arXiv subjects

Ambuj Pandey

Publications and source records attributed to Ambuj Pandey.

8 recordsLinked to original sources

arXiv2026

"Truncated Fourier Filtering" method for fast and high-order evaluation of integrals and convolutions in general domains

This paper introduces and analyzes a novel algorithm---Truncated Fourier Filtering (TFF)---for the fast, high-order accurate evaluation of standard integrals and convolutions involving piecewise-smooth (possibly discontinuous) integrands over general $m$-dimensional domains ($m \ge 1$) employing an $m$-dimensional Cartesian grid. For an $N$-point discretization, the method runs at a computational cost of $\mathcal{O}(N)$ operations for standard integrals and $\mathcal{O}(N \log N)$ operations for convolutions, following, in either case, a one-time $\mathcal{O}(N \log N)$ precomputation step (not required in dimension $m = 1$). The core idea underlying TFF is to approximate the characteristic function of the integration domain by a truncated Fourier expansion of it over a suitably extended periodic domain, and to evaluate the resulting integrals via trapezoidal quadrature on a Cartesian grid with an appropriately chosen discretization size. Despite its conceptual simplicity, TFF attains high-order accuracy even for complex, possibly non-smooth or even non-Lipschitz geometries. A complete theoretical analysis is provided that establishes the superalgebraic convergence (i.e., convergence faster than any negative power of $N$) of the overall approach.

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arXiv2025

Decay of Chebyshev coefficients and error estimates of associated quadrature on shrinking intervals

We analyze decay of Chebyshev coefficients and local Chebyshev approximations for functions of finite regularity on finite intervals, focusing on the framework where the interval length tends to zero while the number of approximation nodes remains fixed. For all four families of Chebyshev polynomials, we derive error estimates that quantify the dependence on the interval length for (i) the decay of Chebyshev coefficients, (ii) the approximation error between continuous and discrete Chebyshev coefficients, and (iii) the convergence of Chebyshev-based quadrature rules. These results fill a gap in the existing theory and provide a unified and rigorous description of how approximation accuracy scales on shrinking intervals, offering new theoretical insight and practical guidance for high-order numerical methods on decomposed domains. Numerical experiments corroborate the theoretical results, confirming the decay rates and illustrating the error behavior in practice.

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arXiv2025

Rectangular polar quadrature in 1D and its error analysis

This paper presents a one-dimensional analog of the Rectangular-Polar (RP) integration strategy and its convergence analysis for weakly singular convolution integrals. The key idea of this method is to break the whole integral into integral over non-overlapping patches (subdomains) and achieve convergence by increasing the number of patches while approximating the integral on patches accurately using a fixed number of quadrature points. The non-singular integrals are approximated to high-order using Fej\'er first quadrature, and a specialized integration strategy is designed and incorporated for singular integrals where the kernel singularity is resolved by mean of Polynomial Change of Variable (PCV). We prove that for high-order convergence, it is essential to compute integral weights accurately, and the method's convergence rate depends critically on the degree of the PCV and the singularity of the integral kernel. Specifically, for kernels of the form $|x-y|^{-\alpha}(\alpha>-1)$, the method achieves high-order convergence if and only if $p(1-\alpha) \in \mathbb{N}$. This relationship highlights the importance of choosing an appropriate degree $p$ for the PCV. A new error estimate in a framework where convergence is achieved by reducing the approximating domain size while keeping the number of discretization points fixed is derived for the Fej\'er first quadrature, decay rate of Chebyshev coefficients, and the error in approximating continuous Chebyshev coefficients with discrete ones. Numerical experiments corroborate the theoretical findings, showcasing the effectiveness of the RP strategy for accurately solving singular integral equations. As an application of the method, numerical solutions of the surface scattering problem in the two dimensions are computed for complex domains, and the algorithm's efficacy is demonstrated for large-scale problems.

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arXiv2021

Fast accurate approximation of convolutions with weakly singular kernel and its applications

In this article, we present an $O(N \log N)$ rapidly convergent algorithm for the numerical approximation of the convolution integral with radially symmetric weakly singular kernels and compactly supported densities. To achieve the reduced computational complexity, we utilize the Fast Fourier Transform (FFT) on a uniform grid of size $N$ for approximating the convolution. To facilitate this and maintain the accuracy, we primarily rely on a periodic Fourier extension of the density with a suitably large period depending on the support of the density. The rate of convergence of the method increases with increasing smoothness of the periodic extension and, in fact, approximations exhibit super-algebraic convergence when the extension is infinitely differentiable. Furthermore, when the density has jump discontinuities, we utilize a certain Fourier smoothing technique to accelerate the convergence to achieve the quadratic rate in the overall approximation. Finally, we apply the integration scheme for numerical solution of certain partial differential equations. Moreover, we apply the quadrature to obtain a fast and high-order Nyst\"om solver for the solution of the Lippmann-Schwinger integral equation. We validate the performance of the proposed scheme in terms of accuracy as well as computational efficiency through a variety of numerical experiments.

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arXiv2020

Fourier smoothed pre-corrected trapezoidal rule for solution of Lippmann-Schwinger integral equation

For the numerical solution of the Lippmann-Schwinger equation, while the pre-corrected trapezoidal rule converges with high-order for smooth compactly supported densities, it exhibits only the linear convergence in the case of discontinuity in material properties across the interface. In this short article, we propose a Nyström solver based on "Fourier smoothed pre-corrected trapezoidal rule" that converges with second order for such scattering problems while maintaining the computational complexity of $O(N \log N)$. Moreover, the method is not only very simple to implement, it is also applicable to problems with geometrically complex inhomogeneities including those with corners and cusps. We present a variety of numerical experiments including comparative studies with competing approaches reported in [J. Comput. Phys., 200(2) (2004), 670--694] by Bruno and Hyde, and in [J. Fourier Anal. Appl., 11(4) (2005), 471-487 ] by Andersson and Holst to exemplify its performance in terms of speed and accuracy. This Fourier smoothed numerical integration scheme can also be adapted to other problems of interest where the convolution integral with discontinuous density is required to be computed.

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arXiv2019

Direct/iterative hybrid solver for scattering by inhomogeneous media

This paper presents a fast high-order method for the solution of two-dimensional problems of scattering by penetrable inhomogeneous media, with application to high-frequency configurations containing (possibly) discontinuous refractivities. The method relies on a hybrid direct/iterative combination of 1)~A differential volumetric formulation (which is based on the use of appropriate Chebyshev differentiation matrices enacting the Laplace operator) and, 2)~A second-kind boundary integral formulation. The approach enjoys low dispersion and high-order accuracy for smooth refractivities, as well as second-order accuracy (while maintaining low dispersion) in the discontinuous refractivity case. The solution approach proceeds by application of Impedance-to-Impedance (ItI) maps to couple the volumetric and boundary discretizations. The volumetric linear algebra solutions are obtained by means of a multifrontal solver, and the coupling with the boundary integral formulation is achieved via an application of the iterative linear-algebra solver GMRES. In particular, the existence and uniqueness theory presented in the present paper provides an affirmative answer to an open question concerning the existence of a uniquely solvable second-kind ItI-based formulation for the overall scattering problem under consideration. Relying on a modestly-demanding scatterer-dependent precomputation stage (requiring in practice a computing cost of the order of $O(N^{\alpha})$ operations, with $\alpha \approx 1.07$, for an $N$-point discretization), together with fast ($O(N)$-cost) single-core runs for each incident field considered, the proposed algorithm can effectively solve scattering problems for large and complex objects possibly containing strong refractivity contrasts and discontinuities.

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arXiv2018

Improved convergence of fast integral equation solvers for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

In recent years, several fast solvers for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by penetrable inhomogeneous obstacles, have been proposed. While many of these fast methodologies exhibit rapid convergence for smoothly varying scattering configurations, the rate for most of them reduce to either linear or quadratic when material properties are allowed to jump across the interface. A notable exception to this is a recently introduced Nystr\"{o}m scheme [J. Comput. Phys., 311 (2016), 258--274] that utilizes a specialized quadrature in the boundary region for a high-order treatment of the material interface. In this text, we present a solution framework that relies on the specialized boundary integrator to enhance the convergence rate of other fast, low order methodologies without adding to their computational complexity of $O(N \log N)$ for an $N$-point discretization. In particular, to demonstrate the efficacy of the proposed framework, we explain its implementation to enhance the order to convergence of two schemes, one introduced by Duan and Rokhlin [J. Comput. Phys., 228(6) (2009), 2152--2174] that is based on a pre-corrected trapezoidal rule while the other by Bruno and Hyde [J. Comput. Phys., 200(2) (2004), 670--694] which relies on a suitable decomposition of the Green's function via Addition theorem. In addition to a detailed description of these methodologies, we also present a comparative performance study of the improved versions of these two and the Nystr\"{o}m solver in [J. Comput. Phys., 311 (2016), 258--274] through a wide range of numerical experiments.

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arXiv2015

An efficient high-order Nystr\"om scheme for acoustic scattering by inhomogeneous penetrable media with discontinuous material interface

This text proposes a fast, rapidly convergent Nystr\"{o}m method for the solution of the Lippmann-Schwinger integral equation that mathematically models the scattering of time-harmonic acoustic waves by inhomogeneous obstacles, while allowing the material properties to jump across the interface. The method works with overlapping coordinate charts as a description of the given scatterer. In particular, it employs "partitions of unity" to simplify the implementation of high-order quadratures along with suitable changes of parametric variables to analytically resolve the singularities present in the integral operator to achieve desired accuracies in approximations. To deal with the discontinuous material interface in a high-order manner, a specialized quadrature is used in the boundary region. The approach further utilizes an FFT based strategy that uses equivalent source approximations to accelerate the evaluation of large number of interactions that arise in the approximation of the volumetric integral operator and thus achieves a reduced computational complexity of $O(N \log N)$ for an $N$-point discretization. A detailed discussion on the solution methodology along with a variety of numerical experiments to exemplify its performance in terms of both speed and accuracy are presented in this paper.

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