Accelerated Born series for high-frequency strong scattering by weakly non-radial media

Mirza Karamehmedović; Faouzi Triki

Home/Archive/arXiv

arXiv · 2610.23428

Accelerated Born series for high-frequency strong scattering by weakly non-radial media

Mirza Karamehmedović·Faouzi Triki

View original source ↗Read source document ↗Export metadata ↓

Abstract

We present a practical method for accelerating the modified Born series for high-frequency Helmholtz scattering in one, two, and three dimensions. We consider bounded, nonnegative, compactly supported media, allowing small departures from radial symmetry in two and three dimensions. The method selects a single complex relaxation parameter using an approximation of the scattering operator's spectrum. Starting from precomputed eigenvalues for a constant reference medium, we estimate their evolution towards the prescribed radial profile. A leading-order Wentzel-Kramers-Brillouin (WKB) approximation reduces this spectral propagation to scalar nonlinear equations, followed by a first-order correction for the nonradial part of the medium. The estimated eigenvalues, together with a bound on the omitted spectrum, define a convex minimax problem for selecting the relaxation parameter. This spectral preparation is independent of the source and can therefore be reused for multiple scattering problems in the same medium. Numerical examples demonstrate convergence in strongly scattering cases where the scattering operator has spectral radius greater than one. Comparisons with direct integral-equation and finite-difference solvers assess accuracy, residual convergence, and computation time, and show computational savings at comparable accuracy in selected examples.

Explore related subjects

math.NA cs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Mirza Karamehmedović, Faouzi Triki. 2026-10-16. Accelerated Born series for high-frequency strong scattering by weakly non-radial media. https://arxiv.org/abs/2610.23428

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Derivation and Numerical Simulation of a Thermodynamically Consistent Magneto Two-phase Flow Model

Motivated by Magnetic Drug Targeting (MDT), we derive a thermodynamically consistent two-phase model for the coupled dynamics of superparamagnetic iron oxide nanoparticles (SPIONs), a carrier fluid, and the magnetic field. The model includes a convection-diffusion equation for the SPION dispersion volume density and a modified Navier-Stokes system for the mixture velocity with SPION-dependent density and viscosity, an additional momentum flux due to relative SPION motion, and a magnetic body force. Additionally, the SPIONs interact with a quasi-static Maxwell system through their magnetization which is of Langevin type. The coupled system satisfies a global energy-dissipation balance. Moreover, we develop a semi-implicit finite element scheme with explicit flux-corrected transport stabilization and simulate the attraction of an initially localized SPION dispersion towards an external magnet in a system closed to mass exchange, using parameter values representative of MDT settings. The resulting SPION redistribution and induced flow are analyzed through physical observables, including energy and dissipation contributions and iron mass within a target region. Temporal convergence studies assess the ability of the numerical scheme to reproduce the energy-dissipation balance. Finally, we study the influence of individual coupling mechanisms on particle accumulation, maximum iron density, and peak flow velocity.

math.NA↗

arXiv2026

Optimization of gridding algorithms for FFT by vector optimization

The Fast Fourier Transform (FFT) is widely used in applications such as MRI, CT, and interferometry; however, because of its dependence on uniformly sampled data, it requires the use of gridding techniques for practical implementation. The performance of these algorithms strongly depends on the choice of the gridding kernel, with the first prolate spheroidal wave function (PSWF) regarded as optimal. This work redefines kernel optimality through the lens of vector optimization (VO), introducing a rigorous framework that characterizes optimal kernels as Pareto-efficient solutions of an error shape operator. We establish the continuity of such operator, study the existence of solutions, and propose a novel methodology to construct kernels tailored to a desired target error function. The approach is implemented numerically via interior-point optimization. Comparative experiments demonstrate that the proposed kernels outperform both the PSWF and the state-of-the-art methods (MIRT-NUFFT) in specific regions of interest, achieving orders-of-magnitude improvements in mean absolute errors. These results confirm the potential of VO-based kernel design to provide customized accuracy profiles aligned with application-specific requirements. Future research will extend this framework to multidimensional cases and relative error minimization, with potential integration of machine learning for adaptive target error selection.

math.NA↗

arXiv2026

On the convergence rates of generalized conditional gradient method for fully discretized Mean Field Games

We study convergence rates of the generalized conditional gradient (GCG) method applied to fully discretized Mean Field Games (MFG) systems. While explicit convergence rates of the GCG method have been established at the continuous PDE level, a rigorous analysis that simultaneously accounts for time-space discretization and iteration errors has been missing. In this work, we consider an MFG system with a quadratic Hamiltonian including an advection term. We employ the GCG method to construct an iterative algorithm in which the HJB and FP equations are solved separately. At each iteration, we apply the Cole--Hopf transformation to linearize these equations and discretize the resulting linear PDEs using a finite difference method. We then analyze the resulting fully discrete GCG scheme. Under suitable structural assumptions on coupling terms, we establish discrete maximum principles and derive explicit error estimates that quantify both discretization errors and iteration errors within a unified framework. Our estimates show how the convergence rates depend on the mesh sizes and the iteration number, and they reveal a non-uniform behavior with respect to the iteration. Moreover, we prove that higher convergence rates can be achieved under additional regularity assumptions on the solution. Numerical experiments are presented to illustrate the theoretical results and to confirm the predicted convergence behavior.

math.NA↗