arXiv subjects

Oscar P. Bruno

Publications and source records attributed to Oscar P. Bruno.

At least 19 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.

math.NA↗

arXiv2026

Two Dimensional Fourier Continuation for Domains with Corners

This paper presents a fast "two-dimensional Fourier Continuation" (2D-FC) method for the construction of biperiodic extensions of smooth, non-periodic functions defined over general two-dimensional (2D) domains, including domains with corners. The algorithm operates with an O(N log N) computational cost, for an N-point discretization grid, and it achieves a user-prescribed d-th order of accuracy. The methodology can be generalized to non-smooth domains of arbitrary dimensionality, but such extensions are not considered in the present work. The usefulness and performance of the 2D-FC method are demonstrated through applications to the Poisson problem posed on bounded 2D domains with corners. One illustrative application concerns a Poisson problem on a non-smooth drop-shaped domain with a highly oscillatory forcing term; employing a discretization containing approximately four million degrees of freedom, the method produces the Poisson solution with accuracies of the order of machine precision in computing times on the order of one second on a single core of a present-day computer.

math.NA↗

arXiv2026

Weakly-singular formulation of the Fractional Laplacian operator

This paper presents a new formulation of the fractional Laplacian operator $(-Δ)^s$ in $n$-dimensional space ($n \ge 1$). The proposed formulation expresses $(-Δ)^s$ as a composition of the classical Laplace differential operator and a weakly singular integral operator -- which can be used to reduce e.g. the Dirichlet problem for the fractional Laplacian to a weakly singular integral equation involving both volumetric and boundary integral operators. This reformulation is well suited for efficient and accurate numerical implementation. Although a full description of the associated high-order algorithm is deferred to a subsequent contribution, several numerical examples are included in this paper to demonstrate the high accuracy and computational efficiency achieved by the proposed approach.

math.AP↗

arXiv2025

Efficient time-domain scattering synthesis via frequency-domain singularity subtraction

Fourier transform-based methods enable accurate, dispersion-free simulations of time-domain scattering problems by evaluating solutions to the Helmholtz equation at a discrete set of frequencies sufficient to approximate the inverse Fourier transform. However, in the case of scattering by trapping obstacles, the Helmholtz solution exhibits nearly-real complex resonances -- which significantly slows the convergence of numerical inverse transform. To address this difficulty this paper introduces a frequency-domain singularity subtraction technique that regularizes the integrand of the inverse transform and efficiently computes the singularity contribution via a combination of a straightforward and inexpensive numerical technique together with a large-time asymptotic expansion. Crucially, all relevant complex resonances and their residues are determined via rational approximation of integral equation solutions at real frequencies. An adaptive algorithm is employed to ensure that all relevant complex resonances are properly identified.

math.NA↗

arXiv2025

Multi-patch/multiple-scattering frequency-time hybrid solver for interior and exterior wave equation problems

This paper proposes a new multiple-scattering frequency-time hybrid (FTH-MS) integral equation solver for problems of wave scattering by obstacles in two dimensional space, including interior problems in closed cavities and problems exterior to a set of disconnected open or closed scattering obstacles. The multiple-scattering FTH-MS method is based on a partition of the domain boundary into a user-prescribed set of overlapping open arcs, along with a corresponding sequence of multiple-scattering problems that effectively decompose the interior problem into a series of open-arc wave equation subproblems. The new strategy provides a significant extension of the original FTH-MS algorithm originally presented in [22], in that (1) By allowing for use of an arbitrary of number of component arcs, and not just two as in the previous contribution, the new approach affords (1a) A significantly increased geometric flexibility, as well as, (1b) The use of partitions for which each open arc leads to small numbers of iterations if iterative linear-algebra solvers are employed; and, (2) It facilitates parallelization -- as the subproblem solutions that are needed at each multiple scattering step can be evaluated in an embarrassingly parallel fashion. Utilizing a suitably-implemented Fourier transformation, each sub-problem is reduced to a Helmholtz frequency-domain problem that is tackled via a uniquely-solvable boundary integral equation. Similar FTH-MS methods are also presented for problems exterior to a number of bounded obstacles. All of the algorithms considered incorporate the previously introduced ``time-windowing and recentering'' methodology (that enables both treatment of incident signals of long duration and long time simulation), as well as a high-frequency Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for arbitrarily long times.

math.NA↗

arXiv2025

General-domain FC-based shock-dynamics solver II: Non-smooth domains, accuracy and parallel performance

This contribution Part II of a two-part series, extends the general-domain FC-SDNN (Fourier Continuation Shock-Detecting Neural Network) introduces in Part I to enable treatment of non-smooth domains, it introduces a parallel implementation of the scheme with high-quality weak and strong scalability properties, and it illustrates the overall methodology for a variety of tests for the 2D Euler equations--including supersonic and hypersonic flows and shocks past obstacles with corners. The results produces by the new methods are compared to previous theoretical and experimental results, and the high parallel scalability of the algorithm is demonstrated in both weak and strong scaling cases. Thanks to its use of a localized yet smooth artificial viscosity term--whose support is confined to regions near flow discontinuities identified by an artificial neural network--the algorithm maintains minimal numerical dissipation away from discontinuities. Overall, the method delivers accurate, sharp resolution of shocks and contact discontinuities, while producing smooth numerical solutions in regular flow regions--as evidences by the near-complete absence of spurious oscillations in level-set contours, even under strong shocks and high-speed flow conditions.

math.NA↗

arXiv2025

General-domain FC-based shock-dynamics solver I: Basic elements

This contribution, Part I in a two-part article series, presents a general-domain version of the FC-SDNN (Fourier Continuation Shock-detecting Neural Network) spectral scheme for the numerical solution of nonlinear conservation laws, which is applicable under arbitrary boundary conditions and in general domains. Like the previous simple-domain contribution (Journal of Computational Physics X 15, (2022)), the present approach relies on the use of the Fourier Continuation method for accurate spectral representation of non-periodic functions in conjunction with smooth artificial viscosity assignments localized in regions detected by means of a Shock-Detecting Neural Network (SDNN). Relying on such techniques, the present Part I paper introduces a novel multi-patch/subpatch artificial viscosity-capable domain decomposition strategy for complex domains with smooth boundaries, and it illustrates the methodology by means of a variety of computational results produced by an associated parallel implementation of the resulting shock-capturing algorithm in a present-day computing cluster. The subsequent Part II contribution then extends the algorithm to enable treatment of obstacles with non-smooth boundaries, it considers questions concerning parallelization and accuracy, and it presents comparisons with physical theory and prior experimental and computational results. The resulting multi-patch FC-SDNN algorithm does not require use of problem-dependent algorithmic parameters or positivity-preserving limiters, and, on account of its use of an overlapping-patch discretization, it is geometrically flexible and efficiently parallelized. A variety of numerical tests for the 2D Euler equations are presented, including the simulation of supersonic and hypersonic flows and shocks past physical obstacles at high speeds, such as Mach 25 re-entry flow speeds.

math.NA↗

arXiv2025

High order-accurate solution of scattering integral equations with unbounded solutions at corners

Although high-order Maxwell integral equation solvers provide significant advantages in terms of speed and accuracy over corresponding low-order integral methods, their performance significantly degrades in presence of non-smooth geometries--owing to field enhancement and singularities that arise at sharp edges and corners which, if left untreated, give rise to significant accuracy losses. The problem is particularly challenging in cases in which the "density" (i.e., the solution of the integral equation) tends to infinity at corners and edges--a difficulty that can be bypassed for 2D configurations, but which is unavoidable in 3D Maxwell integral formulations, wherein the component tangential to an edge of the electrical-current integral density vector tends to infinity at the edge. In order to tackle the problem this paper restricts attention to the simplest context in which the unbounded-density difficulty arises, namely, integral formulations in 2D space whose integral density blows up at corners; the strategies proposed, however, generalize directly to the 3D context. The novel methodologies presented in this paper yield high-order convergence for such challenging equations and achieve highly accurate solutions (even near edges and corners) without requiring a priori analysis of the geometry or use of singular bases.

math.NA↗

arXiv2024

Can Efficient Fourier-Transform Techniques Favorably Impact on Broadband Computational Electromagnetism?

In view of recently demonstrated joint use of novel Fourier-transform techniques and effective high-accuracy frequency domain solvers related to the Method of Moments, it is argued that a set of transformative innovations could be developed for the effective, accurate and efficient simulation of problems of wave propagation and scattering of broadband, time-dependent wavefields. This contribution aims to convey the character of these methods and to highlight their applicability in computational modeling of electromagnetic configurations across various fields of science and engineering.

physics.comp-ph↗

arXiv2024

Evaluation of resonances: adaptivity and AAA rational approximation of randomly scalarized boundary integral resolvents

This paper presents a novel algorithm, based on use of rational approximants of a randomly scalarized boundary integral resolvent in conjunction with an adaptive search strategy and an exponentially convergent secant-method termination stage, for the evaluation of acoustic and electromagnetic resonances in open and closed cavities. The desired cavity resonances are obtained as the poles of associated rational approximants; both the approximants and their poles are obtained by means of the recently introduced AAA rational-approximation algorithm. In fact, the proposed resonance-search method applies to any nonlinear eigenvalue problem associated with a given function $F: U \to \mathbb{C}^{d\times d}$, wherein, denoting $F(k) = F_k$, a complex value $k$ is sought for which $F_kw = 0$ for some nonzero $w\in \mathbb{C}^d$. For the scattering problems considered in this paper, $F_k$ is taken to equal a spectrally discretized version of a Green function-based boundary integral operator at spatial frequency $k$. In all cases, the scalarized resolvent is given by an expression of the form $u^* F_k^{-1} v$, where $u,v \in \mathbb{C}^d$ are fixed random vectors. The proposed adaptive search strategy relies on use of a rectangular subdivision of the resonance search domain which is locally refined to ensure that all resonances in the domain are captured. The approach works equally well in the case in which the search domain is an interval of the real line, in which case the rectangles used degenerate into subintervals of the search domain. A variety of numerical results are presented, including comparisons with well-known methods based on complex contour integration, and a discussion of the asymptotics that result as open cavities approach closed cavities -- in all, demonstrating the accuracy provided by the method, for low- and high-frequency states alike.

math.NA↗

arXiv2024

WKB Across Caustics: The Screened-WKB Method

We present a new methodology, based on the WKB approximation and Fast Fourier Transforms, for the evaluation of wave propagation through inhomogeneous media. This method can accurately resolve fields containing caustics, while still enjoying the computational advantages of the WKB approximation, namely, the ability to resolve arbitrarily high-frequency problems in computing times which are orders-of-magnitude shorter than those required by other algorithms presently available. For example, the proposed approach can simulate with high accuracy (with errors such as e.g. 0.1\%--0.001\%) the propagation of 5 cm radar signals across two-dimensional configurations resembling atmospheric ducting conditions, spanning hundreds of kilometers and millions of wavelengths in electrical size, in computing times of a few minutes in a single CPU core. [Preliminary version]

physics.comp-ph↗

arXiv2023

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 $O(N^α)$ operations, with $α\approx 1.07$, 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.

math.NA↗

arXiv2023

Multiple-scattering frequency-time hybrid solver for the wave equation in interior domains

This paper proposes a frequency-time hybrid solver for the time-dependent wave equation in two-dimensional interior spatial domains. The approach relies on four main elements, namely, 1) A multiple scattering strategy that decomposes a given interior time-domain problem into a sequence of limited-duration time-domain problems of scattering by overlapping open arcs, each one of which is reduced (by means of the Fourier transform) to a sequence of Helmholtz frequency-domain problems; 2) Boundary integral equations on overlapping boundary patches for the solution of the frequency-domain problems in point 1); 3) A smooth "Time-windowing and recentering" methodology that enables both treatment of incident signals of long duration and long time simulation; and, 4) A Fourier transform algorithm that delivers numerically dispersionless, spectrally-accurate time evolution for given incident fields. By recasting the interior time-domain problem in terms of a sequence of open-arc multiple scattering events, the proposed approach regularizes the full interior frequency domain problem-which, if obtained by either Fourier or Laplace transformation of the corresponding interior time-domain problem, must encapsulate infinitely many scattering events, giving rise to non-uniqueness and eigenfunctions in the Fourier case, and ill conditioning in the Laplace case. Numerical examples are included which demonstrate the accuracy and efficiency of the proposed methodology.

math.NA↗

arXiv2022

"Bootstrap Domain of Dependence": Bounds and Time Decay of Solutions of the Wave Equation

This article introduces a novel "bootstrap domain-of-dependence" concept, according to which, for all time following a given illumination period of arbitrary duration, the wave field scattered by an obstacle is encoded in the history of boundary scattering events for a time-length equal to the diameter of the obstacle, measured in time units. Resulting solution bounds provide estimates on the solution values in terms of a short-time history record, and they establish super-algebraically fast decay (i.e., decay faster than any negative power of time) for a wide range of scattering obstacle--including certain types of "trapping" obstacles whose periodic trapped orbits span a set of positive volumetric measure, and for which no previous fast-decay theory was available. The results, which do not rely on consideration of the Lax-Phillips complex-variables scattering framework and associated resonance-free regions in the complex plane, utilize only real-valued frequencies, and follow from use of Green functions and boundary integral equation representations in the frequency and time domains, together with a certain $q$-growth condition on the frequency-domain operator resolvent.

math.AP↗

arXiv2022

IFGF-accelerated integral equation solvers for acoustic scattering

We present an accelerated and hardware parallelized integral-equation solver for the problem of acoustic scattering by a two-dimensional surface in three-dimensional space. The approach is based, in part, on the novel Interpolated Factored Green Function acceleration method (IFGF) that, without recourse to the Fast Fourier Transform (FFT), evaluates the action of Green function-based integral operators for an $N$-point surface discretization at a complexity of $\Ord(N\log N)$ operations instead of the $\Ord(N^2)$ cost associated with nonaccelerated methods. The IFGF algorithm exploits the slow variations of factored Green functions to enable the fast evaluation of fields generated by groups of sources on the basis of a recursive interpolation scheme. In the proposed approach, the IFGF method is used to account for the vast majority of the computations, while, for the relatively few singular, nearly-singular and neighboring non-singular integral operator evaluations, a high-order rectangular-polar quadrature approach is employed instead. Since the overall approach does not rely on the FFT, it is amenable to efficient shared- and distributed-memory parallelization; this paper demonstrates such a capability by means of an OpenMP parallel implementation of the method. A variety of numerical examples presented in this paper demonstrate that the proposed methods enable the efficient solution of large problems over complex geometries on small parallel hardware infrastructures. Numerical examples include acoustic scattering by a sphere of up to $128$ wavelengths, an $80$-wavelength submarine, and a turbofan nacelle that is more than $80$ wavelengths in size, requiring, on a 28-core computer, computing times of the order of a few minutes per iteration and a few tens of iterations of the GMRES iterative solver.

math.NA↗

arXiv2022

Massively Parallelized Interpolated Factored Green Function Method

This paper presents the first parallel implementation of the novel "Interpolated Factored Green Function" (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in highly efficient data communication, and it exhibits in practice excellent parallel scaling up to large numbers of cores - without any hard limitations on the number of cores concurrently employed with high efficiency. Moreover, on any given number of cores, the proposed parallel approach preserves the O(N log N) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited than other methods for efficient parallelization in distributed-memory computer systems. In particular, the IFGF method relies on a "peer-to-peer" strategy wherein, at every level, field propagation is directly enacted via "exchanges" between "peer" polynomials of low and constant degree, without data accumulation in large-scale "telephone-central" mathematical constructs such as those in the Fast Multipole Method (FMM) or pure FFT-based approaches. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, including excellent weak and strong parallel scaling properties in all cases considered - for problems of up to 4,096 wavelengths in acoustic size, and scaling tests spanning from 1 compute core to all 1,680 cores available in the High Performance Computing cluster used.

math.NA↗

arXiv2022

Vector potential-based MHD solver for non-periodic flows using Fourier continuation expansions

A high-order method to evolve in time electromagnetic and velocity fields in conducting fluids with non-periodic boundaries is presented. The method has a small overhead compared with fast FFT-based pseudospectral methods in periodic domains. It uses the magnetic vector potential formulation for accurately enforcing the null divergence of the magnetic field, and allowing for different boundary conditions including perfectly conducting walls or vacuum surroundings, two cases relevant for many astrophysical, geophysical, and industrial flows. A spectral Fourier continuation method is used to accurately represent all fields and their spatial derivatives, allowing also for efficient solution of Poisson equations with different boundaries. A study of conducting flows at different Reynolds and Hartmann numbers, and with different boundary conditions, is presented to study convergence of the method and the accuracy of the solenoidal and boundary conditions.

physics.comp-ph↗

arXiv2021

FC-based shock-dynamics solver with neural-network localized artificial-viscosity assignment

This paper presents a spectral scheme for the numerical solution of nonlinear conservation laws in non-periodic domains under arbitrary boundary conditions. The approach relies on the use of the Fourier Continuation (FC) method for spectral representation of non-periodic functions in conjunction with smooth localized artificial viscosity assignments produced by means of a Shock-Detecting Neural Network (SDNN). Like previous shock capturing schemes and artificial viscosity techniques, the combined FC-SDNN strategy effectively controls spurious oscillations in the proximity of discontinuities. Thanks to its use of a localized but smooth artificial viscosity term, whose support is restricted to a vicinity of flow-discontinuity points, the algorithm enjoys spectral accuracy and low dissipation away from flow discontinuities, and, in such regions, it produces smooth numerical solutions -- as evidenced by an essential absence of spurious oscillations in level set lines. The FC-SDNN viscosity assignment, which does not require use of problem-dependent algorithmic parameters, induces a significantly lower overall dissipation than other methods, including the Fourier-spectral versions of the previous entropy viscosity method. The character of the proposed algorithm is illustrated with a variety of numerical results for the linear advection, Burgers and Euler equations in one and two-dimensional non-periodic spatial domains.

math.NA↗