arXiv subjects

S. M. Mallikarjunaiah

Publications and source records attributed to S. M. Mallikarjunaiah.

At least 19 recordsLinked to original sources

arXiv2026

Neural Network Solvers for Nonlinear ODEs: Benchmarking MLPs and B-Spline/Chebyshev KANs

Three neural networks are compared as solvers of nonlinear ordinary differential equations: a multilayer perceptron (MLP), a Kolmogorov-Arnold network (KAN) whose edges are B-splines, and a KAN whose edges are Chebyshev series. Each network is trained only on the residual of the equation and on its initial or boundary conditions. The exact solution, or a numerical solution accurate to $10^{-13}$, is used only afterward to measure the error. Five test problems of differential orders one to four are investigated: the logistic equation, a singular Lane-Emden type initial value problem, a singular boundary value problem, the Falkner-Skan equation with suction, a fourth-order problem. The same training budget and the same choice of five activation functions are allocated to all networks. No numerical failures are encountered in any of the \nRuns{} training runs. For every problem, the best KAN is found to be at least as accurate as the best MLP while \paramsKANmin{} to \paramsKANmax{} parameters are used instead of \paramsMLP{}; the margin ranges from a tie to a factor of three. In the Falkner-Skan setting of an earlier study, the B-spline KAN is shown to be 34 to 220 times more accurate at every published test point. Much smaller MLPs are also found to be competitive, whereas at orders three and four, failures are experienced more frequently by KANs, and a coarse basis is required. The derivatives in the equation are distorted by the usual grid extension of a B-spline KAN; consequently, it is replaced with knot insertion, by which the logistic error is lowered more than a hundredfold, though no improvement is provided at order four.

math.GM↗

arXiv2026

Space-Time Finite Element Approximation of Quasilinear Hyperbolic Equations Arising in Dynamic Strain-Limiting Elasticity

The numerical approximation of a class of quasilinear hyperbolic equations arising in dynamic strain-limiting elasticity-whose nonlinear constitutive law relates stress and strain nonlinearly-is investigated. Ellipticity may be lost by such problems in regions of large strain, leading to significant challenges in the design of stable and accurate numerical methods. To address these difficulties, a fully discrete continuous Galerkin finite element framework is developed, combining continuous linear finite elements for spatial discretization with the Hilber-Hughes-Taylor (HHT-alpha) time integration scheme. A lifting technique is incorporated within the proposed formulation to treat non-homogeneous Dirichlet boundary conditions, and a consistent Newton iteration is employed for the efficient solution of the resulting nonlinear systems. Robustness is enhanced by the algorithmic dissipation introduced through the HHT-alpha method by suppressing nonphysical high-frequency oscillations near degenerate regions, while consistency and accuracy are preserved. Under suitable structural assumptions on the nonlinear constitutive coefficient, the weak formulation is established in appropriate energy spaces. Theoretical second-order convergence in the L^2-norm and first-order convergence in the H^1-norm, rapid nonlinear convergence with nearly mesh-independent Newton iterations, physically consistent wave-speed evolution, and stable energy dissipation are demonstrated through comprehensive numerical experiments. Furthermore, it is confirmed by the results that an accurate, stable, and computationally efficient framework for simulating nonlinear strain-limiting wave propagation is provided, offering a solid foundation for future extensions to multidimensional nonlinear elastodynamics, adaptive finite element methods, and fracture and damage mechanics.

math.NA↗

arXiv2026

Uncertainty-Aware Crack Growth Forecasting via Conditional Denoising Diffusion Models for Phase-Field Fracture

The accurate prediction of brittle crack initiation, propagation, and complex topological evolution remains computationally prohibitive when utilizing traditional high-fidelity phase-field finite element methods. To overcome these computational bottlenecks, a physics-informed conditional Denoising Diffusion Probabilistic Model (DDPM) is proposed for the full-field spatiotemporal forecasting of fracture evolution across diverse loading regimes and energy decomposition methods. The generative architecture is conditioned on rolling historical damage states and explicitly derived kinematic proxies -- phase-field velocity and gradient magnitude -- ensuring temporal coherence without non-physical artifacts. The principal contribution is spatially-localized uncertainty quantification without modification to the trained model. Ensemble variance concentrates at crack branching junctions ($σ_\mathrm{max} = 0.222$ at Y-junction bifurcations; zero high-uncertainty pixels in four deterministic propagation cases), while the high-$σ$ tail identifies high-error predictions with 90\% precision -- an 18-fold improvement over random selection. One-step crack tip localization achieves sub-pixel accuracy (0.12 px mean error) across both held-out validation subsets (shear-star and tension-spect), confirming cross-regime generalization. In closed-loop autoregressive rollout over 50 steps, the DDPM maintains Dice = 0.929 $\pm$ 0.010 while a deterministic U-Net collapses to Dice = 0.423 under error accumulation, a 2.2$\times$ gap that establishes the value of stochastic re-sampling for long-horizon stability. Per-step inference requires approximately 3.6 s on an H100 GPU, approximately 28$\times$ faster than the FEM reference and 1{,}000$\times$ slower than a deterministic U-Net a cost that buys the stochastic diversity enabling uncertainty quantification.

math.GM↗

arXiv2026

CrackMorph-XAI-Net: A Topology-Preserving and Explainable Framework for Automated Crack Morphology

Automated crack inspection is increasingly recognized as a critical component of infrastructure monitoring; however, cracks continue to be reported primarily as binary segmentation masks by many current vision-based systems. While localization is facilitated by such masks, limited structural information is provided for robust engineering interpretation. For practical crack assessment, measurable morphological features -- including centerline geometry, branching behavior, junction locations, topology, and severity-related indicators -- are required. In this work, \textit{CrackMorph-XAI-Net}, an explainable morphology-aware framework for image-based crack analysis, is presented. Crack image and region-mask data are converted into a sequence of interpretable structural outputs through four distinct stages: topology-preserving skeleton extraction, junction detection via Gaussian heatmap regression, morphology descriptor computation, and severity-oriented screening. To support rigorous stage-wise evaluation, the standard \textit{CRACK500} benchmark is extended with aligned skeleton maps, junction heatmaps, and topology labels. Experimental validation demonstrates that a mean Dice coefficient of 0.991 is achieved by the learned skeleton extraction stage, with topology preserved in 98.5\% of test images. Furthermore, a recall of 0.964 and an F1-score of 0.887 are obtained in the junction detection stage, highlighting the efficacy of heatmap regression for sparse structural targets. Strong agreement between predicted and reference morphology values is revealed by descriptor-level evaluation, with correlations exceeding 0.95 for length, width, orientation, junction count, and tortuosity.

math.GM↗

arXiv2026

Computational modeling of crack-tip fields in transversely isotropic strain-limiting solids subjected to piecewise linear slope loads

Crack-tip fields within a transversely isotropic strain-limiting elastic body are investigated under the influence of piecewise linear slope boundary loads. The mechanical response is characterized via a nonlinear constitutive framework relating the Cauchy stress to the linearized strain, by which non-physical strain singularities at the crack tip are eliminated. The governing system is formulated as a quasi-linear elliptic boundary value problem in terms of the displacement field and is solved utilizing a continuous Galerkin finite element method coupled with a Picard linearization scheme. Boundary conditions are prescribed such that the vertical displacement varies piecewise linearly along the top and bottom edges, exhibiting opposite slopes on each half of the boundary. Numerical results are derived for two distinct fiber orientations. It is demonstrated that piecewise slope loads provide a flexible and realistic configuration for elucidating the interplay between strain-limiting behavior and crack-tip mechanics in transversely isotropic media.

math.GM↗

arXiv2026

Learning constitutive laws under explicit strain limits: An interpretable strain-limiting elasticity--Kolmogorov Arnold neural network framework

A physically consistent framework for modeling materials with saturating deformation, such as elastomers and biological tissues, is provided by strain-limiting elasticity. Fundamental limitations of classical elasticity are addressed through the enforcement of bounded strains; however, significant challenges for data-driven learning are posed by the strong nonlinearity of these laws. In this work, an interpretable hybrid constitutive modeling framework integrating strain-limiting elasticity (SLE) with Kolmogorov-Arnold Networks (KANs) is proposed to balance mechanical admissibility with data-driven flexibility. The dominant nonlinear response is captured by the SLE backbone, while smooth residual corrections are learned exclusively via a KAN. Essential mechanical principles-including symmetry, monotonicity, and bounded strain-are embedded directly into the model structure to ensure physical admissibility. The framework is assessed on synthetic benchmarks, where near-exact recovery is achieved in smooth regimes and consistency is retained under sharp transitions. Application to Treloar's rubber elasticity data demonstrates systematic improvement in stress-stretch agreement while preserving explicit strain limits. A regime-based analysis reveals a transparent trade-off between data fidelity and mechanical admissibility, demonstrating that deviations arise from deliberately imposed physical restrictions rather than unconstrained model expressivity. This SLE-KAN framework offers a robust, physics-consistent alternative to black-box neural networks for constitutive modeling.

math.GM↗

arXiv2026

An HHT-$α$-based finite element framework for wave propagation in constitutively nonlinear elastic materials

This paper presents a computational framework for modeling wave propagation in geometrically linear elastic materials characterized by algebraically nonlinear constitutive relations. We derive a specific form of the nonlinear wave equation in which the nonlinearity explicitly appears in the time-derivative terms that govern the evolution of the mechanical fields. The numerical solution is established using a fully discrete formulation that combines the standard finite element method for spatial discretization with the implicit Hilber-Hughes-Taylor (HHT)-$α$ scheme for time integration. To address the nonlinear nature of the discrete system, we employ Newton's method to iteratively solve the linearized equations at each time step. The accuracy and robustness of the proposed framework are rigorously verified through convergence analyses, which demonstrate optimal convergence rates in both space and time. Furthermore, a detailed parametric study is conducted to elucidate the influence of the model's constitutive parameters. The results reveal that the magnitude parameter of the stress-dependent variation in wave speed leads to wavefront steepening and the formation of shock discontinuities. Conversely, the exponent parameter acts as a nonlinearity filter; high values suppress nonlinear effects in small-strain regimes, whereas low values allow significant dispersive behavior. This work provides a validated tool for analyzing shock formation in advanced nonlinear materials.

math.NA↗

arXiv2025

An $hp$-adaptive finite element framework for static cracks: The impact of pointwise density variations on mode I, mode II, and mixed-mode fracture

A robust $hp$-adaptive finite element framework is presented for the investigation of static cracks in materials characterized by complex, pointwise density variations. Within such heterogeneous media, the equilibrium equation governed by the divergence of the stress tensor is reduced to a vector-valued quasilinear partial differential equation, wherein significant gradients and nonlinearities are introduced into the governing constitutive relations by spatial density variations. To ensure that these localized field variables are accurately captured, an automated $hp$-refinement strategy is implemented so that element sizes ($h$) and polynomial degrees ($p$) are concurrently optimized. A dual-indicator approach is employed by the framework. In this approach, $h$-refinement is driven by the Kelly error estimator, which utilizes the jump of the normal derivative across element interfaces, while $p$-refinement is determined by the decay of Legendre-Fourier coefficients to assess the local smoothness of the solution. Optimal convergence rates are ensured by this dual-refinement strategy, particularly within the singular regions surrounding the crack tip where material heterogeneities and geometric discontinuities intersect. The performance of the proposed method is rigorously evaluated across three distinct loading regimes: pure tensile (Mode I), pure shear (Mode II), and mixed-mode loading. It is demonstrated through numerical simulations that the intricate interaction between the fracture process zone and the underlying material heterogeneities is effectively resolved by the $hp$-adaptive scheme. Furthermore, the influence of specific density-dependent constitutive relations on the resulting crack-tip stress fields and strain energy density distributions is analyzed in detail.

math.NA↗

arXiv2025

Neural Networks as Physics-Consistent Surrogates: An \textit{Explainable AI} Validation Framework for Learning Constitutive Relations

This paper presents a Physics-\textit{Explainable AI} (XAI) framework to validate and interpret neural networks for the constitutive modeling of solid materials. The study bridges the gap between data-driven models and continuum mechanics by applying a suite of explainability methods to neural networks trained on three distinct material behaviors: hyperelasticity (\textit{Mooney-Rivlin}), elastoplasticity (\textit{Chaboche}), and viscoelasticity (\textit{Fractional Zener}). First, high-fidelity surrogate models, including dense feed-forward networks, LSTMs, and GRUs, are trained on synthetically generated data to accurately capture complex material responses. The core of the work then employs XAI techniques to "open the black box" and confirm that the networks learn physically meaningful principles. For hyperelasticity, gradient-based attributions (\textit{Grad Input} (GI)) successfully match the analytical tangent modulus, proving the network learned material stiffness. For elastoplasticity, \textit{SHapley Additive exPlanations} (SHAP) and \textit{Principal Component Analysis} (PCA) demonstrate the \textit{Recurrent Neural Network} (RNN) internalizes path-dependent memory, with SHAP identifying \textit{plastic strain} as the dominant feature governing the stress prediction. For viscoelasticity, latent-space and wavelet analyses of the \textit{Gated Recurrent Unit. } GRU layers reveal a clear temporal hierarchy, with different layers encoding instantaneous elastic response, intermediate relaxation, and long-term fractional memory. Ultimately, the study demonstrates that the XAI framework can verify that the neural networks are not merely curve-fitting but are, in fact, learning the underlying physical mechanisms of stiffness, history-dependence, and temporal damping.

cond-mat.mtrl-sci↗

arXiv2025

Convergence Analysis of Adaptive Finite Element Algorithms for a Regularized Variational Model of Quasi-Static Brittle Fracture in "Strain-Limiting" Elastic Solids

The rigorous convergence analysis of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in strain-limiting elastic solids is presented. This work introduces two novel adaptive mesh refinement algorithms, based on robust local error indicators, designed to solve the underlying energy minimization problem efficiently. A comprehensive convergence analysis is provided for minimizer sequences generated by these distinct adaptive strategies. It is rigorously demonstrated that sequences from the first algorithm converge to a prescribed tolerance. Notably, the second algorithm is proven to yield inherently convergent sequences without requiring an explicit stopping criterion. The practical efficacy of the proposed adaptive framework is validated through extensive numerical simulations, where critical comparisons of energy components (bulk, surface, and total) demonstrate the performance of the two adaptive algorithms in the case of an edge crack in a strain-limiting solid subjected to anti-plane shear-type loading.

math.NA↗

arXiv2025

Numerical Approximation of Electrohydrodynamics Model: A Comparative Study of PINNs and FEM

The accurate representation of numerous physical, chemical, and biological processes relies heavily on differential equations (DEs), particularly nonlinear differential equations (NDEs). While understanding these complex systems necessitates obtaining solutions to their governing equations, the derivation of precise approximations for NDEs remains a formidable task in computational mathematics. Although established techniques such as the finite element method (FEM) have long been foundational, remarkable promise for approximating continuous functions with high efficacy has recently been demonstrated by advancements in physics-informed deep-learning feedforward neural networks. In this work, a novel application of PINNs is presented for the approximation of the challenging Electrohydrodynamic (EHD) problem. A specific $L^2$-type \textit{total loss function} is employed, notably without reliance on any prior knowledge of the exact solution. A comprehensive comparative study is conducted, juxtaposing the approximation capabilities of the proposed neural network with those of the conventional FEM. The PINN training regimen is composed of two critical steps: forward propagation for adjustments to gradient and curvature, and backpropagation for the refinement of hyperparameters. The critical challenge of identifying optimal neural network architectures and hyperparameter configurations for efficient optimization is meticulously investigated. Excellent performance is shown to be delivered by the neural network even with a limited training dataset. Simultaneously, it is demonstrated that the accuracy of the FEM can be substantially enhanced through the judicious selection of smaller mesh sizes.

math.NA↗

arXiv2025

A finite element model for thermomechanical stress-strain fields in transversely isotropic strain-limiting materials

This paper presents a comprehensive computational framework for investigating thermo-elastic fracture in transversely isotropic materials, where classical linear elasticity fails to predict physically realistic behavior near stress concentrations. We address the challenge of unphysical strain singularities at crack tips by employing a strain-limiting theory of elasticity. This theory is characterized by an algebraically nonlinear constitutive relationship between stress and strain, which intrinsically enforces a limit on the norm of the strain tensor. This approach allows the development of very large stresses, as expected near a crack tip, while ensuring that the corresponding strains remain physically bounded. A loosely coupled system of linear and nonlinear partial differential equations governing the response of a thermo-mechanical transversely isotropic solid is formulated. We develop a robust numerical solution based on the finite element method, utilizing a conforming finite element discretization within a continuous Galerkin framework to solve the two-dimensional boundary value problem. The model is applied to analyze the stress and strain fields near an edge crack under severe thermo-mechanical loading. Our numerical results reveal a significant departure from classical predictions: while stress concentrates intensely at the crack tip, the strain grows at a substantially slower rate and remains bounded throughout the domain. This work validates the efficacy of the strain-limiting model in regularizing thermo-elastic crack-tip fields and establishes a reliable computational foundation for the predictive modeling of thermally driven crack initiation and evolution in advanced anisotropic materials.

math.NA↗

arXiv2025

Computational investigation of crack-tip fields in a compressed nonlinear strain-limiting material

A finite element framework is presented for the analysis of crack-tip phenomena in an elastic material containing a single edge crack under compressive loading. The mechanical response of the material is modeled by a nonlinear constitutive relationship that algebraically relates stress to linearized strain. This approach serves to mitigate non-physical strain singularities and ensures that the crack-tip strains don't grow, unlike singular stresses. A significant advancement is thus achieved in the formulation of boundary value problems (BVPs) for such complex scenarios. The governing equilibrium equation, derived from the balance of linear momentum and the nonlinear constitutive model, is formulated as a second-order, vector-valued, quasilinear elliptic BVP. A classical traction-free boundary condition is imposed on the crack face. The problem is solved using a robust numerical scheme in which a Picard-type linearization is combined with a continuous Galerkin finite element method for the discretization. Analyses are performed for both an isotropic and a transversely isotropic elastic solid containing a crack subjected to compressive loads. The primary crack-tip variables**-stress, strain, and strain energy density-**are examined in detail. It is demonstrated that while high concentrations of compressive stress and strain energy density are observed at the crack tip, the growth of strain is substantially lower than that of stress. These findings are shown to be consistent with the predictions of linear elastic fracture mechanics, but a more physically meaningful representation of the crack-tip field is provided by the nonlinear approach. A rigorous basis is thus established for investigating fundamental processes like crack propagation and damage in anisotropic, strain-limiting solids under various loading conditions, including compression.

math.NA↗

arXiv2025

$hp$-adaptive finite element simulation of a static anti-plane shear crack in a nonlinear strain-limiting elastic solid

An $hp$-adaptive continuous Galerkin finite element method is developed to analyze a static anti-plane shear crack embedded in a nonlinear, strain-limiting elastic body. The geometrically linear material is described by a constitutive law relating stress and strain that is algebraically nonlinear. In this investigation, the constitutive relation utilized is \textit{uniformly bounded}, \textit{monotone}, \textit{coercive}, and \textit{Lipschitz continuous}, ensuring the well-posedness of the mathematical model. The governing equation, derived from the balance of linear momentum coupled with the nonlinear constitutive relationship, is formulated as a second-order quasi-linear elliptic partial differential equation. For a body with an edge crack, this governing equation is augmented with a classical traction-free boundary condition on the crack faces. An $hp$-adaptive finite element scheme is proposed for the numerical approximation of the resulting boundary value problem. The adaptive strategy is driven by a dual-component error estimation scheme: mesh refinement ($h$-adaptivity) is guided by a residual-based a posteriori error indicator of the \textit{Kelly type}, while the local polynomial degree ($p$-adaptivity) is adjusted based on an estimator of the local solution regularity. The performance, accuracy, and convergence characteristics of the proposed method are demonstrated through numerical experiments. The structure of the regularized crack-tip fields is examined for various modeling parameters. Furthermore, the presented framework establishes a robust foundation for extension to more complex and computationally demanding problems, including quasi-static and dynamic crack propagation in brittle materials.

math.NA↗

arXiv2025

Crack-tip field characterization in nonlinearly constituted and geometrically linear elastoporous solid containing a star-shaped crack: A finite element study

This paper introduces a three-dimensional (3-D) mathematical and computational framework for the characterization of crack-tip fields in star-shaped cracks within porous elastic solids. A core emphasis of this model is its direct integration of density-dependent elastic moduli, offering a more physically realistic representation of engineering materials where intrinsic porosity and density profoundly influence mechanical behavior. The governing boundary value problem, formulated for the static equilibrium of a 3-D, homogeneous, and isotropic material, manifests as a system of second-order, quasilinear partial differential equations. This system is meticulously coupled with classical traction-free boundary conditions imposed at the complex crack surface. For the robust numerical solution of this intricate nonlinear problem, we employ a continuous trilinear Galerkin-type finite element discretization. The inherent strong nonlinearities arising within the discrete system are effectively managed through a powerful and stable {Picard-type linearization scheme}. The proposed model demonstrates a remarkable ability to accurately describe the full stress and strain states in a diverse range of materials, crucially recovering the well-established classical singularities observed in linearized elastic fracture mechanics. A comprehensive numerical examination of tensile stress, strain, and strain energy density fields consistently reveals that these quantities attain their peak values in the immediate vicinity of the crack tip, an observation that remarkably aligns with established findings in standard linearized elastic fracture mechanics.

math.NA↗

arXiv2025

Finite element modeling of V-notched thermoelastic strain-limiting solids containing inclusions

A precise domain triangulation is recognized as indispensable for the accurate numerical approximation of differential operators within collocation methods, leading to a substantial reduction in discretization errors. An efficient finite element method (FEM) is presented in this paper, meticulously developed to solve a complex mathematical model. This model governs the behavior of thermoelastic solids containing both a V-notch and inclusions. The system of partial differential equations underlying this model consists of two primary components: a linear elliptic equation, which is used to describe the temperature distribution, and a quasilinear equation, which governs the mechanical behavior of the body. Through the application of this specifically tailored FEM, accurate and efficient solutions are able to be obtained for these intricate thermoelastic problems. The algebraically nonlinear constitutive equation, alongside the balance of linear momentum, is effectively reduced to a second-order quasi-linear elliptic partial differential equation. Complex curved boundaries are represented through the application of a smooth, distinctive point transformation. Furthermore, higher-order shape functions are employed to ensure the accurate computation of entries within the FEM matrices and vectors, from which a highly precise approximate solution to the BVP is subsequently obtained. The inherent nonlinearities in the governing differential equation are addressed through the implementation of a Picard-type linearization scheme. Numerical results, derived from a series of test cases, have consistently demonstrated a significant enhancement in accuracy, a crucial achievement for the nuanced analysis of thermoelastic solids.

math.NA↗

arXiv2025

Analyzing mob dynamics in social media networks using epidemiology model

Epidemiological models, traditionally used to study disease spread, can effectively analyze mob behavior on social media by treating ideas, sentiments, or behaviors as ``contagions" that propagate through user networks. In this research, we introduced a mathematical model to analyze social behavior related to COVID-19 spread by examining Twitter activity from April 2020 to June 2020. Our analysis focused on key terms such as ``lockdown" and ``quarantine" to track public sentiment and engagement trends during the pandemic. The threshold number $\Re_{0}$ is derived, and the stability of the steady states is established. Numerical simulations and sensitivity analysis of applicable parameters are carried out. The results show that negative sentiment on Twitter has less influence on COVID-19 spread compared to positive sentiment. However, the effect of negative sentiment on the spread of COVID-19 remains remarkably strong. Moreover, we use the Caputo operator with different parameter values to study the impact of social media platforms on the transmission of COVID-19 diseases.

physics.soc-ph↗

arXiv2025

Adaptive finite element convergence analysis of AT1 phase-field model for quasi-static fracture in strain-limiting solids

This research rigorously investigates the convergence of adaptive finite element methods for regularized variational models of quasi-static brittle fracture in elastic solids. We specifically examine a novel Ambrosio-Tortorelli (AT1) phase-field model within the framework of elasticity theories, particularly for material models characterized by an algebraically nonlinear stress-strain relationship. Two distinct and novel adaptive mesh refinement algorithms, underpinned by robust local error indicators, were introduced to efficiently solve the underlying nonlinear energy minimization problem. A detailed convergence analysis was conducted on the sequences of minimizers produced by these strategies. Our findings rigorously demonstrate that the minimizer sequences from the first adaptive algorithm achieve convergence to a predefined tolerance. Crucially, the second algorithm is proven to generate inherently convergent sequences, thereby eliminating the need for an explicit stopping criterion. The practical effectiveness of this proposed adaptive framework is thoroughly validated through extensive numerical simulations. A case study involving an edge crack in an elastic body, governed by an algebraically nonlinear strain-limiting relationship and subjected to anti-plane shear-type loading, is presented. Critical comparisons of the energy components-bulk, surface, and total-showcase the superior performance of both adaptive algorithms.

math.NA↗