Analysis Of Partial Differential Equations Research Articles

Mini-Workshop: Multivariate Orthogonal Polynomials: New synergies with Numerical Analysis

Multivariate polynomials and, in particular, multivariate orthogonal polynomials (MOPs) are research areas within the fields of special functions, Lie groups, quantum groups, computer algebra to name only some of them. However, there are many important areas in the field of numerical analysis where multivariate polynomials (of high order) play a crucial role: approximation by spectral methods and finite elements, discrete calculus, polynomial trace liftings, exact sequence properties, sparsity, efficient and stable recursions, analysis of the geometry of the zeros. The miniworkshop brought together experts from the fields of MOPs and numerical analysis of partial differential equations.

Semiclassical theory and the Koopman-van Hove equation * *Contribution to the Special Collection ‘Koopman Methods in Classical and Quantum-Classical Mechanics’.

The phase space Koopman-van Hove (KvH) equation can be derived from the asymptotic semiclassical analysis of partial differential equations. Semiclassical theory yields the Hamilton–Jacobi equation for the complex phase factor and the transport equation for the amplitude. These two equations can be combined to form a nonlinear semiclassical version of the KvH equation in configuration space. There is a natural injection of configuration space solutions into phase space and a natural projection of phase space solutions onto configuration space. Hence, every solution of the configuration space KvH equation satisfies both the semiclassical phase space KvH equation and the Hamilton–Jacobi constraint. For configuration space solutions, this constraint resolves the paradox that there are two different conserved densities in phase space. For integrable systems, the KvH spectrum is the Cartesian product of a classical and a semiclassical spectrum. If the classical spectrum is eliminated, then, with the correct choice of Jeffreys–Wentzel–Kramers–Brillouin (JWKB) matching conditions, the semiclassical spectrum satisfies the Einstein–Brillouin–Keller quantization conditions which include the correction due to the Maslov index. However, semiclassical analysis uses different choices for boundary conditions, continuity requirements, and the domain of definition. For example, use of the complex JWKB method allows for the treatment of tunneling through the complexification of phase space. Finally, although KvH wavefunctions include the possibility of interference effects, interference is not observable when all observables are approximated as local operators on phase space. Observing interference effects requires consideration of nonlocal operations, e.g. through higher orders in the asymptotic theory.

An efficient accurate scheme for solving the three-dimensional Bratu-type problem

In this manuscript, we present an innovative discretization algorithm designed to address the challenges posed by the three-dimensional (3D) Bratu problem, a well-known problem characterized by non-unique solutions. Our algorithm aims to achieve exceptional precision and accuracy in determining all potential solutions, as previous studies in the literature have only managed to produce limited accurate results. Additionally, our computational scheme approximates the critical values of the transition parameter. Moreover, we establish a rigorous proof demonstrating the uniform convergence of the approximation sequence to the exact solution of the original problem, under the condition that the initial guess is sufficiently close to the true solution. This theoretical result further establishes the reliability of our algorithm. To evaluate the effectiveness of our proposed approach, we conduct extensive numerical simulations, which convincingly demonstrate its capability to accurately solve the 3D Bratu problem while effectively determining the critical values of the transition parameter. Furthermore, we investigate the bifurcated behavior of the solution by analyzing the infinity norm for various values of the transition parameter. The outcomes of our study offer a robust and efficient method for tackling the 3D Bratu problem, making a significant contribution to the field of numerical analysis of partial differential equations (PDEs) where non-unique solutions are commonly encountered. Our algorithm's ability to produce accurate results for this challenging problem showcases its potential for broader applications in diverse scientific and engineering domains.

LPF: a framework for exploring the wing color pattern formation of ladybird beetles in Python.

Biological pattern formation is one of the complex system phenomena in nature, requiring theoretical analysis based on mathematical modeling and computer simulations for in-depth understanding. We propose a Python framework named LPF to systematically explore the highly diverse wing color patterns of ladybirds using reaction-diffusion models. LPF supports GPU-accelerated array computing for numerical analysis of partial differential equation (PDE) models, concise visualization of ladybird morphs, and evolutionary algorithms for searching mathematical models with deep learning models for computer vision. LPF is is available on GitHub at https://github.com/cxinsys/lpf. Supplementary data are available at Bioinformatics online.

On the Global Convergence of Particle Swarm Optimization Methods

In this paper we provide a rigorous convergence analysis for the renowned particle swarm optimization method by using tools from stochastic calculus and the analysis of partial differential equations. Based on a continuous-time formulation of the particle dynamics as a system of stochastic differential equations, we establish convergence to a global minimizer of a possibly nonconvex and nonsmooth objective function in two steps. First, we prove consensus formation of an associated mean-field dynamics by analyzing the time-evolution of the variance of the particle distribution, which acts as Lyapunov function of the dynamics. We then show that this consensus is close to a global minimizer by employing the asymptotic Laplace principle and a tractability condition on the energy landscape of the objective function. These results allow for the usage of memory mechanisms, and hold for a rich class of objectives provided certain conditions of well-preparation of the hyperparameters and the initial datum. In a second step, at least for the case without memory effects, we provide a quantitative result about the mean-field approximation of particle swarm optimization, which specifies the convergence of the interacting particle system to the associated mean-field limit. Combining these two results allows for global convergence guarantees of the numerical particle swarm optimization method with provable polynomial complexity. To demonstrate the applicability of the method we propose an efficient and parallelizable implementation, which is tested in particular on a competitive and well-understood high-dimensional benchmark problem in machine learning.

Strong form mesh-free hp-adaptive solution of linear elasticity problem

We present an algorithm for hp-adaptive collocation-based mesh-free numerical analysis of partial differential equations. Our solution procedure follows a well-established iterative solve–estimate–mark–refine paradigm. The solve phase relies on the Radial Basis Function-generated Finite Differences (RBF-FD) using point clouds generated by advancing front node positioning algorithm that supports variable node density. In the estimate phase, we introduce an Implicit-Explicit (IMEX) error indicator, which assumes that the error relates to the difference between the implicitly obtained solution (from the solve phase) and a local explicit re-evaluation of the PDE at hand using a higher order approximation. Based on the IMEX error indicator, the modified Texas Three Step marking strategy is used to mark the computational nodes for h-, p- or hp-(de-)refinement. Finally, in the refine phase, nodes are repositioned and the order of the method is locally redefined using the variable order of the augmenting monomials according to the instructions from the mark phase. The performance of the introduced hp-adaptive method is first investigated on a two-dimensional Peak problem and further applied to two- and three-dimensional contact problems. We show that the proposed IMEX error indicator adequately captures the global behaviour of the error in all cases considered and that the proposed hp-adaptive solution procedure significantly outperforms the non-adaptive approach. The proposed hp-adaptive method stands for another important step towards a fully autonomous numerical method capable of solving complex problems in realistic geometries without the need for user intervention.

Recent progress on limit theorems for large stochastic particle systems

This article presents a selection of recent results in the mathematical study of physical systems described by a large number of particles, with various types of interactions (mean-field, moderate, nearest-neighbor). Limit theorems are obtained concerning either the large-scale or the long-time behavior of these systems. These results rely on the use of a large range of mathematical tools, arising from both probability theory and the analysis of partial differential equations, and thereby illustrate fruitful interactions between these two disciplines.

Mathematical modeling of the population dynamics of a distinct interactions type system with local dispersal

Distinct biotic interactions in multi-species communities are a ubiquitous force in the natural ecosystem, and this force is an essential determinant of community stability and species coexistence outcomes. We conduct numerical simulations and bifurcation analysis of partial differential equations to gain better understanding and ecological insights into how predation (a), predator handling time (h), and local dispersal affect multi-species community dynamics. This system consists of resource-mutualist-exploiter-competitor interactions and local dispersal. From the inspection of our numerical simulations and co-dimension one bifurcation analysis findings, we discover several critical values that correspond to transcritical bifurcation, subcritical and supercritical Hopf bifurcations. This occurs as we vary the bifurcation parameters a and h in this complex ecological system under symmetric and asymmetric dispersal scenarios. Furthermore, the interplay between these local bifurcation points results in an exciting co-dimension two bifurcations, i.e., Bogdanov-Takens and cusp bifurcation points, respectively, which act as the synchronization points in this complex ecological system. From an ecological viewpoint, we find that (i) the effect of the no-dispersal scenario supports the maintenance of species biodiversity when the predation strength is moderate; (ii) symmetric dispersal induces both subcritical and supercritical Hopf bifurcation and support species diversity for moderate predation strength; and (iii) asymmetric dispersal promotes species diversity as it simplifies the bifurcation changes in dynamics by eliminating the subcritical bifurcations that trigger uncertainty, and this dispersal mechanism mediates species coexistence outcomes. Fundamentally, stable limit cycles have been reported as predator handling time varies in some ecological models; however, we observed in our bifurcation analysis the emergence of the unstable limit cycle as predator handling time changes. We discover that intense predator handling time destabilizes this complex ecological community. In general, our results demonstrate the influential roles of predation, predator handling time, and local dispersal in determining this system’s coexistence dynamics. This knowledge provides a better understanding of species conservation and biological control management.

Direct FE numerical simulation for dynamic instability of frame structures

Dynamic instability analysis of structures has been a critical issue in the application of engineering and mechanical designs. Can we directly simulate the dynamic stability of structures like buckling analysis in finite element software? To address this question, in this study, we explore the dynamic instability of frame structures, discretized into finite element structures, through developing a direct FE numerical simulation procedure. Benefiting from the advances of eigenvalue analysis of partial differential equations and finite element technology, a direct numerical simulation procedure is developed by using Floquet theorem, harmonic balance methods and finite element method. Dynamic instability characteristics of frame structures, concerning fundamental vibration, multi-mode coupling vibration and flexural-torsional vibration deformation due to their spatial characteristics, are thoroughly investigated. The results reveal that, the developed procedure can achieve dynamic stability analysis of damped and non-damped structures. Frame's flexural-torsional vibration deformation can significantly enlarge the size and instability intensity of unstable regions, threatening dynamic stability of frame structures. Moreover, damping ratio has a slight impact on instability mitigation of the frame induced by flexural-torsional vibration. Unstable regions of mode-coupling vibration of frame have a hidden attribute. No existing approximate formula can be available to estimate instability regions of frame's flexural-torsional vibration and mode-coupling vibration. The dynamic instability index presented here can characterize unstable regions intuitively. The proposed insights can provide design guidelines for frame structures, the developed direct FE simulation procedure may be insightful for extending to explore instability mechanisms of generalized structures with complex forms owing to parametric resonance.

Phase-space dynamics of a single-atom laser: Nonclassicality and bistability

We investigate the dynamics of a single-qubit single-mode laser with continuous incoherent pumping of a qubit, the field mode being initially prepared in a coherent state. Analysis of partial differential equations for quasiprobability distributions helps us distinguish two stages of evolution (coherent and incoherent ones) with their characteristic features. The system can exhibit bistabilitylike behavior, similar to that reported for a single-atom laser with continuous coherent pumping, and is capable of generating Schr\"odinger cat states. Quantitative analysis of the field state nonclassicality shows that its maximum is reached at intermediate stages of evolution.

Remarks on anomalous dissipation for passive scalars.

We consider the problem of anomalous dissipation for passive scalars advected by an incompressible flow. We review known results on anomalous dissipation from the point of view of the analysis of partial differential equations, and present simple rigorous examples of scalars that admit a Batchelor-type energy spectrum and exhibit anomalous dissipation in the limit of zero scalar diffusivity. This article is part of the theme issue 'Scaling the turbulence edifice (part 1)'.

Fractional view analysis of partial differential equation via residual power series transform method

Fractional view analysis of partial differential equation via residual power series transform method

Regularity of a weak solution to a linear fluid-composite structure interaction problem

In this manuscript, we deal with the regularity of a weak solution to the fluid-composite structure interaction problem introduced in [12]. The problem describes a linear fluid-structure interaction between an incompressible, viscous fluid flow, and an elastic structure composed of a cylindrical shell supported by a mesh-like elastic structure. The fluid and the mesh-supported structure are coupled via the kinematic and dynamic boundary coupling conditions describing continuity of velocity and balance of contact forces at the fluid-structure interface. In [12], it is shown that there exists a weak solution to the described problem. By using the standard techniques from the analysis of partial differential equations we prove that such a weak solution possesses an additional regularity in both time and space variables for initial and boundary data satisfying the appropriate regularity and compatibility conditions imposed on the interface.

Parametrizing diffusion‐taxis equations from animal movement trajectories using step selection analysis

Abstract Mathematical analysis of partial differential equations (PDEs) has led to many insights regarding the effect of organism movements on spatial population dynamics. However, their use has mainly been confined to the community of mathematical biologists, with less attention from statistical and empirical ecologists. We conjecture that this is principally due to the inherent difficulties in fitting PDEs to data. To help remedy this situation, in the context of movement ecology, we show how the popular technique of step selection analysis (SSA) can be used to parametrize a class of PDEs, called diffusion‐taxis models, from an animal's trajectory. We examine the accuracy of our technique on simulated data, then demonstrate the utility of diffusion‐taxis models in two ways. First, for non‐interacting animals, we derive the steady‐state utilization distribution in a closed analytic form. Second, we give a recipe for deriving spatial pattern formation properties that emerge from interacting animals: specifically, do those interactions cause heterogeneous spatial distributions to emerge and if so, do these distributions oscillate at short times or emerge without oscillations? The second question is applied to data on concurrently tracked bank voles Myodes glareolus. Our results show that SSA can accurately parametrize diffusion‐taxis equations from location data, providing the frequency of the data is not too low. We show that the steady‐state distribution of our diffusion‐taxis model, where it is derived, has an identical functional form to the utilization distribution given by resource selection analysis (RSA), thus formally linking (fine scale) SSA with (broad scale) RSA. For the bank vole data, we show how our SSA‐PDE approach can give predictions regarding the spatial aggregation and segregation of different individuals, which are difficult to predict purely by examining results of SSA. Our methods provide a user‐friendly way into the world of PDEs, via a well‐used statistical technique, which should lead to tighter links between the findings of mathematical ecology and observations from empirical ecology. By providing a non‐speculative link between observed movement behaviours and space use patterns on larger spatio‐temporal scales, our findings will also aid integration of movement ecology into understanding spatial species distributions.

Geometric analysis of partial differential equations and several complex variables: in honor of Nick Hanges

Geometric analysis of partial differential equations and several complex variables: in honor of Nick Hanges

Anisotropic multiscale systems on bounded domains

We establish a construction of hybrid multiscale systems on a bounded domain ${\Omega } \subset \mathbb {R}^{2}$ consisting of shearlets and boundary-adapted wavelets, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct hybrid shearlet-wavelet systems that form frames for the Sobolev spaces $H^{s}({\Omega }),~s\in \mathbb {N} \cup \{0\}$ with controllable frame bounds and admit optimally sparse approximations for functions which are smooth apart from a curve-like discontinuity. Per construction, these systems allow to incorporate boundary conditions.

Existence of solutions of an explicit energy-conserving scheme for a fractional Klein–Gordon–Zakharov system

In the article Martínez et al. (2019) [7], the authors assumed the triviality of the existence of solutions of a finite-difference model for a fractional Klein–Gordon–Zakharov equation. It turns out that property is not trivial at all. In this short communication, we provide a proof of that result and, in the way, we mention some wrong proofs of similar results available in the literature. We believe that the report on the demonstration of this result will be a useful tool for researchers working on the numerical analysis of partial differential equations.

Mean-field models in swarm robotics: a survey

We present a survey on the application of fluid approximations, in the form of mean-field models, to the design of control strategies in swarm robotics. Mean-field models that consist of ordinary differential equations, partial differential equations, and difference equations have been used in the swarm robotics literature, depending on whether the state of each agent and the time variable take values from a discrete or continuous set. These macroscopic models are independent of the number of agents in the swarm, and hence can be used to synthesize robot control strategies in a scalable manner, in contrast to individual-based microscopic models of swarms that represent finite numbers of discrete agents. Moreover, mean-field models are amenable to rigorous investigation using tools from dynamical systems theory, control theory, stochastic processes, and analysis of partial differential equations, enabling new insights and provable guarantees on the dynamics of collective behaviors. In this paper, we survey the applications of these models to problems in swarm robotics that include coverage, task allocation, self-assembly, consensus, and environmental mapping.

Error bounds for approximations with deep ReLU neural networks in Ws,p norms

We analyze to what extent deep Rectified Linear Unit (ReLU) neural networks can efficiently approximate Sobolev regular functions if the approximation error is measured with respect to weaker Sobolev norms. In this context, we first establish upper approximation bounds by ReLU neural networks for Sobolev regular functions by explicitly constructing the approximate ReLU neural networks. Then, we establish lower approximation bounds for the same type of function classes. A trade-off between the regularity used in the approximation norm and the complexity of the neural network can be observed in upper and lower bounds. Our results extend recent advances in the approximation theory of ReLU networks to the regime that is most relevant for applications in the numerical analysis of partial differential equations.

Theoretical developments in the study of partial differential equations

In this article we review some of the main contributions of Indian mathematicians in the theoretical analysis of partial differential equations in the last decade.