Vishal Vaibhav

Vishal Vaibhav

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.

Vishal Vaibhav

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.

Vishal Vaibhav

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.

Vishal Vaibhav

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.

Vishal Vaibhav

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.

Vishal Vaibhav

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.

Nilmadhab Das, Vishal Vaibhav, Yash Sunil Choudhary et al.

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.

Vishal Vaibhav

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.