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Interpreting data with mathematical models is an important aspect of real-world applied mathematical modeling. Very often we are interested to understand the extent to which a particular data set informs and constrains model parameters. This question is closely related to the concept of parameter identifiability, and in this article we present a series of computational exercises to introduce tools that can be used to assess parameter identifiability, estimate parameters and generate model predictions. Taking a likelihood-based approach, we show that very similar ideas and algorithms can be used to deal with a range of different mathematical modelling frameworks. The exercises and results presented in this article are supported by a suite of open access codes that can be accessed on GitHub.

Sequences of parametrized Lyapunov equations can be encountered in many application settings. Moreover, solutions of such equations are often intermediate steps of an overall procedure whose main goal is the computation of $\text{trace}(EX)$ where $X$ denotes the solution of a Lyapunov equation and $E$ is a given matrix. We are interested in addressing problems where the parameter dependency of the coefficient matrix is encoded as a low-rank modification to a \emph{seed}, fixed matrix. We propose two novel numerical procedures that fully exploit such a common structure. The first one builds upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov techniques, and it is well-suited for small dimensional problems as it makes use of dense numerical linear algebra tools. The second algorithm can instead address large-scale problems by relying on state-of-the-art projection techniques based on the extended Krylov subspace. We test the new algorithms on several problems arising in the study of damped vibrational systems and the analyses of output synchronization problems for multi-agent systems. Our results show that the algorithms we propose are superior to state-of-the-art techniques as they are able to remarkably speed up the computation of accurate solutions.

We present and analyze a simple numerical method that diagonalizes a complex normal matrix A by diagonalizing the Hermitian matrix obtained from a random linear combination of the Hermitian and skew-Hermitian parts of A.

For boundary value problem of an elliptic equation with variable coefficients describing the physical field distribution in inhomogeneous media, the Levi function can represent the solution in terms of volume and surface potentials, with the drawback that the volume potential involving in the solution expression requires heavy computational costs as well as the solvability of the integral equations with respect to the density pair. We introduce an modified integral expression for the solution to an elliptic equation in divergence form under the Levi function framework. The well-posedness of the linear integral system with respect to the density functions to be determined is rigorously proved. Based on the singularity decomposition for the Levi function, we propose two schemes to deal with the volume integrals so that the density functions can be solved efficiently. One method is an adaptive discretization scheme (ADS) for computing the integrals with continuous integrands, leading to the uniform accuracy of the integrals in the whole domain, and consequently the efficient computations for the density functions. The other method is the dual reciprocity method (DRM) which is a meshless approach converting the volume integrals into boundary integrals equivalently by expressing the volume density as the combination of the radial basis functions determined by the interior grids. The proposed schemes are justified numerically to be of satisfactory computation costs. Numerical examples in 2-dimensional and 3-dimensional cases are presented to show the validity of the proposed schemes.

In this paper, a comparative study between the Coco-Russo scheme (based on finite-difference scheme) and the $\mathghost$-FEM (based on finite-element method) is presented when solving the Poisson equation in arbitrary domains. The comparison between the two numerical methods is carried out by presenting analytical results from the literature \cite{cocoStissi,astuto2024nodal}, together with numerical tests in various geometries and boundary conditions.

This paper focuses on the numerical scheme for delay-type stochastic McKean-Vlasov equations (DSMVEs) driven by fractional Brownian motion with Hurst parameter $H\in (0,1/2)\cup (1/2,1)$. The existence and uniqueness of the solutions to such DSMVEs whose drift coefficients contain polynomial delay terms are proved by exploting the Banach fixed point theorem. Then the propagation of chaos between interacting particle system and non-interacting system in $\mathcal{L}^p$ sense is shown. We find that even if the delay term satisfies the polynomial growth condition, the unmodified classical Euler-Maruyama scheme still can approximate the corresponding interacting particle system without the particle corruption. The convergence rates are revealed for $H\in (0,1/2)\cup (1/2,1)$. Finally, as an example that closely fits the original equation, a stochastic opinion dynamics model with both extrinsic memory and intrinsic memory is simulated to illustrate the plausibility of the theoretical result.

In this paper, we reconsider two new iterative methods for solving absolute value equations (AVE), which is proposed by Ali and Pan (Jpn. J. Ind. Appl. Math. 40: 303--314, 2023). Convergence results of the two iterative schemes and new sufficient conditions for the unique solvability of AVE are presented. In addition, for a special case, the optimal iteration parameters of the two algorithms are analyzed, respectively. Numerical results demonstrate our claims.

The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for parametrically modelling the dynamics of the hazard function using systems of autonomous ordinary differential equations (ODEs). This modelling approach can be used to provide qualitative and quantitative analyses of the evolution of the hazard function over time. Our proposal capitalises on the extensive literature of ODEs which, in particular, allow for establishing basic rules or laws on the dynamics of the hazard function via the use of autonomous ODEs. We show how to implement the proposed modelling framework in cases where there is an analytic solution to the system of ODEs or where an ODE solver is required to obtain a numerical solution. We focus on the use of a Bayesian modelling approach, but the proposed methodology can also be coupled with maximum likelihood estimation. A simulation study is presented to illustrate the performance of these models and the interplay of sample size and censoring. Two case studies using real data are presented to illustrate the use of the proposed approach and to highlight the interpretability of the corresponding models. We conclude with a discussion on potential extensions of our work and strategies to include covariates into our framework. Although we focus on examples on Medical Statistics, the proposed framework is applicable in any context where the interest lies on estimating and interpreting the dynamics hazard function.

Nevanlinna-Pick interpolation problem has been widely studied in recent decades, however, the known algorithm is not simplistic and robust enough. This paper provide a new method to solve the Nevanlinna-Pick interpolation problem with degree constraint. It is based on the covariance extension equation proposed by Byrnes and Lindquist. A reformulation of the Nevanlinna-Pick interpolation problem is achieved and then solved by continuation method. This method need not calculate the initial value and a numerical example illustrates robustness and effciency of the proposed procedure

In this paper, we investigate nonlinear optimization problems whose constraints are defined as fuzzy relational equations (FRE) with max-min composition. Since the feasible solution set of the FRE is often a non-convex set and the resolution of the FREs is an NP-hard problem, conventional nonlinear approaches may involve high computational complexity. Based on the theoretical aspects of the problem, an algorithm (called FRE-ACO algorithm) is presented which benefits from the structural properties of the FREs, the ability of discrete ant colony optimization algorithm (denoted by ACO) to tackle combinatorial problems, and that of continuous ant colony optimization algorithm (denoted by ACOR) to solve continuous optimization problems. In the current method, the fundamental ideas underlying ACO and ACOR are combined and form an efficient approach to solve the nonlinear optimization problems constrained with such non-convex regions. Moreover, FRE-ACO algorithm preserves the feasibility of new generated solutions without having to initially find the minimal solutions of the feasible region or check the feasibility after generating the new solutions. FRE-ACO algorithm has been compared with some related works proposed for solving nonlinear optimization problems with respect to maxmin FREs. The obtained results demonstrate that the proposed algorithm has a higher convergence rate and requires a less number of function evaluations compared to other considered algorithms.