Abstract In angles-only initial relative orbit determination for non-cooperative targets, many existing methods require multiple angle measurements over a long time span or rely on iterative optimization. As a result, it is difficult to achieve fast initialization from a small number of measurements at the beginning of on-orbit navigation. To address this issue, this paper proposes a non-iterative initial relative orbit determination algorithm in a spherical coordinate framework. First, the relative motion equations are derived in detail in a spherical coordinate system centered on the line-of-sight direction, including the exact nonlinear two-body model and a linear time-invariant model applicable to near-circular orbits. Second, a non-iterative estimation algorithm that requires no prior state information is developed. The algorithm only needs to perform singular value decomposition on a single $6 \times 6$ matrix and solve a univariate polynomial. In the absence of camera offsets or orbital maneuvers, the initial relative state of the target can be obtained rapidly using only four angle measurements. Simulation results show that, compared with the classical iterative method, the proposed algorithm significantly improves computational efficiency while maintaining comparable or even superior estimation accuracy. It also exhibits excellent numerical stability and noise robustness, making it suitable for fast initialization in angles-only relative navigation of non-cooperative targets.


Computing matrix gradient has become a key aspect in modern signal processing/machine learning, with the recent use of matrix neural networks requiring matrix backpropagation. In this field, two main methods exist to calculate the gradient of matrix functions for symmetric positive definite (SPD) matrices, namely, the Daleckiǐ-Kreǐn/Bhatia formula and the Ionescu method. However, there appear to be a few errors. This brief aims to demonstrate each of these formulas in a self-contained and unified framework, to prove theoretically their equivalence, and to clarify inaccurate results of the literature. A numerical comparison of both methods is also provided in terms of computational speed and numerical stability to show the superiority of the Daleckiǐ-Kreǐn/Bhatia approach. We also extend the matrix gradient to the general case of diagonalizable matrices. Convincing results with the two backpropagation methods are shown on the EEG-based BCI competition dataset with the implementation of an SPDNet, yielding around 80% accuracy for one subject. Daleckiǐ-Kreǐn/Bhatia formula achieves an 8% time gain during training and handles degenerate cases.

Abstract. Precise, standardized assessment of spatial resolution is essential for optical remote sensing. We present an adaptable Siemens-star methodology aligned with DIN 18740-8 that enables reliable, reproducible modulation transfer function (MTF) measurements across simulation, laboratory, and aerial (UAV) scenarios. The target geometry is parameterized by star frequency and ground sampling distance (GSD), providing a simple sizing rule for practical deployment. Design variants include multiple segment counts (8–48 Hz) and four gray-level contrasts to mitigate overexposure, a frequent field issue that biases MTF. A phase-based center-detection algorithm — applicable to square- and sinusoidal-type stars — ensures robust centering, a prerequisite for valid MTF estimation. Results show highly consistent MTFs across all configurations: average deviations of 0.007, 0.006, and 0.011 line/pixel for simulation, DSLR, and UAV data, respectively. Center localization exhibits an average two-dimensional error of 0.074 pixels over twenty target variants, confirming numerical stability and implementation accuracy. The scalable vector graphics (SVG) design ensures pixel-exact reproducibility and straightforward scaling to different GSDs while offering practical advantages in production cost, transport, and deployment. Overall, the approach establishes Siemens-star targets as a cost-effective and field-ready alternative to conventional methods, providing standardized procedures, robustness to non-ideal imaging conditions, and direct applicability to both RGB and monochrome systems in operational photogrammetry and remote sensing.SVG-designs of all Siemens stars as well as the ”Resolving-Power” tool to perform the measurements can be downloaded from https://macs.dlr.de/box/resolvingpower.html.

Abstract Thermodynamic inconsistencies, particularly the prominent spurious currents near three-phase contact lines, present a major obstacle in pseudopotential lattice Boltzmann (LB) simulations of contact angles. Previous solutions have predominantly focused on equations of state (EOSs) and forcing schemes (FSs). In this work, a hierarchical force computation architecture (HFCA) is established through a comprehensive assessment of EOSs, interparticle interaction force terms (IFTs), and FSs, aiming to suppress spurious currents at three-phase contact lines in lattice Boltzmann simulations. The results demonstrate that the Peng-Robinson EOS shows excellent numerical stability at high density ratios, yet it leads to elevated spurious currents when used alone. More significantly, Gong’s term, when incorporated into the HFCA, ensures universal stability, full compatibility with all EOSs and FSs, and systematically reduces the maximum spurious currents through hierarchical coordinated optimization. Additionally, the relaxation time τ is identified as a sensitive system-level parameter whose optimal value depends on the configuration established by the higher layers of the HFCA. By leveraging the synergistic effects within the HFCA, specifically by optimizing τ in a configured framework, spurious currents can be significantly reduced. Evidently, this approach is more effective than optimizations based solely on schemes and EOSs. Based on these findings, specific combinations of τ, forcing terms, schemes, and EOSs are recommended to minimize spurious currents in LB simulations.

This work presents an information system designed to determine the mechanical characteristics of elastomeric materials through the integration of artificial neural networks with classical numerical analysis methods. A hybrid framework is proposed in which a neural network module approximate the experimental loading-unloading curve, including the Mullins effect as a key manifestation of material hysteresis. The predicted stress-strain response is incorporated into an iterative Newton-Raphson scheme to compute the tangent modulus of elasticity in the absence of analytical derivatives. Numerical stability is ensured by implementing an adaptive step-control mechanism, constraints on deformation corrections, and a local recovery procedure that prevents the propagation of divergence throughout the global solution. The system architecture is implemented as a cross-platform desktop application based on PySide6, combining interactive editing of the finite-element model structure, dynamic data validation, and maintenance of topological consistency through the delegation pattern. The model supports serialization and deserialization in CSV format, including field validation, automatic type conversion, and restoration of empty entries, thereby enabling reproducible calculations and reliable model exchange between users. The outputs of the system include both a graphical representation of the deformed structural configuration and a detailed analytical report containing nodal displacements and reactions, internal forces, strain measures, and resulting effective elastic moduli. The proposed system provides high computational stability and demonstrates practical applicability for modelling elastomers with pronounced nonlinear behaviour. The results highlight the potential of combining machine learning with classical mechanical techniques to advance modern tools for the engineering analysis.

ABSTRACT This paper introduces a novel method for numerically stabilizing sequential continuous adjoint flow solvers utilizing an elliptic relaxation strategy. Unlike previous stabilization approaches, the proposed approach is formulated as a Partial Differential Equation (PDE) containing a single user‐defined parameter, which analytical investigations reveal to represent the filter width of a probabilistic density function or Gaussian kernel. Key properties of the approach include smoothing features with redistribution capabilities while preserving integral properties. The technique targets explicit adjoint cross‐coupling terms, such as the Adjoint Transpose Convection (ATC) term, which frequently causes numerical instabilities, especially on unstructured grids common in industrial applications. A trade‐off is made by sacrificing sensitivity consistency to achieve enhanced numerical robustness. The method is validated on a two‐phase, laminar, two‐dimensional cylinder flow test case at a Reynolds number of and Froude number of , focusing on minimizing resistance or maximizing lift. A range of homogeneous and inhomogeneous filter widths is evaluated. Subsequently, the relaxation method is employed to stabilize adjoint simulations during shape optimizations that aim at drag reduction of ship hulls. Two case studies are considered: A model‐scale bulk carrier traveling at and as well as a harbor ferry cruising at and in full‐scale conditions. Both cases, characterized by unstructured grids prone to adjoint divergence, demonstrate the effectiveness of the proposed method in overcoming stability challenges. The resulting optimizations achieve superior outcomes compared to approaches that omit problematic coupling terms, yielding stable and adjoint solutions of improved consistency even for complex, unstructured, two‐phase flow configurations. This demonstrates that the proposed elliptic relaxation strategy provides a practical and broadly applicable means to enhance the numerical robustness of segregated continuous adjoint solvers in industrial CFD environments.

ABSTRACT Most spin density functionals are collinear, assuming the spin magnetization has only one nonzero component. However, a fully defined functional should be noncollinear, treating all three components of the spin magnetization vector as variables. The multicollinear approach is introduced to bridge this gap by generalizing an arbitrary collinear functional to a noncollinear one. In contrast to the traditional scheme, which adopts the local projection of the spin magnetization vector field, the multicollinear method employs a global projection scheme. It offers several key advantages, including recovering the collinear limit, ensuring global spin rotational invariance, maintaining numerical stability, and providing nonzero local torque. Its broad applicability spans relativistic and nonrelativistic cases, molecular and periodic systems, ground and excited states, as well as static and dynamic simulations. Furthermore, for collinear systems, it provides capabilities that go beyond standard collinear functionals by establishing a rigorous framework for spin‐flip TDDFT. This makes it a powerful tool for treating challenging problems such as double excitations, conical intersections, bond dissociation, and diradicals. Overall, the multicollinear approach provides a unified and versatile framework for quantum chemistry. This article is categorized under: Electronic Structure Theory > Density Functional Theory

This study presents a numerical analysis of airfoil aerodynamic characteristics at low angles of attack using a structured multizone meshing approach. The computational model was developed in ANSYS Workbench, with simulations conducted using ANSYS Fluent on a two-dimensional airfoil enclosed within a far-field domain. The mesh configuration consists of approximately 540,000 elements and 541,000 nodes, achieving a maximum skewness below 0.26, which indicates high mesh quality and numerical stability. Steady-state simulations were performed for angles of attack of −5°, 0°, 5°, and 10° to evaluate lift and drag behaviour, as well as pressure and velocity distributions around the airfoil surface. The numerical results show a consistent increase in lift coefficient with increasing angle of attack, accompanied by a corresponding rise in drag coefficient. At moderate angles of attack, particularly around 5°, the airfoil demonstrates an optimal aerodynamic performance with a favourable lift-to-drag ratio. These findings highlight the capability of structured multizone meshing to accurately capture key aerodynamic trends while maintaining computational efficiency. The results confirm that this meshing strategy is suitable for preliminary aerodynamic analysis and early-stage design of airfoil-based applications, such as small-scale wind turbine blades operating under low to moderate inflow conditions

Background Traditional fuzzy regression approaches, such as Tanaka’s fuzzy minimum method and fuzzy least squares, often lack theoretical guarantees of existence, uniqueness, and numerical stability. These limitations are of paramount importance in engineering applications involving uncertainty, such as predicting the compressive strength of concrete. This study addresses these issues by presenting a mathematically rigorous fuzzy regression model, based on fixed-point theory, formulated within the full metric space of trapezoidal fuzzy numbers using the metric scale d ∞ . Methodology We define a shrinkage coefficient on the trapezoidal fuzzy coefficient vector space and prove, using Banach’s fixed-point theory, the existence and uniqueness of the regression solution. An iterative algorithm is constructed to estimate the coefficients, using alpha computation and Lipschitz continuity to ensure convergence. The University of California, Irvine concrete compressive strength dataset was amplified using ASTM and ACI-based uncertainty coefficients, and the proposed fixed-point model was evaluated against the Tanaka method and least-squares fuzzy regression. Performance was assessed by mean squared error (MSE), coefficient ambiguity, convergence behavior, and toughness under ±5% noise. Results The proposed method demonstrated consistent geometric convergence with an average of 12.3 iterations and zero divergence across all experiments. It also reduced the overall mean squared error by an average of 12.5%, and by up to 25.1% at best, compared to the comparator methods. Coefficient ambiguity—measured by ambiguity width—was reduced by 18.3% compared to the Tanaka method and by 13.3% compared to ambiguity squares. Under noise perturbation, the model exhibits a significantly smaller increase in the mean error (+6.2%) compared to the Tanaka approach (+24.7%) and the LS-based approach (+18.3%), indicating a substantial improvement in robustness. Conclusions Incorporating fuzzy regression within a fixed-point theoretical framework helps resolve the stability, existence, and singularity challenges that have long plagued classical fuzzy regression models. The proposed approach offers a mathematically consistent, computationally stable, uncertainty-aware regression tool suitable for engineering applications involving imprecise measurements. Future work includes extending the model to nonlinear fuzzy structures, Gaussian/LR fuzzy representations, and broader applications in data-driven prediction under uncertainty.

In this paper, we present a new analytical approximation for the mathematical constant π, derived from an unconventional relationship involving the sixth root of 31 and a rapidly convergent factorial-based infinite series. Starting from the equation ⁶(31)^(1/6) = √π + ∑(k = 1 to ∞) [ k / (8k)! ] − 1/9801, we algebraically isolate √π and obtain an explicit symbolic expression for π. The proposed formulation combines radical expressions with a highly convergent factorial series, ensuring fast numerical stability and computational efficiency. A step-by-step derivation is provided, followed by numerical evaluation demonstrating that the resulting approximation yields π ≈ 3.14187, which is remarkably close to the accepted value of π ≈ 3.14159. The small deviation arises primarily from truncation of the infinite series, indicating that higher-order terms can further enhance accuracy. This work highlights a novel pathway for π-approximation, distinct from classical geometric, trigonometric, and Ramanujan-type series, and contributes to ongoing research in number theory and mathematical constants. The proposed method enriches the landscape of π-approximations and opens new directions for exploring factorial-series structures linked with radical expressions.

In recent decades, ray tracing methods in gradient-index (GRIN) lenses have gained significant attention due to their diverse applications. However, most of these methods have been developed for idealized lenses—namely, normalized, symmetric, and centered configurations in which the isoindicial surfaces (surfaces of constant refractive index) are concentric. This limits their applicability to more complex lens geometries. In this work, we present a ray tracing method suitable for both idealized and non-idealized GRIN lenses, which relies on a fixed optical path length step size, thereby eliminating the need to discretize the medium in advance. Its simple implementation and versatility make it a valuable tool for modeling light propagation in arbitrary GRIN media, offering strong numerical stability and high accuracy.

This paper presents a numerical approach for solving Fredholm integral equations of the first kind using the Bubnov–Galerkin method with Alpert wavelet bases. These equations are well-known for being ill-posed, meaning that small changes in input data can lead to large deviations in the solution. Therefore, robust and accurate numerical methods are essential. The proposed method utilizes orthonormal and compactly supported Alpert wavelets, which offer excellent localization properties and yield well-conditioned, sparse system matrices when projecting the integral operator. This enhances numerical stability and reduces computational complexity. A series of computational experiments was carried out using various refinement levels and polynomial degrees. The accuracy of the method was evaluated by comparing approximate solutions to the exact analytical solution. The results demonstrate exceptionally small absolute errors, often approaching machine precision. Additionally, a comparative analysis with power polynomial bases confirms the superiority of the Alpert wavelet approach in terms of convergence and approximation quality. Overall, the method proves to be efficient, stable, and suitable for further extension to more complex integral equations, including multidimensional and noisy-data problems. This confirms the potential of Alpert wavelet-based Galerkin schemes as a reliable tool for the numerical treatment of inverse and ill-posed problems in applied sciences.

Abstract Accurate spectral analysis of high-energy astrophysical sources often relies on comparing observed data to incident spectral models convolved with the instrument response. However, for gamma-ray bursts and other high-energy transient events observed at high count rates, significant distortions (e.g., pile-up, dead time, and large signal trailing) are introduced, complicating this analysis. We present a method framework to address the model dependence problem, especially to solve the problem of energy spectrum distortion caused by instrument signal pile-up due to high counting rate and high-rate effects, applicable to X-ray, gamma-ray, and particle detectors. Our approach combines physics-based Monte Carlo (MC) simulations with a model-independent spectral inversion technique. The MC simulations quantify instrumental effects and enable correction of the distorted spectrum. Subsequently, the inversion step reconstructs the incident spectrum using an inverse response matrix approach, conceptually equivalent to deconvolving the detector response. The inversion employs a convolutional neural network, selected for its numerical stability and effective handling of complex detector responses. Validation using simulations across diverse input spectra demonstrates high fidelity. Specifically, for 27 different parameter sets of the brightest gamma-ray bursts, goodness-of-fit tests confirm the reconstructed spectra are in excellent statistical agreement with the input spectra and residuals are typically within ±2 σ . This method enables precise analysis of intense transients and other high-flux events, overcoming limitations imposed by instrumental effects in traditional analyses.

We introduce a family of all-electron Gaussian basis sets, augmented MOLOPT, optimized for excited-state calculations on large molecules. We generate these basis sets by augmenting existing STO-3G, STO-6G, and MOLOPT basis sets optimized for ground state energy calculations. The augmented MOLOPT basis sets achieve fast convergence of GW gaps and Bethe-Salpeter excitation energies, while maintaining low condition numbers of the overlap matrix to ensure numerical stability. For GW HOMO-LUMO gaps, the double-ζ augmented MOLOPT basis yields a mean absolute deviation of 60 meV to the complete basis set limit. The basis set convergence for excitation energies from time-dependent density functional theory and the Bethe-Salpeter equation is similar. We use our smallest generated augmented MOLOPT basis (aug-SZV-MOLOPT-ae-mini) to demonstrate GW calculations on nanographenes with 9224 atoms requiring only 34300 core hours of computational resources.

Proton exchange membrane fuel cells (PEMFCs) have emerged as a promising technology due to their high efficiency, adaptability, and potential for integration into various applications, ranging from portable devices to large-scale power grids. A critical aspect of PEMFC research is the accurate modeling of its electrical characteristics. While traditional modeling approaches focus on the voltage-current relationship, there is a growing need to develop models that define current as a function of output voltage, particularly for control system applications. This study introduces a novel approach to PEMFC modeling using the g-function, a logarithmic transformation of the Lambert W function, which has been successfully applied in solar cell modeling. The g-function overcomes numerical limitations associated with the Lambert W function, ensuring greater stability and accuracy in computational applications. Additionally, this research proposes the self-adaptive differential evolution (SaDE) algorithm, a new metaheuristic optimization technique for estimating PEMFC parameters, addressing the need for robust and efficient parameter determination. To validate the proposed approach, a comprehensive comparative analysis is conducted against existing modeling and parameter estimation methods. Furthermore, sensitivity analysis is performed to assess the impact of parameter variations on model performance. The comparative evaluation across three different PEMFC systems (Ballard Mark V, BCS 500, and NedStack PS6) demonstrates consistent improvements, with RMSE reductions of up to 6.65% and SSE gains as high as 12.87%. These quantitative results highlight not only the enhanced accuracy but also the robustness and transferability of the proposed methodology across different fuel cell types. The results demonstrate that the proposed methodology offers improved numerical stability, enhanced accuracy, and efficient parameter estimation compared to conventional approaches. This study contributes to advancing PEMFC modeling techniques, providing a reliable framework for optimizing fuel cell performance and supporting their integration into sustainable energy systems. Overall, this study contributes a validated and versatile framework for advancing PEMFC modeling and optimization, supporting their integration into sustainable and real-world energy systems.

Continued fractions and their generalization, branched continued fractions, are the effective tools used to study special functions. In this aspect, an important problem of continued fractions and branched continued fractions is the study of their numerical stability. The backward recurrence algorithm is one of the main tools for computing approximants of both continued fraction and branched continued fractions. Like most recursive processes, it is prone to error growth. Each cycle of the recursive process not only generates its own rounding errors but also inherits the rounding errors made in all the previous cycles. This paper considers numerical stability of branched continued fraction expansion of the one ratio of Horn's hypergeometric functions $H_4$ in the special case, namely, $H_4(a,d+1;c,d;\mathbf{z})/H_4(a,d+2;c,d+1;\mathbf{z}).$ For this purpose, the backward recurrence algorithm is investigated. It is proven that under certain conditions on the parameters $a,$ $c,$ and $d$ the some open bi-disc is the set of numerical stability for branched continued fraction expansion, and it is found the estimate of relative rounding error, produced by the backward recurrence algorithm in calculating an $n$th approximant of this expansion. The results of this paper provide a toolkit for analyzing the numerical stability of algorithms that use branched continued fractions of the studied structure. Error estimates can be used to choose computation parameters, control accuracy, and ensure the reliability of results in applied problems that will use the aforementioned branched continued fractions.

Spectral flow cytometry has evolved from a contentious idea into a mainstay of high-parameter single-cell analysis, yet its vocabulary (and the statistical reasoning behind it) remains a patchwork of overlapping, sometimes contradictory terms. This position paper highlights the terminological fragmentation and translation gap, and aims to harmonize the field's lexicon while providing a cohesive information-theoretic framework for panel optimization. We expose the limits of popular ad hoc heuristics, matrix condition numbers, and pairwise cosine similarities, and promote stronger surrogates: effective rank and information efficiency, which together capture spectral independence and numerical stability, and the Cramér-Rao lower bound (CRLB) matrix, which directly predicts fluorochrome-specific spreading error. Building on this foundation, we revive the optimal-design criteria: D-optimality maximizes total information; A-optimality minimizes the average parameter variance; E-optimality constrains worst-direction inflation. By aligning precise definitions with actionable design rules, we provide a roadmap for consistent terminology, next-generation panel construction, and objective instrument benchmarking.

ABSTRACT Background Quantum federated learning presents a promising paradigm for privacy‐preserving collaborative training across distributed quantum devices. However, its scalability is hindered by the significant communication overhead associated with transmitting high‐dimensional, high‐precision quantum model parameters over classical networks. Methods To address this bottleneck, this paper proposes QFI‐Opt (Quantum Fisher Information‐guided Adaptive Optimization), a quantum adaptive communication optimization framework based on Quantum Fisher Information. QFI‐Opt establishes a “sensing‐compression‐regulation” pipeline that achieves communication efficiency while preserving quantum model fidelity. The framework uses QFI as a physically interpretable metric to dynamically assess quantum state sensitivity to parameter perturbations, enabling a progressive pruning strategy that removes low‐sensitivity parameters during training while retaining critical quantum features. Additionally, a dynamic bit‐width quantization mechanism adapts precision based on parameter importance, maximizing compression without compromising numerical stability. This is further complemented by a physics‐aware aggregation method that weighs client updates based on both local data volume and quantum information quality derived from QFI scores, improving global model robustness. Results Extensive evaluation on quantum convolutional neural networks demonstrates that QFI‐Opt significantly reduces per‐round communication overhead compared to baseline methods. Conclusions Simultaneously, the proposed framework maintains competitive model accuracy and convergence performance across diverse quantum architectures.

High-dimensional medical datasets pose a persistent challenge for artificial intelligence because traditional classification algorithms often incur escalating computational costs and reduced predictive accuracy. As healthcare systems generate increasingly complex clinical records, imaging outputs, and genomic profiles, scalable analytic methods that balance precision and efficiency are critical. This study proposes a Quantum-Inspired Optimization (QIO) framework for efficient and accurate classification of high-dimensional healthcare data. Leveraging the exploratory power of variational quantum algorithms, specifically techniques analogous to the Quantum Approximate Optimization Algorithm, the framework integrates quantum-style search strategies with classical computation to achieve global optimization and numerical stability. Publicly available medical datasets with hundreds of features were used to evaluate the approach. Classification models were trained and tested across varying feature dimensionalities, and performance was assessed using accuracy, runtime, and scalability metrics. Empirical results demonstrate that QIO achieves up to 95.4% classification accuracy and reduces computational time by 40% compared with state-of-the-art classical baselines. The method demonstrates stable convergence and clear decision boundaries even as feature dimensionality grows, highlighting its resilience to the curse of dimensionality. These results indicate that QIO can enable fast and reliable healthcare analytics in data-rich clinical environments. Future research may examine domain-specific adaptations, real-time deployment, and integration with emerging quantum hardware to enhance the impact of quantum-inspired artificial intelligence further.

ABSTRACT Interval-valued time series (ITS), frequently encountered in financial applications, often arise from cross-sectional aggregation (e.g. daily highest and lowest returns across a portfolio) or temporal aggregation (e.g. intraday highs and lows). Despite the extreme-value nature of ITS components, existing models seldom exploit extreme value theory (EVT). This article introduces a new EVT-based framework for modelling ITS using the Fréchet distribution, a heavy-tailed case of the generalized extreme value (GEV) family well suited to financial extremes. Extending the autoregressive conditional Fréchet (AcF) model of Zhao et al. (2018), we propose a novel specification in which the range, the difference between the upper and lower bounds, drives the scale and tail parameters of the Fréchet distribution. This structure incorporates information from both bounds, improves numerical stability, and captures the empirical link between volatility clustering and the clustering of extreme events. The model is estimated via maximum likelihood, and simulations confirm strong finite-sample performance. Empirical studies using daily returns of DJI30 constituents and 5-minute S&P 500 returns show that the proposed model achieves superior in-sample fit and out-of-sample predictive accuracy compared with competing approaches. We also identify significant tail connectedness between interval bounds. A trading application demonstrates model’s practical value.