Methods For Differential Equations Research Articles

Multi‐indice B$B$‐series

AbstractWe propose a novel way to study numerical methods for ordinary differential equations in one dimension via the notion of multi‐indice. The main idea is to replace rooted trees in Butcher's ‐series by multi‐indices. The latter were introduced recently in the context of describing solutions of singular stochastic partial differential equations. The combinatorial shift away from rooted trees allows for a compressed description of numerical schemes. Furthermore, such multi‐indices ‐series uniquely characterize the Taylor expansion of 1‐dimensional local and affine equivariant maps.

Spinning waveforms in cubic effective field theories of gravity

We derive analytic all-order-in-spin expressions for the leading-order time-domain waveforms generated in the scattering of two Kerr black holes with arbitrary masses and spin vectors in the presence of all independent cubic deformations of Einstein-Hilbert gravity. These are the two parity-even interactions I1 and G3, and the parity-odd ones Ĩ1 and G~3. Our results are obtained using three independent methods: a particularly efficient direct integration and tensor reduction approach; integration by parts combined with the method of differential equations; and finally a residue computation. For the case of the G3 and G~3 deformations we can express the spinning waveform in terms of the scalar waveform with appropriately shifted impact parameters, which are reminiscent of Newman-Janis shifts. For I1 and Ĩ1 similar shifts occur, but are accompanied by additional contributions that cannot be captured by simply shifting the scalar I1 and Ĩ1 waveforms. We also show the absence of leading-order corrections to gravitational memory. Our analytic results are notably compact, and we compare the effectiveness of the three methods used to obtain them. We also briefly comment on the magnitude of the corrections to observables due to cubic deformations.

Artificial neural networks for classifying the productive capacity of a forest stand in the Jari region, western Pará

The growth and production of a forest stand depend on indicators such as the site's productive capacity, which can be defined as the site's potential to produce wood, or another product in each location, for a given species or clone. This study compared traditional methods and artificial neural networks in the classification of the productive capacity of sites in Eucalyptus urograndis plantations located in the Jari region, western Pará state. For this, 593 permanent circular plots of 500 m² each were randomized, where 38,072 trees were monitored annually from 2013 to 2017. In each plot, the diameter at the height of 1.30 from the ground (DBH, in cm) and the total height (Ht, in meters) of nine trees were measured. Three traditional methods were evaluated: guide-curve method (GC), difference equation method (DE), and parameter prediction method (PP). The statistical criteria used to assess the quality of the classification were: BIAS (%), mean squared error (RMSE (%)), and Pearson correlation coefficient (r). Artificial neural networks were superior to traditional methods, indicating the potential of the algorithm to model the productive capacity of a site for forest stands.

Mixed QCD-EW corrections to W-pair production at electron-positron colliders

The discrepancy between the CDF measurement and the Standard Model theoretical prediction for the W-boson mass underscores the importance of conducting high-precision studies on the W boson, which is one of the predominant objectives of proposed future e+e− colliders. We investigate in detail the production of W-boson pairs at e+e− colliders, and compute the next-to-next-to-leading order mixed QCD-EW corrections to both the integrated cross section and various kinematic distributions. By employing the method of differential equations, we analytically calculate the two-loop master integrals for the mixed QCD-EW virtual corrections to e+e− → W+W−. Utilizing the Magnus transformation, we derive a set of canonical master integrals for each integral family. This canonical basis satisfies a system of differential equations in which the dependence on the dimensional regulator is linearly factorized from the kinematics. We then express all these canonical master integrals as Taylor series in ϵ up to ϵ4, with coefficients articulated in terms of Goncharov polylogarithms up to weight four. Upon applying our analytic expressions of these master integrals to the phenomenological analysis of W-pair production, we observe that the O(ααs) corrections are significantly impactful in the α(0) scheme, particularly in certain phase-space regions. However, these mixed QCD-EW corrections can be heavily suppressed by adopting the Gμ scheme.

Dilated convolution neural operator for multiscale partial differential equations

This paper introduces a data-driven operator learning method for multiscale partial differential equations, with a particular emphasis on preserving high-frequency information. Drawing inspiration from the representation of multiscale parameterized solutions as a combination of low-rank global bases (such as low-frequency Fourier modes) and localized bases over coarse patches (analogous to dilated convolution), we propose the Dilated Convolutional Neural Operator (DCNO). The DCNO architecture effectively captures both high-frequency and low-frequency features while maintaining a low computational cost through a combination of convolution and Fourier layers. We conduct experiments to evaluate the performance of DCNO on various datasets, including the multiscale elliptic equation, its inverse problem, Navier–Stokes equation, and Helmholtz equation. Our results demonstrate that DCNO achieves an optimal balance between accuracy and computational efficiency, making it a promising approach for multiscale operator learning.

Balanced implicit two-step Maruyama methods for stochastic differential equations

Balanced implicit two-step Maruyama methods for stochastic differential equations

Direct estimates for adaptive time-stepping finite element methods

We study direct estimates for adaptive time-stepping finite element methods for time-dependent partial differential equations. Our results generalize previous findings from “On approximation classes for adaptive time-stepping finite element methods” by Actis et al. (2023), where the approximation error was only measured in L2([0,T],L2(Ω)). In particular, we now also cover the error norms L∞([0,T],L2(Ω)) and L2([0,T],H1(Ω)) which are more natural in this context.

Chemical Reaction Neural Networks for fitting Accelerating Rate Calorimetry data

As the demand for lithium-ion batteries rapidly increases there is a need to design these cells in a safe manner to mitigate thermal runaway. Thermal runaway in batteries leads to an uncontrollable temperature rise and potentially battery fires, which is a major safety concern. Typically, when modeling the chemical kinetics of thermal runaway calorimetry data (e.g. Accelerating Rate Calorimetry (ARC)) is needed to determine the temperature-driven decomposition kinetics. Conventional methods of fitting Arrhenius Ordinary Differential Equation (ODE) thermal runaway models to ARC data make several assumptions that reduce the fidelity and generalizability of the obtained model. In this paper, Chemical Reaction Neural Networks (CRNNs) are trained to fit the kinetic parameters of N-equation Arrhenius ODEs to ARC data obtained from a Molicel 21700 P45B. The models are found to be better approximations of the experimental data. The flexibility of the method is demonstrated by experimenting with two-equation and four-equation models. Thermal runaway simulations are conducted in 3D using the obtained kinetic parameters, showing the applicability of the obtained thermal runaway models to large-scale simulations.

Bifurcation analysis and exact solutions of the conformable time fractional Symmetric Regularized Long Wave equation

This paper investigates exact solutions for the conformable time fractional Symmetric Regularized Long Wave equation by applying the bifurcation analysis method and the exp(−Φ(ξ))-expansion method. By analyzing the long behaviors of the exact solutions plotted in 3D and 2D figures, we can model weakly nonlinear ion acoustic and space-charge waves. The bifurcation of the equation is analyzed based on the condition where the first integral constant is zero and the second integral constant is not zero. Based on different parameter conditions, many phase portraits and exact solutions including dark soliton, bright soliton, breaking wave, periodic and singular solutions for the equation are obtained. It has been proven that bifurcation method provides a wider range of solutions compared with other methods. Then the exp(−Φ(ξ))−expansion method is utilized to get the more solutions. Next graphical representations are presented that show physical characteristics of the solutions and the significance of the methods for fractional partial differential equations. Finally, we make a comprehensive comparison with other literatures.

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

We develop a registration-based nonlinear model-order reduction (MOR) method for partial differential equations (PDEs) with applications to transonic Euler and Reynolds-averaged Navier–Stokes (RANS) equations in aerodynamics. These PDEs exhibit discontinuous features, namely shocks, whose location depends on problem configuration parameters, and the associated parametric solution manifold exhibits a slowly decaying Kolmogorov N-width. As a result, conventional linear MOR methods, which use linear reduced approximation spaces, do not yield accurate low-dimensional approximations. We present a registration-based nonlinear MOR method to overcome this challenge. Our formulation builds on the following key ingredients: (i) a geometrically transformable parametrized PDE discretization; (ii) localized spline-based parametrized transformations which warp the domain to align discontinuities; (iii) an efficient dilation-based shock sensor and metric to compute optimal transformation parameters; (iv) hyperreduction and online-efficient output-based error estimates; and (v) simultaneous transformation and adaptive finite element training. Compared to existing methods in the literature, our formulation is efficiently scalable to larger problems and is equipped with error estimates and hyperreduction. We demonstrate the effectiveness of the method on two-dimensional inviscid and turbulent flows modeled by the Euler and RANS equations, respectively.

Convergence and stability of an explicit numerical method for stochastic differential equations with piecewise continuous arguments

Convergence and stability of an explicit numerical method for stochastic differential equations with piecewise continuous arguments

Collocation methods for nonlinear differential equations on low-rank manifolds

We introduce new methods for integrating nonlinear differential equations on low-rank manifolds. These methods rely on interpolatory projections onto the tangent space, enabling low-rank time integration of vector fields that can be evaluated entry-wise. A key advantage of our approach is that it does not require the vector field to exhibit low-rank structure, thereby overcoming significant limitations of traditional dynamical low-rank methods based on orthogonal projection. To construct the interpolatory projectors, we develop a sparse tensor sampling algorithm based on the discrete empirical interpolation method (DEIM) that parameterizes tensor train manifolds and their tangent spaces with cross interpolation. Using these projectors, we propose two time integration schemes on low-rank tensor train manifolds. The first scheme integrates the solution at selected interpolation indices and constructs the solution with cross interpolation. The second scheme generalizes the well-known orthogonal projector-splitting integrator to interpolatory projectors. We demonstrate the proposed methods with applications to several tensor differential equations arising from the discretization of partial differential equations.

On the electron self-energy to three loops in QED

We compute the electron self-energy in Quantum Electrodynamics to three loops in terms of iterated integrals over kernels of elliptic type. We make use of the differential equations method, augmented by an ϵ-factorized basis, which allows us to gain full control over the differential forms appearing in the iterated integrals to all orders in the dimensional regulator. We obtain compact analytic expressions, for which we provide generalized series expansion representations that allow us to evaluate the result numerically for all values of the electron momentum squared. As a by-product, we also obtain ϵ-resummed results for the self-energy in the on-shell limit p2 = m2, which we use to recompute the known three-loop renormalization constants in the on-shell scheme.

Discontinuous galerkin method of the first order parabolic differential equation

The objective of this paper is to propose a modest numerical error analysis by applying DG finite element method for the parabolic differential equation. The DG method is an imperative numerical method with much mass compensation and more flexible meshing than other numerical methods. The DG method starts by discretizing the domain into a set of non-overlapping elements. This study gives a general introduction and discuss about the discontinuous Galerkin Method of first order parabolic problem. The parabolic problem satisfies the condition of the existence and uniqueness of DG solution. The main goal of this study is to theoretically explore the convergence of the solution of the above methods and show the validity of the results.

Three loop QCD corrections to the heavy-light form factors: fermionic contributions

We present analytic results for three-loop fermionic corrections to the heavy-light form factors in perturbative quantum chromodynamics. Specifically, we present all light quark contributions and contributions from two heavy quark loops. We use the method of differential equations to compute all relevant three-loop master integrals. The results for all these contributions are expressed in terms of harmonic polylogarithms and generalized harmonic polylogarithms.

Application of Sumudu Transform and New Iterative Method to Solve Certain Nonlinear Ordinary Caputo Fractional Differential Equations

By combining the new iteration method (NIM) and Sumudu transform (ST) methods, together known as the new iteration ST method (NISTM), a semi-analytical or series solution for various fractional differential equations, in which new iteration method (NIM) is used to decompose the nonlinear operator prior to performing the ST, is produced in this script. Using the Caputo sense to account for the fractional derivative, this method computes series solutions for a variety of nonlinear fractional differential equation instances. While in many convergence issues a solution is achieved for just a small number of series terms, our series solution merges with the exact solution differential equation of fractional order in many example problems.

Conformable bilinear neural network method: a novel method for time-fractional nonlinear partial differential equations in the sense of conformable derivative

Conformable bilinear neural network method: a novel method for time-fractional nonlinear partial differential equations in the sense of conformable derivative

Option pricing under stochastic volatility on a quantum computer

We develop quantum algorithms for pricing Asian and barrier options under the Heston model, a popular stochastic volatility model, and estimate their costs, in terms of T-count, T-depth and number of logical qubits, on instances under typical market conditions. These algorithms are based on combining well-established numerical methods for stochastic differential equations and quantum amplitude estimation technique. In particular, we empirically show that, despite its simplicity, weak Euler method achieves the same level of accuracy as the better-known strong Euler method in this task. Furthermore, by eliminating the expensive procedure of preparing Gaussian states, the quantum algorithm based on weak Euler scheme achieves drastically better efficiency than the one based on strong Euler scheme. Our resource analysis suggests that option pricing under stochastic volatility is a promising application of quantum computers, and that our algorithms render the hardware requirement for reaching practical quantum advantage in financial applications less stringent than prior art.

Multigrid Reduction‐In‐Time Convergence for Advection Problems: A Fourier Analysis Perspective

ABSTRACTA long‐standing issue in the parallel‐in‐time community is the poor convergence of standard iterative parallel‐in‐time methods for hyperbolic partial differential equations (PDEs), and for advection‐dominated PDEs more broadly. Here, a local Fourier analysis (LFA) convergence theory is derived for the two‐level variant of the iterative parallel‐in‐time method of multigrid reduction‐in‐time (MGRIT). This closed‐form theory allows for new insights into the poor convergence of MGRIT for advection‐dominated PDEs when using the standard approach of rediscretizing the fine‐grid problem on the coarse grid. Specifically, we show that this poor convergence arises, at least in part, from inadequate coarse‐grid correction of certain smooth Fourier modes known as characteristic components, which was previously identified as causing poor convergence of classical spatial multigrid on steady‐state advection‐dominated PDEs. We apply this convergence theory to show that, for certain semi‐Lagrangian discretizations of advection problems, MGRIT convergence using rediscretized coarse‐grid operators cannot be robust with respect to CFL number or coarsening factor. A consequence of this analysis is that techniques developed for improving convergence in the spatial multigrid context can be re‐purposed in the MGRIT context to develop more robust parallel‐in‐time solvers. This strategy has been used in recent work to great effect; here, we provide further theoretical evidence supporting the effectiveness of this approach.

A study on generalized balanced split drift stochastic Runge- Kutta methods for stochastic differential equations

To reduce computational complexity, the balanced numerical approximations of the general split drift stochastic Runge-Kutta methods are analyzed. The primary reasons for considering the numerical approximations of these balanced split stochastic Runge-Kutta methods are their improved stability characteristics and lower mean square error compared to other methods. By balancing the drift and diffusion components, the splitting techniques outperform the mean square error over longer time increments. For Ito multi-dimensional stochastic differential equations, we propose a novel family of balanced universal split stochastic Runge-Kutta procedures. The Kronecker product concept is utilized to derive the mean-square stability conditions. We conduct numerical tests to evaluate these methods against an existing weak order 2 split drift method. Ultimately, a specific numerical example validates the theoretical outcomes of the balanced general split stochastic Runge-Kutta procedures.