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Differential equation models are crucial to scientific processes. The values of model parameters are important for analyzing the behaviour of solutions. A parameter is called globally identifiable if its value can be uniquely determined from the input and output functions. To determine if a parameter estimation problem is well-posed for a given model, one must check if the model parameters are globally identifiable. This problem has been intensively studied for ordinary differential equation models, with theory and several efficient algorithms and software packages developed. A comprehensive theory of algebraic identifiability for PDEs has hitherto not been developed due to the complexity of initial and boundary conditions. Here, we provide theory and algorithms, based on differential algebra, for testing identifiability of polynomial PDE models. We showcase this approach on PDE models arising in the sciences.

We introduce free probability analogues of the stochastic theta methods for free stochastic differential equations, which generalize the free Euler-Maruyama method introduced by Schl\"{u}chtermann and Wibmer [27]. Under some mild conditions, we prove the strong convergence and exponential stability in mean square of the numerical solution. The free stochastic theta method with $\theta=1$ can inherit the exponential stability of original equations for any given step size. Our method can offer better stability and efficiency than the free Euler-Maruyama method. Moreover, numerical results are reported to confirm these theoretical findings.

The interest in network analysis of bibliographic data has grown substantially in recent years, yet comprehensive statistical models for examining the complete dynamics of scientific networks based on bibliographic data are generally lacking. Current empirical studies often focus on models restricting analysis either to paper citation networks (paper-by-paper) or author networks (author-by-author). However, such networks encompass not only direct connections between papers, but also indirect relationships between the references of papers connected by a citation link. In this paper, we extend recently developed relational hyperevent models (RHEM) for analyzing scientific networks. We introduce new covariates representing theoretically meaningful and empirically interesting sub-network configurations. The model accommodates testing hypotheses considering: (i) the polyadic nature of scientific publication events, and (ii) the interdependencies between authors and references of current and prior papers. We implement the model using purpose-built, publicly available open-source software, demonstrating its empirical value in an analysis of a large publicly available scientific network dataset. Assessing the relative strength of various effects reveals that both the hyperedge structure of publication events, as well as the interconnection between authors and references significantly improve our understanding and interpretation of collaborative scientific production.

We study the approximation by a Voronoi finite-volume scheme of the Gross-Pitaevskii equation with time-dependent potential in two and three dimensions. We perform an explicit splitting scheme for the time integration alongside a two-point flux approximation scheme in space. We rigorously analyze the error bounds relying on discrete uniform Sobolev inequalities. We also prove the convergence of the pseudo-vorticity of the wave function. We finally perform some numerical simulations to illustrate our theoretical results.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can be efficiently implemented via the (inverse) fast Fourier transforms, and the computational cost at each time step is ${\mathcal O}(N\log N)$ with $N$ the total number of spatial points. However, this fast algorithm fails in the variable-order cases due to the spatial dependence of the Fourier multiplier. On the other hand, the direct matrix-vector multiplication approach becomes impractical due to excessive memory requirements. To address this challenge, we proposed an accelerated matrix-free approach for the efficient computation of variable-order cases. The computational cost is ${\mathcal O}(MN\log N)$ and storage cost ${\mathcal O}(MN)$, where $M \ll N$. Moreover, our method can be easily parallelized to further enhance its efficiency. Numerical studies show that our methods are effective in solving the variable-order space fractional wave equations, especially in high-dimensional cases. Wave propagation in heterogeneous media is studied in comparison to homogeneous counterparts. We find that wave dynamics in fractional cases become more intricate due to nonlocal interactions. Specifically, dynamics in heterogeneous media are more complex than those in homogeneous media.

This paper introduces several depths for random sets with possibly non-convex realisations, proposes ways to estimate the depths based on the samples and compares them with existing ones. The depths are further applied for the comparison between two samples of random sets using a visual method of DD-plots and statistical testing. The advantage of this approach is identifying sets within the sample that are responsible for rejecting the null hypothesis of equality in distribution and providing clues on differences between distributions. The method is justified using a simulation study and applied to real data consisting of histological images of mastopathy and mammary cancer tissue.

This manuscript investigates the problem of locational complexity, a type of complexity that emanates from a companys territorial strategy. Using an entropy-based measure for supply chain structural complexity ( pars-complexity), we develop a theoretical framework for analysing the effects of locational complexity on the profitability of service/manufacturing networks. The proposed model is used to shed light on the reasons why network restructuring strategies may result ineffective at reducing complexity-related costs. Our contribution is three-fold. First, we develop a novel mathematical formulation of a facility location problem that integrates the pars-complexity measure in the decision process. Second, using this model, we propose a decomposition of the penalties imposed by locational complexity into (a) an intrinsic cost of structural complexity; and (b) an avoidable cost of ignoring such complexity in the decision process. Such a decomposition is a valuable tool for identifying more effective measures for tackling locational complexity, moreover, it has allowed us to provide an explanation to the so-called addiction to growth within the locational context. Finally, we propose three alternative strategies that attempt to mimic different approaches used in practice by companies that have engaged in network restructuring processes. The impact of those approaches is evaluated through extensive numerical experiments. Our experimental results suggest that network restructuring efforts that are not accompanied by a substantial reduction on the target market of the company, fail at reducing complexity-related costs and, therefore, have a limited impact on the companys profitability.

An efficient approximate version of implicit Taylor methods for initial-value problems of systems of ordinary differential equations (ODEs) is introduced. The approach, based on an approximate formulation of Taylor methods, produces a method that requires less evaluations of the function that defines the ODE and its derivatives than the usual version. On the other hand, an efficient numerical solution of the equation that arises from the discretization by means of Newton's method is introduced for an implicit scheme of any order. Numerical experiments illustrate that the resulting algorithm is simpler to implement and has better performance than its exact counterpart.

The stochastic heat equation on the sphere driven by additive isotropic Wiener noise is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. Optimal strong convergence rates for a given regularity of the initial condition and driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown, where the approximation of the second moment converges with twice the strong rate. Numerical simulations confirm the theoretical results.