An application of generalized matrix inversion to sequential least squares parameter estimation
Generalized matrix inversion theory used in formulating recursive algorithm for least squares parameter estimation
INVERSION↗
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Machine dependence of degrees of difficulty.
Machine dependence of degree-of-difficulty relation for recursive functions, considering Turing machine as programmed computer
RECURSION FORMULA↗
Inversion of the N-dimensional Laplace transform.
Series expansion of inversion of n-dimensional Laplace transforms in terms of ultraspherical polynomials having coefficient computed recursively
RECURSION FORMULA↗
An algorithm to obtain series expansions in the three-body problem
Algorithm and recursion formulas for series expansion of three-body problem
ALGORITHM↗
Open string amplitudes: singularities, asymptotics and new representations
Open string amplitudes at tree level have been studied for over fifty years. However, there is no known analytic form for general n-point amplitudes, and their conventional representation in terms of worldsheet integrals does not make many of their most basic physical properties manifest. Recently, a formulation of these amplitudes exposing the underlying “binary geometry” via the use of “u” variables has given us many insights into their basic features. In this paper, we initiate a systematic exploration of fundamental aspects of open string amplitudes from this new point of view. We begin by finding explicit expressions for the factorization of amplitudes at general massive levels, which are seen to be determined by products of lower-point massless amplitudes with shifted kinematics. We then study the asymptotic behavior when subsets of kinematic variables become large, delineating regimes with exponential (generalized hard scattering) and power-law (generalized Regge) behavior. We also give precise expressions for the asymptotics, which reveal another example of the recently observed property of factorization away from poles. We derive new recursion relations for the amplitude, which when repeatedly applied reduce to infinite series representations with a wider domain of convergence than the usual integral representations. For the five-point case, we present a new closed-form expression for the amplitude that for the first time gives its analytic continuation to all of kinematic space. We also discuss novel relations between amplitudes at different kinematic points following from the recently observed “split” factorizations.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Improving the five-point bootstrap
We present a new algorithm for the numerical evaluation of five-point conformal blocks in d-dimensions, greatly improving the efficiency of their computation. To do this we use an appropriate ansatz for the blocks as a series expansion in radial coordinates, derive a set of recursion relations for the unknown coefficients in the ansatz, and evaluate the series using a Padé approximant to accelerate its convergence. We then study the 〈σσϵσσ〉 correlator in the 3d critical Ising model by truncating the operator product expansion (OPE) and only including operators with conformal dimension below a cutoff ∆ ⩽ ∆cutoff. We approximate the contributions of the operators above the cutoff by the corresponding contributions in a suitable disconnected five-point correlator. Using this approach, we compute a number of OPE coefficients with greater accuracy than previous methods.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Jet classification using high-level features from anatomy of top jets
Recent advancements in deep learning models have significantly enhanced jet classification performance by analyzing low-level features (LLFs). However, this approach often leads to less interpretable models, emphasizing the need to understand the decision-making process and to identify the high-level features (HLFs) crucial for explaining jet classification. To address this, we consider the top jet tagging problems and introduce an analysis model (AM) that analyzes selected HLFs designed to capture important features of top jets. Our AM mainly consists of the following three modules: a relation network analyzing two-point energy correlations, mathematical morphology and Minkowski functionals for generalizing jet constituent multiplicities, and a recursive neural network analyzing subjet constituent multiplicity to enhance sensitivity to subjet color charges. We demonstrate that our AM achieves performance comparable to the Particle Transformer (ParT) while requiring fewer computational resources in a comparison of top jet tagging using jets simulated at the hadronic calorimeter angular resolution scale. Furthermore, as a more constrained architecture than ParT, the AM exhibits smaller training uncertainties because of the bias-variance tradeoff. We also compare the information content of AM and ParT by decorrelating the features already learned by AM. Lastly, we briefly comment on the results of AM with finer angular resolution inputs.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
Integrating the full four-loop negative geometries and all-loop ladder-type negative geometries in ABJM theory
The decomposition of the four-point ABJM amplituhedron into negative geometries produces compact integrands of logarithmic of amplitudes such that the infrared divergence only comes from the last loop integration, from which we can compute the cusp anomalous dimension of the ABJM theory. In this note, we integrate L – 1 loop momenta of the L-loop negative geometries for all four-loop negative geometries and a special class of all-loop ladder-type negative geometries by a method based on Mellin transformation, and from these finite quantities we extract the corresponding contribution to the cusp anomalous dimension. We find that the infrared divergence of a box-type negative geometry at L = 4 is weaker than other negative geometries, then only tree-type negative geometries contribute to the cusp anomalous dimension at L = 4. For the all-loop ladder-type negative geometries, we prove and conjecture some recursive structures as integral equations in Mellin space and find that they cannot contribute zeta values like ζ 3 , ζ 5 to the cusp anomalous dimension.
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
ZERNIPAX: A fast and accurate Zernike polynomial calculator in Python
Zernike polynomials serve as an orthogonal basis on the unit disc, and have proven to be effective in optics simulations, astrophysics, and more recently in plasma simulations. Unlike Bessel functions, Zernike polynomials are inherently finite and smooth at the disc center (r=0), ensuring continuous differentiability along the axis. This property makes them particularly suitable for simulations, requiring no additional handling at the origin. We developed ZERNIPAX, an open-source Python package capable of utilizing CPU/GPUs, leveraging Google's JAX package and available on GitHub as well as the Python software repository PyPI. Furthermore, our implementation of the recursion relation between Jacobi polynomials significantly improves computation time compared to alternative methods by use of parallel computing while still performing more accurately for high-mode numbers.
Astrophysics↗
On the computation of moments in the Super-Transition-Arrays model for radiative opacity calculations
In the Super-Transition-Array statistical method for the computation of radiative opacity of hot dense matter, the moments of the absorption or emission features involve partition functions with reduced degeneracies, occurring through the calculation of averages of products of subshell populations. Here, in the present work, we discuss several aspects of the computation of such peculiar partition functions, insisting on the precautions that must be taken in order to avoid numerical difficulties. In a previous work, we derived a formula for supershell partition functions, which takes the form of a functional of the distribution of energies within the supershell and allows for fast and accurate computations, truncating the number of terms in the expansion. The latter involves coefficients for which we obtained a recursion relation and an explicit formula. We show that such an expansion can be combined with the recurrence relation for shifted partition functions. We also propose, neglecting the effect of fine structure as a first step, a positive-definite formula for the Super-Transition-Array moments of any order, providing an insight into the asymmetry and sharpness of the latter. The corresponding formulas are free of alternating sums. Several ways to speed up the calculations are also presented.
74 ATOMIC AND MOLECULAR PHYSICS↗
Permutation-adapted complete and independent basis for atomic cluster expansion descriptors
Atomic cluster expansion (ACE) methods provide a systematic way to describe particle local environments of arbitrary body order. For practical applications it is often required that the basis of cluster functions be symmetrized with respect to rotations and permutations. Existing methodologies yield sets of symmetrized functions that are over-complete. These methodologies thus require an additional numerical procedure, such as singular value decomposition (SVD), to eliminate redundant functions. In this work, it is shown that analytical linear relationships for subsets of cluster functions may be derived using recursion and permutation properties of generalized Wigner symbols. From these relationships, subsets (blocks) of cluster functions can be selected such that, within each block, functions are guaranteed to be linearly independent. It is conjectured that this block-wise independent set of permutation-adapted rotation and permutation invariant (PA-RPI) functions forms a complete, independent basis for ACE. Along with the first analytical proofs of block-wise linear dependence of ACE cluster functions and other theoretical arguments, numerical results are offered to demonstrate this. The utility of the method is demonstrated in the development of an ACE interatomic potential for tantalum. Using the new basis functions in combination with Bayesian compressive sensing sparse regression, some high degree descriptors are observed to persist and help achieve high-accuracy models.
Angular momentum↗
Multi-frequency progressive refinement for learned inverse scattering
Interpreting scattered acoustic and electromagnetic wave patterns is a computational task that enables remote imaging in a number of important applications, including medical imaging, geophysical exploration, sonar and radar detection, and nondestructive testing of materials. However, accurately and stably recovering an inhomogeneous medium from far-field scattered wave measurements is a computationally difficult problem, due to the nonlinear and non-local nature of the forward scattering process. We design a neural network, called Multi-Frequency Inverse Scattering Network (MFISNet), and a training method to approximate the inverse map from far-field scattered wave measurements at multiple frequencies. We consider three variants of MFISNet, with the strongest performing variant inspired by the recursive linearization method — a commonly used technique for stably inverting scattered wavefield data — that progressively refines the estimate with higher frequency content. MFISNet outperforms past methods in regimes with high-contrast, heterogeneous large objects, and inhomogeneous unknown backgrounds.
97 MATHEMATICS AND COMPUTING↗
Joint state-parameter estimation for the reduced fracture model via the united filter
Here, in this paper, we introduce an effective United Filter method for jointly estimating the solution state and physical parameters in flow and transport problems within fractured porous media. Fluid flow and transport in fractured porous media are critical in subsurface hydrology, geophysics, and reservoir geomechanics. Reduced fracture models, which represent fractures as lower-dimensional interfaces, enable efficient multi-scale simulations. However, reduced fracture models also face accuracy challenges due to modeling errors and uncertainties in physical parameters such as permeability and fracture geometry. To address these challenges, we propose a United Filter method, which integrates the Ensemble Score Filter (EnSF) for state estimation with the Direct Filter for parameter estimation. EnSF, based on a score-based diffusion model framework, produces ensemble representations of the state distribution without deep learning. Meanwhile, the Direct Filter, a recursive Bayesian inference method, estimates parameters directly from state observations. The United Filter combines these methods iteratively: EnSF estimates are used to refine parameter values, which are then fed back to improve state estimation. Numerical experiments demonstrate that the United Filter method surpasses the state-of-the-art Augmented Ensemble Kalman Filter, delivering more accurate state and parameter estimation for reduced fracture models. This framework also provides a robust and efficient solution for PDE-constrained inverse problems with uncertainties and sparse observations.
Bayesian inference↗
A score-based diffusion model approach for adaptive learning of stochastic partial differential equation solutions
In this paper, we propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.
97 MATHEMATICS AND COMPUTING↗
Development and validation of a software for simulating γ-γ coincidence emission and detection probabilities
Gamma-gamma coincidence spectrometers have the potential to significantly enhance detection sensitivity for ultra-trace radionuclide measurements. The implementation of these spectrometers, however, is limited by the complexity of acquisition hardware, data processing and quantification. This work reports development of a novel radionuclide quantification software for γ-γ coincidence measurements. For any radionuclide, the software parses the Evaluated Nuclear Structure Data File (ENSDF) database, recursively simulating all possible γ-γ coincidence signatures and their respective emission and detection probabilities. Implemented using Python programming language, the software employs several strategies to boost overall computational performance. Since coincidence-based spectrometers are of notable interest in monitoring compliance for the Comprehensive Nuclear-Test-Ban Treaty (CTBT), the software’s execution was tested for 84 CTBT-relevant radionuclides. To date, the software has been experimentally validated for 15 radionuclides using the Advanced Radionuclide Gamma spectrOmeter (ARGO) at Pacific Northwest National Laboratory, USA (PNNL). Notably, the software can be operated in convergence mode, whereby coincidence detection efficiency’s convergence behavior can help avoid unreliable radionuclide activity estimates. With growing number of coincidence spectrometers worldwide, this paper aims to assist the radiation metrology community in developing similar software for their system.
46 INSTRUMENTATION RELATED TO NUCLEAR SCIENCE AND ↗
ν-point energy correletors with F AST EEC: Small-x physics from LHC jets
In recent years, energy correlators have emerged as a powerful tool for studying jet substructure, with promising applications such as probing the hadronization transition, analyzing the quark-gluon plasma, and improving the precision of top quark mass measurements. The projected N-point correlator measures correlations between N final-state particles by tracking the largest separation between them, showing a scaling behavior related to DGLAP splitting functions. These correlators can be analytically continued in N, commonly referred to as ν-correlators, allowing access to non-integer moments of the splitting functions. Of particular interest is the ν → 0 limit, where the small momentum fraction behavior of the splitting functions requires resummation. Originally, the computational complexity of evaluating ν-correlators for M particles scaled as 2 2M , making it impractical for real-world analyses. However, by using recursion, we reduce this to M 2M , and through the FastEEC method of dynamically resolving subjets, M is replaced by the number of subjets. This breakthrough enables, for the first time, the computation of ν-correlators for LHC data. In practice, limiting the number of subjets to 16 is sufficient to achieve percent-level precision, which we validate using known integer-ν results and convergence tests for non-integer ν. We have implemented this in an update to FastEEC and conducted an initial study of power-law scaling in the perturbative regime as a function of ν, using CMS Open Data on jets. The results agree with DGLAP evolution, except at small ν, where the anomalous dimension saturates to a value that matches the BFKL anomalous dimension.
Energy correlators↗
Efficient Mixed-Precision Matrix Factorization of the Inverse Overlap Matrix in Electronic Structure Calculations with AI-Hardware and GPUs
In recent years, a new kind of accelerated hardware has gained popularity in the artificial intelligence (AI) community which enables extremely high-performance tensor contractions in reduced precision for deep neural network calculations. In this article, we exploit Nvidia Tensor cores, a prototypical example of such AI-hardware, to develop a mixed precision approach for computing a dense matrix factorization of the inverse overlap matrix in electronic structure theory, S –1 . This factorization of S –1 , written as ZZT = S –1 , is used to transform the general matrix eigenvalue problem into a standard matrix eigenvalue problem. Here we present a mixed precision iterative refinement algorithm where Z is given recursively using matrix–matrix multiplications and can be computed with high performance on Tensor cores. To understand the performance and accuracy of Tensor cores, comparisons are made to GPU-only implementations in single and double precision. Additionally, we propose a nonparametric stopping criteria which is robust in the face of lower precision floating point operations. The algorithm is particularly useful when we have a good initial guess to Z, for example, from previous time steps in quantum-mechanical molecular dynamics simulations or from a previous iteration in a geometry optimization.
36 MATERIALS SCIENCE↗
Exact-Two-Component Complete Active Space Method with Variational Treatment of Magnetic Field and Spin–Orbit Coupling: Application to X-ray Magnetic Circular Dichroism Spectroscopy
We introduce an exact-two-component complete active space self-consistent-field (X2C-CASSCF) method formulated under the restricted-magnetic-balance condition. This framework allows for the nonperturbative treatment of static magnetic fields using gauge-including atomic orbitals (GIAOs). The GIAO-X2C-CASSCF methodology effectively captures all microstates within the same 2J + 1-degenerate manifold and their splitting in a static magnetic field, which are not accessible through single-reference-based methods. We also present mathematical recursive expressions for evaluating one-electron relativistic integrals by using GIAOs in the presence of a finite magnetic field. Benchmark studies include oxygen and nitrogen K-edge X-ray magnetic circular dichroism spectroscopy (XMCD) for closed-shell organic compounds, as well as L-edge XMCD spectroscopy for the high-spin open-shell transition metal ion Mn 2+ and the tetrahedral Mn(II)O 4 6– complex.
Chemical calculations↗