arXiv subjects

Pietro Benedusi

Publications and source records attributed to Pietro Benedusi.

17 recordsLinked to original sources

arXiv2026

Validity of the CRD limit for modeling scattering polarization in the photospheric Sr I 4607 Å line

Context. Scattering polarization in the Sr I 4607 Å line is a key diagnostic for small-scale, unresolved magnetic fields in the quiet solar photosphere, inaccessible to Zeeman-based techniques. The complete frequency redistribution (CRD) limit is commonly used to model this line, but its validity and impact on magnetic sensitivity have not been systematically investigated. This is timely given new facilities observing this line with unprecedented accuracy, and future synoptic programs targeting long-term quiet-Sun magnetism. Aims. We assess the CRD limit against the general partial frequency redistribution (PRD) description of scattering for the Sr I 4607 Å polarization, focusing on its magnetic sensitivity via the Hanle effect. Methods. We solved the non-LTE radiative transfer problem for polarized radiation in a semi-empirical 1D plane-parallel static atmosphere, with CRD and PRD scattering, across magnetic field configurations relevant for Hanle diagnostics. Results. For small-scale unresolved fields, where the field acts only via Hanle depolarization of Q/I, CRD and PRD emergent profiles and Hanle sensitivity are essentially identical. For deterministic fields, CRD remains accurate when polarization signals exceed about 0.3%; for weaker signals, PRD effects can appreciably affect the emergent profiles. Conclusions. CRD is adequate for modeling Sr I 4607 Å scattering polarization in the vast majority of observationally relevant cases; PRD effects matter only for weak polarization signals.

astro-ph.SR↗

arXiv2026

Analysis of eigenvalue clustering leads to optimal scaling in numerical radiative transfer

We consider a multidimensional polychromatic radiative transfer (RT) problem, accounting for scattering processes in a general form, i.e. anisotropic (dipole) scattering with partial frequency redistribution. Given a discrete ordinates discretization, we report the corresponding matrix structures, depending on model and discretization parameters. Despite the possibly dense nature of these matrices, the use of Krylov methods is effective (especially in the matrix-free context) and robust. We propose a theoretical analysis, using the spectral tools of the symbol theory, explaining why Krylov convergence is robust w.r.t. all the discretization parameters, even in the unpreconditioned case. In fact, the compactness of the continuous operators used in the modeling leads to zero-clustered dense matrix sequences plus identity, so that the clustering at the unity of the spectra is deduced. Numerical experiments confirm the theoretical results, which have a direct application, for example, in the simulation of radiative transfer in stellar atmospheres, a key problem in astrophysical research. In general, we demonstrate that optimal scaling with respect to RT discretization parameters is expected for Krylov solution strategies.

math.NA↗

arXiv2026

3D radiative transfer modeling of scattering polarization with partial frequency redistribution I. Verification and disk-center results for the solar Ca I 4227 Å line

Several strong solar resonance lines show observable linear scattering polarization signals, holding a great potential for investigating the magnetism of the outer solar atmosphere. Accurately modeling these signals requires solving the radiative transfer (RT) problem for polarized radiation in comprehensive 3D models of the solar atmosphere, in non-local thermodynamic equilibrium, accounting for partial frequency redistribution (PRD) effects. This problem has so far been computationally inaccessible. We present the first scientific application of TRIP, a novel software for the massively parallel solution of the 3D non-LTE RT problem for polarized radiation, including scattering polarization and PRD. We aim to verify the code and explore the combined action of PRD and the 3D structure of the solar atmosphere on scattering polarization. We run TRIP to synthesize the Stokes profiles of the Ca I line at 4227 Å in a 3D model of the solar atmosphere extracted from a radiation magneto-hydrodynamic simulation. We efficiently solve the resulting large-scale problem, with up to $4 \times 10^{10}$ degrees of freedom, with a state-of-the-art preconditioned Krylov method, using up to 20 thousand parallel CPUs. After including verification tests, we find that the joint impact of PRD effects and the detailed 3D structure of the atmospheric model produce disk-center scattering polarization signals in the line wings. These signals are sensitive to the magnetic field, via magneto-optical effects, and to bulk velocity gradients. We also show that the CRD approximation underestimates the amplitude of disk-center line-core signals. This achievement represents a crucial step forward for diagnosing the magnetism of the solar chromosphere and transition region through the quantitative comparisons of synthetic and observational data.

astro-ph.SR↗

arXiv2025

Adaptive Multidimensional Quadrature on Multi-GPU Systems

We introduce a distributed adaptive quadrature method that formulates multidimensional integration as a hierarchical domain decomposition problem on multi-GPU architectures. The integration domain is recursively partitioned into subdomains whose refinement is guided by local error estimators. Each subdomain evolves independently on a GPU, which exposes a significant load imbalance as the adaptive process progresses. To address this challenge, we introduce a decentralised load redistribution schemes based on a cyclic round-robin policy. This strategy dynamically rebalance subdomains across devices through non-blocking, CUDA-aware MPI communication that overlaps with computation. The proposed strategy has two main advantages compared to a state-of-the-art GPU-tailored package: higher efficiency in high dimensions; and improved robustness w.r.t the integrand regularity and the target accuracy.

cs.DC↗

arXiv2025

A numerical approach for modelling the polarisation signals of strong resonance lines with partial frequency redistribution. Numerical applications to two-term atoms and plane-parallel atmospheres

Aims. The main goal of this paper is to present an accurate and efficient numerical strategy for solving the radiative transfer problem for polarised radiation in strong resonance lines forming out of local thermodynamic equilibrium, taking angle-dependent (AD) partial frequency redistribution (PRD) effects and J-state interference into account. We consider the polarisation produced both by the Zeeman effect and by the scattering of anisotropic radiation, along with its sensitivity to the Hanle and magneto-optical effects. Methods. We introduce a formalism that allows treating both a two-level and a two-term atom in the presence of arbitrary magnetic and bulk velocity fields. The problem is formulated by treating the population of the lower level/term as a fixed input parameter. This approach makes the problem linear with respect to the radiation field, enabling the application of efficient matrix-free preconditioned iterative methods for its solution. Additionally, the computation of the scattering emissivity in the comoving frame, together with a careful choice of the angular and spectral quadrature nodes, allow us to speed up the calculations by reducing the number of evaluations of the redistribution functions. Results. The proposed solution strategy is applied to synthesise the Stokes profiles of the Mg ii h&k doublet and the H i Ly-α line in 1D semi-empirical models. The results demonstrate that the method is both fast and accurate. A comparison with calculations from HanleRT-TIC displays an overall good agreement, thereby validating our solution strategy. Moreover, for the wavelength-integrated polarisation profiles of the H i Ly-α line, we find an excellent agreement between the results obtained including PRD effects in their general AD description and those obtained considering the angle-averaged simplifying approximation.

astro-ph.SR↗

arXiv2024

Dense cell-by-cell systems of PDEs: approximation, spectral analysis, and preconditioning

In the present study, we consider the Extra-Membrane-Intra model (EMI) for the simulation of excitable tissues at the cellular level. We provide the (possibly large) system of partial differential equations (PDEs), equipped with ad hoc boundary conditions, relevant to model portions of excitable tissues, composed of several cells. In particular, we study two geometrical settings: computational cardiology and neuroscience. The Galerkin approximations to the considered system of PDEs lead to large linear systems of algebraic equations, where the coefficient matrices depend on the number $N$ of cells and the fineness parameters. We give a structural and spectral analysis of the related matrix-sequences with $N$ fixed and with fineness parameters tending to zero. Based on the theoretical results, we propose preconditioners and specific multilevel solvers. Numerical experiments are presented and critically discussed, showing that a monolithic multilevel solver is efficient and robust with respect to all the problem and discretization parameters. In particular, we include numerical results increasing the number of cells $N$, both for idealized geometries (with $N$ exceeding $10^5$) and for realistic, densely populated 3D tissue reconstruction.

math.NA↗

arXiv2024

A splitting, discontinuous Galerkin solver for the cell-by-cell electroneutral Nernst-Planck framework

Mathematical models for excitable tissue with explicit representation of individual cells are highly detailed and can, unlike classical homogenized models, represent complex cellular geometries and local membrane variations. However, these cell-based models are challenging to approximate numerically, partly due to their mixed-dimensional nature with unknowns both in the bulk and at the lower-dimensional cellular membranes. We here develop and evaluate a novel solution strategy for the cell-based KNP-EMI model describing ionic electrodiffusion in and between intra- and extracellular compartments with explicit representation of individual cells. The strategy is based on operator splitting, a multiplier-free formulation of the coupled dynamics across sub-regions, and a discontinuous Galerkin discretization. In addition to desirable theoretical properties, such as local mass conservation, the scheme is practical as it requires no specialized functionality in the finite element assembly and order optimal solvers for the resulting linear systems can be realized with black-box algebraic multigrid preconditioners. Numerical investigations show that the proposed solution strategy is accurate, robust with respect to discretization parameters, and that the parallel scalability of the solver is close to optimal - both for idealized and realistic two and three dimensional geometries.

cs.CE↗

arXiv2024

Accurate PRD modeling of the forward-scattering Hanle effect in the chromospheric CaI 4227 Å line

Measurable linear scattering polarization signals have been predicted and detected at the solar disk center in the core of chromospheric lines. These forward-scattering polarization signals, which are of high interest for magnetic field diagnostics, have always been modeled either under the assumption of complete frequency redistribution (CRD), or taking partial frequency redistribution (PRD) effects into account under the angle-averaged (AA) approximation. This work aims at assessing the suitability of the CRD and PRD-AA approximations for modeling the forward-scattering polarization signals produced by the presence of an inclined magnetic field, the so-called forward-scattering Hanle effect, in the chromospheric CaI 4227 A line. Radiative transfer calculations are performed in semi-empirical 1D solar atmospheres, out of local thermodynamic equilibrium (LTE). A two-step solution strategy is applied: the non-LTE RT problem is first solved considering a multilevel atom and neglecting polarization phenomena. The same problem is then solved including polarization, considering a two-level atom and keeping fixed the lower-level population calculated at the previous step. The emergent linear polarization signals calculated under the CRD and PRD-AA approximations are analyzed and compared to those obtained by modeling PRD effects in their general angle-dependent (AD) formulation. With respect to the PRD-AD case, the CRD and PRD-AA calculations significantly underestimate the amplitude of the line-center polarization signals produced by the forward-scattering Hanle effect. The results of this work suggest that a PRD-AD modeling is required in order to develop reliable diagnostic techniques exploiting the forward-scattering polarization signals observed in the CaI 4227 A line. These results need to be confirmed by full 3D calculations including non-magnetic symmetry-breaking effects.

astro-ph.SR↗

arXiv2024

Scalable approximation and solvers for ionic electrodiffusion in cellular geometries

The activity and dynamics of excitable cells are fundamentally regulated and moderated by extracellular and intracellular ion concentrations and their electric potentials. The increasing availability of dense reconstructions of excitable tissue at extreme geometric detail pose a new and clear scientific computing challenge for computational modelling of ion dynamics and transport. In this paper, we design, develop and evaluate a scalable numerical algorithm for solving the time-dependent and nonlinear KNP-EMI equations describing ionic electrodiffusion for excitable cells with an explicit geometric representation of intracellular and extracellular compartments and interior interfaces. We also introduce and specify a set of model scenarios of increasing complexity suitable for benchmarking. Our solution strategy is based on an implicit-explicit discretization and linearization in time, a mixed finite element discretization of ion concentrations and electric potentials in intracellular and extracellular domains, and an algebraic multigrid-based, inexact block-diagonal preconditioner for GMRES. Numerical experiments with up to $10^8$ unknowns per time step and up to 256 cores demonstrate that this solution strategy is robust and scalable with respect to the problem size, time discretization and number of cores.

math.NA↗

arXiv2023

Assessment of the CRD approximation for the observer's frame RIII redistribution matrix

Approximated forms of the RII and RIII redistribution matrices are frequently applied to simplify the numerical solution of the radiative transfer problem for polarized radiation, taking partial frequency redistribution (PRD) effects into account. A widely used approximation for RIII is to consider its expression under the assumption of complete frequency redistribution (CRD) in the observer frame (RIII CRD). The adequacy of this approximation for modeling the intensity profiles has been firmly established. By contrast, its suitability for modeling scattering polarization signals has only been analyzed in a few studies, considering simplified settings. In this work, we aim at quantitatively assessing the impact and the range of validity of the RIII CRD approximation in the modeling of scattering polarization. Methods. We first present an analytic comparison between RIII and RIII CRD. We then compare the results of radiative transfer calculations, out of local thermodynamic equilibrium, performed with RIII and RIII CRD in realistic 1D atmospheric models. We focus on the chromospheric Ca i line at 4227 A and on the photospheric Sr i line at 4607 A.

astro-ph.SR↗

arXiv2023

Modeling excitable cells with the EMI equations: spectral analysis and iterative solution strategy

In this work, we are interested in solving large linear systems stemming from the Extra-Membrane-Intra (EMI) model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations (PDEs) equipped with proper boundary conditions, we provide numerical approximation schemes for the EMI PDEs and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate (preconditioned) Krylov solvers. We show, through numerical experiments, that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.

math.NA↗

arXiv2021

Numerical solutions to linear transfer problems of polarized radiation I. Algebraic formulation and stationary iterative methods

Context. The numerical modeling of the generation and transfer of polarized radiation is a key task in solar and stellar physics research and has led to a relevant class of discrete problems that can be reframed as linear systems. In order to solve such problems, it is common to rely on efficient stationary iterative methods. However, the convergence properties of these methods are problem-dependent, and a rigorous investigation of their convergence conditions, when applied to transfer problems of polarized radiation, is still lacking. Aims. After summarizing the most widely employed iterative methods used in the numerical transfer of polarized radiation, this article aims to clarify how the convergence of these methods depends on different design elements, such as the choice of the formal solver, the discretization of the problem, or the use of damping factors. The main goal is to highlight advantages and disadvantages of the different iterative methods in terms of stability and rate of convergence. Methods. We first introduce an algebraic formulation of the radiative transfer problem. This formulation allows us to explicitly assemble the iteration matrices arising from different stationary iterative methods, compute their spectral radii and derive their convergence rates, and test the impact of different discretization settings, problem parameters, and damping factors. Conclusions. The general methodology used in this article, based on a fully algebraic formulation of linear transfer problems of polarized radiation, provides useful estimates of the convergence rates of various iterative schemes. Additionally, it can lead to novel solution approaches as well as analyses for a wider range of settings, including the unpolarized case.

astro-ph.SR↗

arXiv2021

Numerical solutions to linear transfer problems of polarized radiation II. Krylov methods and matrix-free implementation

Context. Numerical solutions to transfer problems of polarized radiation in solar and stellar atmospheres commonly rely on stationary iterative methods, which often perform poorly when applied to large problems. In recent times, stationary iterative methods have been replaced by state-of-the-art preconditioned Krylov iterative methods for many applications. However, a general description and a convergence analysis of Krylov methods in the polarized radiative transfer context are still lacking. Aims. We describe the practical application of preconditioned Krylov methods to linear transfer problems of polarized radiation, possibly in a matrix-free context. The main aim is to clarify the advantages and drawbacks of various Krylov accelerators with respect to stationary iterative methods. Methods. We report the convergence rate and the run time of various Krylov-accelerated techniques combined with different formal solvers when applied to a 1D benchmark transfer problem of polarized radiation. In particular, we analyze the GMRES, BICGSTAB, and CGS Krylov methods, preconditioned with Jacobi, or (S)SOR. Results. Krylov methods accelerate the convergence, reduce the run time, and improve the robustness of standard stationary iterative methods. Jacobi-preconditioned Krylov methods outperform SOR-preconditioned stationary iterations in all respects. In particular, the Jacobi-GMRES method offers the best overall performance for the problem setting in use. Conclusions. Krylov methods can be more challenging to implement than stationary iterative methods. However, an algebraic formulation of the radiative transfer problem allows one to apply and study Krylov acceleration strategies with little effort. Furthermore, many available numerical libraries implement matrix-free Krylov routines, enabling an almost effortless transition to Krylov methods.

math.NA↗

arXiv2021

Space-time multilevel Monte Carlo methods and their application to cardiac electrophysiology

We present a novel approach aimed at high-performance uncertainty quantification for time-dependent problems governed by partial differential equations. In particular, we consider input uncertainties described by a Karhunen-Loeeve expansion and compute statistics of high-dimensional quantities-of-interest, such as the cardiac activation potential. Our methodology relies on a close integration of multilevel Monte Carlo methods, parallel iterative solvers, and a space-time discretization. This combination allows for space-time adaptivity, time-changing domains, and to take advantage of past samples to initialize the space-time solution. The resulting sequence of problems is distributed using a multilevel parallelization strategy, allocating batches of samples having different sizes to a different number of processors. We assess the performance of the proposed framework by showing in detail its application to the solution of nonlinear equations arising from cardiac electrophysiology. Specifically, we study the effect of spatially-correlated perturbations of the heart fibers conductivities on the mean and variance of the resulting activation map. As shown by the experiments, the theoretical rates of convergence of multilevel Monte Carlo are achieved. Moreover, the total computational work for a prescribed accuracy is reduced by an order of magnitude with respect to standard Monte Carlo methods.

cs.CE↗

arXiv2021

An experimental comparison of a space-time multigrid method with PFASST for a reaction-diffusion problem

We consider two parallel-in-time approaches applied to a (reaction) diffusion problem, possibly non-linear. In particular, we consider PFASST (Parallel Full Approximation Scheme in Space and Time) and space-time multilevel strategies. For both approaches, we start from an integral formulation of the continuous time-dependent problem. Then, a collocation form for PFASST and a discontinuous Galerkin discretization in time for the space-time multigrid are employed, resulting in the same discrete solution at the time nodes. Strong and weak scaling of both multilevel strategies is compared for varying order of the temporal discretization. Moreover, we investigate the respective convergence behavior for non-linear problems and highlight quantitative differences.

math.NA↗

arXiv2016

Time parallel gravitational collapse simulation

This article demonstrates the applicability of the parallel-in-time method Parareal to the numerical solution of the Einstein gravity equations for the spherical collapse of a massless scalar field. To account for the shrinking of the spatial domain in time, a tailored load balancing scheme is proposed and compared to load balancing based on number of time steps alone. The performance of Parareal is studied for both the sub-critical and black hole case; our experiments show that Parareal generates substantial speedup and, in the super-critical regime, can reproduce Choptuik's black hole mass scaling law.

gr-qc↗

arXiv2016

An Iterative Approach for Time Integration Based on Discontinuous Galerkin Methods

We present a new class of iterative schemes for solving initial value problems (IVP) based on discontinuous Galerkin (DG) methods. Starting from the weak DG formulation of an IVP, we derive a new iterative method based on a preconditioned Picard iteration. Using this approach, we can systematically construct explicit, implicit and semi-implicit schemes with arbitrary order of accuracy. We also show that the same schemes can be constructed by solving a series of correction equations based on the DG weak formulation. The accuracy of the schemes is proven to be $\min\{2p+1, K+1\}$ with $p$ the degree of the DG polynomial basis and $K$ the number of iterations. The stability is explored numerically; we show that the implicit schemes are $A$-stable at least for $0 \leq p \leq 9$. Furthermore, we combine the methods with a multilevel strategy to accelerate their convergence speed. The new multilevel scheme is intended to provide a flexible framework for high order space-time discretizations and to be coupled with space-time multigrid techniques for solving partial differential equations (PDEs). We present numerical examples for ODEs and PDEs to analyze the performance of the new methods. Moreover, the newly proposed class of methods, due to its structure, is also a competitive and promising candidate for parallel in time algorithms such as Parareal, PFASST, multigrid in time, etc.

math.NA↗