Oscar P. Bruno

Adarsh Ganeshram, Haydn Maust, Valentin Duruisseaux et al.

The physics-informed neural operator (PINO) is a machine learning paradigm that has demonstrated promising results for learning solutions to partial differential equations (PDEs). It leverages the Fourier Neural Operator to learn solution operators in function spaces and leverages physics losses during training to penalize deviations from known physics laws. Spectral differentiation provides an efficient way to compute derivatives for the physics losses, but it inherently assumes periodicity. When applied to non-periodic functions, this assumption can lead to significant errors, including Gibbs phenomena near domain boundaries which degrade the accuracy of both function representations and derivative computations. To overcome this limitation, we introduce the FC-PINO (Fourier-Continuation-based Physics-Informed Neural Operator) architecture which extends the accuracy and efficiency of PINO and spectral differentiation to non-periodic and non-smooth PDEs. In FC-PINO, we propose integrating Fourier continuation into the PINO framework, and test two different continuation approaches: FC-Legendre and FC-Gram. By transforming non-periodic signals into periodic functions on extended domains in a well-conditioned manner, Fourier continuation enables fast and accurate derivative computations. This approach avoids the discretization sensitivity of finite differences and the memory overhead of automatic differentiation. We demonstrate that standard PINO fails (without padding) or struggles (even with padding) to solve non-periodic and non-smooth PDEs with high precision, across challenging benchmarks. In contrast, the proposed FC-PINO provides accurate, robust, and scalable solutions, substantially outperforming PINO alternatives, and demonstrating that Fourier continuation is critical for extending PINO to a wider range of PDE problems when high-precision solutions are needed.

Faisal Amlani, Oscar P. Bruno, José Carlos López-Vázquez et al.

We study the scattering of transient, high-frequency, narrow-band quasi-Rayleigh elastic waves by through-thickness holes in aluminum plates, in the framework of ultrasonic nondestructive testing (NDT) based on full-field optical detection. Sequences of the instantaneous two-dimensional (2-D) out-of-plane displacement scattering maps are measured with a self-developed PTVH system. The corresponding simulated sequences are obtained by means of an FC(Gram) elastodynamic solver introduced recently, which implements a full three-dimensional (3D) vector formulation of the direct linear-elasticity scattering problem. A detailed quantitative comparison between these experimental and numerical sequences, which is presented here for the first time, shows very good agreement both in the amplitude and the phase of the acoustic field in the forward, lateral and backscattering areas. It is thus suggested that the combination of the PTVH system and the FC(Gram) elastodynamic solver provides an effective ultrasonic inspection tool for plate-like structures, with a significant potential for ultrasonic NDT applications.

Oscar P. Bruno, Emmanuel Garza

This paper introduces a high-order-accurate strategy for integration of singular kernels and edge-singular integral densities that appear in the context of boundary integral equation formulations of the problem of acoustic scattering. In particular, the proposed method is designed for use in conjunction with geometry descriptions given by a set of arbitrary non-overlapping logically-quadrilateral patches---which makes the algorithm particularly well suited for treatment of CAD-generated geometries. Fejér's first quadrature rule is incorporated in the algorithm, to provide a spectrally accurate method for evaluation of contributions from far integration regions, while highly-accurate precomputations of singular and near-singular integrals over certain "surface patches" together with two-dimensional Chebyshev transforms and suitable surface-varying "rectangular-polar" changes of variables, are used to obtain the contributions for singular and near-singular interactions. The overall integration method is then used in conjunction with the linear-algebra solver GMRES to produce solutions for sound-soft open- and closed-surface scattering obstacles, including an application to an aircraft described by means of a CAD representation. The approach is robust, fast, and highly accurate: use of a few points per wavelength suffices for the algorithm to produce far-field accuracies of a fraction of a percent, and slight increases in the discretization densities give rise to significant accuracy improvements.

Oscar P. Bruno, Manuel A. Santana, Lloyd N. Trefethen

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.

Oscar P. Bruno, Stephen P. Shipman, Catalin Turc et al.

This paper, Part I in a two-part series, presents (i) A simple and highly efficient algorithm for evaluation of quasi-periodic Green functions, as well as (ii) An associated boundary-integral equation method for the numerical solution of problems of scattering of waves by doubly periodic arrays of scatterers in three-dimensional space. Except for certain "Wood frequencies" at which the quasi-periodic Green function ceases to exist, the proposed approach, which is based on use of smooth windowing functions, gives rise to lattice sums which converge superalgebraically fast--that is, faster than any power of the number of terms used--in sharp contrast with the extremely slow convergence exhibited by the corresponding sums in absence of smooth windowing. (The Wood-frequency problem is treated in Part II.) A proof presented in this paper establishes rigorously the superalgebraic convergence of the windowed lattice sums. A variety of numerical results demonstrate the practical efficiency of the proposed approach.

Oscar P. Bruno, Victor Dominguez, Francisco-Javier Sayas

In this paper we present a convergence analysis for the Nystrom method proposed in [Jour. Comput. Phys. 169 pp. 2921-2934, 2001] for the solution of the combined boundary integral equation formulations of sound-soft acoustic scattering problems in three-dimensional space. This fast and efficient scheme combines FFT techniques and a polar change of variables that cancels out the kernel singularity. We establish the stability of the algorithms in the $L^2$ norm and we derive convergence estimates in both the $L^2$ and $L^\infty$ norms. In particular, our analysis establishes theoretically the previously observed super-algebraic convergence of the method in cases in which the right-hand side is smooth.

O. P. Bruno, A. Prieto

We present fast, spatially dispersionless and unconditionally stable high-order solvers for Partial Differential Equations (PDEs) with variable coefficients in general smooth domains. Our solvers, which are based on (i) A certain "Fourier continuation" (FC) method for the resolution of the Gibbs phenomenon, together with (ii) A new, preconditioned, FC-based solver for two-point boundary value problems (BVP) for variable-coefficient Ordinary Differential Equations, and (iii) An Alternating Direction strategy, generalize significantly a class of FC-based solvers introduced recently for constant-coefficient PDEs. The present algorithms, which are applicable, with high-order accuracy, to variable-coefficient elliptic, parabolic and hyperbolic PDEs in general domains with smooth boundaries, are unconditionally stable, do not suffer from spatial numerical dispersion, and they run at FFT speeds. The accuracy, efficiency and overall capabilities of our methods are demonstrated by means of applications to challenging problems of diffusion and wave propagation in heterogeneous media.

Oscar P. Bruno, Liwei Xu, Tao Yin

We present a novel approach for the numerical solution of problems of elastic scattering by open arcs in two dimensions. Our methodology relies on the composition of weighted versions of the classical operators associated with Dirichlet and Neumann boundary conditions in conjunction with a certain "open-arc elastic Calderón relation" whose validity is demonstrated in this paper on the basis of numerical experiments, but whose rigorous mathematical proof is left for future work. Using this Calderón relation in conjunction with spectrally accurate quadrature rules and the Krylov-subspace linear algebra solver GMRES, the proposed overall open-arc elastic solver produces results of high accuracy in small number of iterations---for low and high frequencies alike. A variety of numerical examples in this paper demonstrate the accuracy and efficiency of the proposed methodology.

Oscar Bruno, Martín Maas

This paper introduces a fast algorithm, applicable throughout the electromagnetic spectrum, for the numerical solution of problems of scattering by periodic surfaces in two-dimensional space. The proposed algorithm remains highly accurate and efficient for challenging configurations including randomly rough surfaces, deep corrugations, large periods, near grazing incidences, and, importantly, Wood-anomaly resonant frequencies. The proposed approach is based on use of certain `shifted equivalent sources' which enable FFT acceleration of a Wood-anomaly-capable quasi-periodic Green function introduced recently (Bruno and Delourme, Jour. Computat. Phys., 262--290, 2014). The Green-function strategy additionally incorporates an exponentially convergent shifted version of the classical spectral series for the Green function. While the computing-cost asymptotics depend on the asymptotic configuration assumed, the computing costs rise at most linearly with the size of the problem for a number of important rough-surface cases we consider. In practice, single-core runs in computing times ranging from a fraction of a second to a few seconds suffice for the proposed algorithm to produce highly-accurate solutions in some of the most challenging contexts arising in applications.

Oscar P. Bruno, Allen Yang

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.

Oscar P. Bruno, Manuel A. Santana

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.