as the exponential integrators. In this paper, we explore the theoretical aspects of exponential Runge-Kutta (RK) and linear multistep (LM) methods, in particular, the stability and convergence of these methods via the transfer matrix formulation. The analysis carried out in the paper shows that while the exponential LM methods are naturally amenable to FFT-based fast polynomial arithmetic, the RK methods require equispaced nodes to achieve that. Therefore, each these family of methods is capable of yielding a family of fast NFT algorithms such that the scattering coefficients can be computed with a complexity of $\mathscr{O}(N\log^2N)$ and a rate of convergence given by $\mathscr{O}(N^{-p})$ where $N$ is the number of samples of the signal and $p$ is order of the underlying discretization scheme. Further, while RK methods can accommodate vanishing as well as periodic boundary conditions, the LM methods can only handle the former type of boundary conditions without requiring a starting procedure. The ideas presented in this paper extend naturally to the family of integrators known as general linear methods which will be explored in a forthcoming paper.

math.NA↗

arXiv2018

Nonlinearly Bandlimited Signals

In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand-Levitan-Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker-Kotelnikov-Shannon sampling series yields an algorithm that converges as $\mathscr{O}\left(N^{-1/2}\right)$ whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as $\mathscr{O}\left(N^{-m-1/2}\right)$ for any $m>0$ provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity $\mathscr{O}\left(N_{\text{iter.}}N^2\right)$ per sample of the signal where $N$ is the number of sampling basis functions used and $N_{\text{iter.}}$ is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity $\mathscr{O}\left(N_{\text{iter.}}N\log N\right)$ (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved.

physics.comp-ph↗

arXiv2018

Discrete Darboux Transformation for Ablowitz-Ladik Systems Derived from Numerical Discretization of Zakharov-Shabat Scattering Problem

The numerical discretization of the Zakharov-Shabat Scattering problem using integrators based on the implicit Euler method, trapezoidal rule and the split-Magnus method yield discrete systems that qualify as Ablowitz-Ladik systems. These discrete systems are important on account of their layer-peeling property which facilitates the differential approach of inverse scattering. In this paper, we study the Darboux transformation at the discrete level by following a recipe that closely resembles the Darboux transformation in the continuous case. The viability of this transformation for the computation of multisoliton potentials is investigated and it is found that irrespective of the order of convergence of the underlying discrete framework, the numerical scheme thus obtained is of first order with respect to the step size.

physics.comp-ph↗

arXiv2018

Nonlinear Fourier Transform of Time-Limited and One-sided Signals

In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform (NFT). In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal has a prescribed support. In our exposition, we assume that the support is a simply connected domain that is either a bounded interval or the half-line, which amounts to studying the class of signals which are either time-limited or one-sided, respectively. Further, it is shown that the analyticity properties of the scattering coefficients of the aforementioned classes of signals can be exploited to improve the numerical conditioning of the differential approach of inverse scattering. Here, we also revisit the integral approach of inverse scattering and provide the correct derivation of the so called Töplitz inner-bordering algorithm. Finally, we conduct extensive numerical tests in order to verify the analytical results presented in the article. These tests also provide us an opportunity to compare the performance of the two aforementioned numerical approaches in terms of accuracy and complexity of computations.

physics.comp-ph↗

arXiv2018

Fast Inverse Nonlinear Fourier Transform

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU$(2)$-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur (AKNS) formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present two fast inverse NFT algorithms with $O(KN+N\log^2N)$ complexity and a convergence rate of $O(N^{-2})$ where $N$ is the number of samples of the signal and $K$ is the number of eigenvalues. These algorithms are realized using a new fast layer-peeling (LP) scheme ($O(N\log^2N)$) together with a new fast Darboux transformation (FDT) algorithm ($O(KN+N\log^2N)$) previously developed by the author. The proposed fast inverse NFT algorithm proceeds in two steps: The first step involves computing the radiative part of the potential using the fast LP scheme for which the input is synthesized under the assumption that the radiative potential is nonlinearly bandlimited, i.e., the continuous spectrum has a compact support and the discrete spectrum is empty. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.

math.NA↗

arXiv2017

Higher Order Convergent Fast Nonlinear Fourier Transform

It is demonstrated is this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of $O{KN+C_pN\log^2N}$ such that the error vanishes as $O{N^{-p}}$ where $p\in\{1,2,3,4\}$ and $K$ is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula ($C_p=p^3$) and the implicit Adams method ($C_p=(p-1)^3$) of which the latter proves to be the most accurate family of methods for fast NFT.

physics.comp-ph↗

arXiv2017

Fast Inverse Nonlinear Fourier Transformation using Exponential One-Step Methods, Part I: Darboux Transformation

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU$(2)$-nonlinear Fourier transformation (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transformation is quite well established in the Ablowitz-Kaup-Newell-Segur (AKNS) formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present a unified framework for the forward and inverse NFT using exponential one-step methods which are amenable to FFT-based fast polynomial arithmetic. Within this discrete framework, we propose a fast Darboux transformation (FDT) algorithm having an operational complexity of $\mathscr{O}\left(KN+N\log^2N\right)$ such that the error in the computed $N$-samples of the $K$-soliton vanishes as $\mathscr{O}\left(N^{-p}\right)$ where $p$ is the order of convergence of the underlying one-step method. For fixed $N$, this algorithm outperforms the the classical DT (CDT) algorithm which has a complexity of $\mathscr{O}\left(K^2N\right)$. We further present extension of these algorithms to the general version of DT which allows one to add solitons to arbitrary profiles that are admissible as scattering potentials in the ZS-problem. The general CDT/FDT algorithms have the same operational complexity as that of the $K$-soliton case and the order of convergence matches that of the underlying one-step method. A comparative study of these algorithms is presented through exhaustive numerical tests.

physics.comp-ph↗