We present a class of simple advection-pressure splitting numerical methods to solve the blood flow equations in compliant arterial vessels. The schemes are inspired by the TV flux vector splitting approach for conservative systems, proposed by Toro and Vázquez [30]. But the reformulated TV-type splitting schemes of this paper have a wider range of applicability, including systems of equations in non-conservative form. The spatial differential operator is split into advection terms, which may be in conservative form, from pressure terms in conservative or non-conservative form. Additionally, unlike the original TV scheme, the reformulated splitting of this paper fully preserves the continuity equation as part of the pressure system. This last feature is consistent with zero-dimensional models for blood flow that are based on neglecting the inertial term in the momentum equation. The schemes are also well suited for systems in which geometric and biomechanical parameters of the problem vary discontinuously. The splitting schemes of this paper are systematically assessed on a carefully designed suite of test problems and compared with several existing, mainstream methods. Overall, the proposed numerical methods perform very satisfactorily and suggest themselves as attractive computational tools for modelling the dynamics of bodily fluids under realistic conditions.

In this paper, we study multiple pole solutions for the focusing Hirota equation under the nonzero boundary conditions via inverse scattering method. The direct scattering problem is based on the spectral analysis and exhibits the Jost solutions, scattering matrix as well as their analyticity, symmetries and asymptotic behaviors. Compared with previous studies, we define a more complex discrete spectrum. The inverse scattering problem is explored by solving the corresponding matrix Riemann-Hilbert problems. Particularly, we solve the scattering problem by a suitable uniformization variable on the complex $z$-plane instead of a two-sheeted Riemann surface. Finally, we deduce general formulas of $N$-double pole and $N$-triple pole solutions with mixed discrete spectra and show some prominent characteristics of these solutions graphically. Our results should be helpful to further explore and enrich breather wave phenomena arising in nonlinear and complex systems.

We propose a kind of reduced Ablowitz-Kaup-Newell-Segur matrix spectral problems under two local group reductions by similarity transformations. Associated integrable hierarchies of matrix mKdV type integrable models are presented, which amend the complex matrix mKdV integrable hierarchies. Zero curvature equations are key objects in generating integrable models.

In this paper, the application of the orthogonality sampling method (OSM) to the real-world microwave imaging for identifying location of small anomalies is addressed. In order to show the feasibility and limitation of the OSM, we theoretically prove that the indicator function can be represented in terms of an infinite series of Bessel functions of integer order and the transmitting and receiving signal antenna configurations. This is based on the application of the Born approximation and the reciprocity of the incident fields. Throughout real-data experiments, it was shown that the OSM works well for identifying single anomaly under the specific location of transmitter while further improvement is needed for identification of multiple anomalies. To improve the imaging performance, we consider traditional indicator function with multiple sources and design a new indicator function with multiple sources weighted by the incident field. Theoretical results are contained to demonstrate the improvement.

The multivariate feedback particle filter (FPF) is formulated from the viewpoint of splitting-up methods. The essential difference between this formulation and the formal derivation is that instead of one-time control at a discrete time instant, we consider the updating stage as a stochastic flow of particles in each time interval. This allows to easily obtain a consistent stochastic flow by comparing the Kolmogorov forward equation of particles and the updating part of the Kushner’s equation in the splitting-up method. Moreover, if an optimal stochastic flow exists, the convergence of the splitting-up method can be studied by passing to an FPF with a continuous time. To guarantee the existence of a stochastic flow, we validate the Poincaré inequality for the alternating distributions, given the time discretization and the observation path, under mild conditions on the nonlinear filtering system and the initial state. Besides, re-examining the original derivation of the FPF, we show that the optimal transport map between the prior and the posterior is an $f$-divergence invariant in the abstract Bayesian inference framework.

We introduce a new subclass of $H$-matrices called partitioned Dashnic-Zusmanovich type (DZT) matrices and present the corresponding scaling matrices for this kind of matrices. There are three major applications. The first application is to provide equivalent eigenvalue localization related to index partition by using the nonsingularity of the new subclass. By taking some specific partitions, we provide other forms of eigenvalue localization sets that generalize and improve some well-known eigenvalue localization sets. The second application is to obtain an upper bound on the infinite norm of the inverse of partitioned DZT matrices using scaling matrices. The third application is to give an error bound of the linear complementarity problems (LCPs) by using scaling matrices. Additionally, we give another upper bound of the infinite norm and error bound of the LCPs by a reduction method, which transforms the given partitioned DZT matrix into the corresponding DZT matrix by partition and summation. The results obtained by the reduction method are generalizations of some known conclusions.

A two-grid method (TGM) for the time-dependent Maxwell’s equations in Cole-Cole dispersive media with a fractional time derivative term is proposed. We employ the lowest Raviart-Thomas-Nédélec mixed finite elements to discrete the space. It is known that for these type of Nédélec edge finite elements, the standard TGM cannot be applied directly. Therefore, we modified the traditional TGM, and the discrete process can be divided into two steps. Firstly, we get the rough discrete solutions on the coarse mesh. At the same time, superconvergence results can be obtained by using a post-processing technique. Secondly, the superconvergent solutions on the coarse grid are added on the fine mesh as a correction, and the optimal error estimates could be obtained accordingly. Finally, the numerical experiments can verify that the theoretical results are correct and reasonable.

In this paper, we propose a nonconforming virtual element method for the elliptic interface problem based on an unfitted polygonal mesh. On interface elements, the intersecting points of the interface and the edges of elements are considered as additional nodes of the mesh, and then we present a virtual element space satisfying the interface conditions. On non-interface elements, we use the usual nonconforming virtual element. By employing a computable operator, we introduce a discrete scheme and obtain optimal convergence results which are independent of the contrast of the coefficients. Numerical examples are presented to validate the theoretical results.

From the Black-Scholes option pricing model, this work evaluates the evolution of the mathematical modelling into the double stochastic volatility model that studies the optimization performance in partial differential equation (PDE) methods. This paper focuses on the calibration and numerical methodology processes to derive the comparison of the Heston and the double Heston models to design a more efficient numerical iterative splitting method. Through Li and Huang’s iterative splitting method, the numerical results conclude that the mixed method reduces the overall computational cost and improves the convergence of the iterative process while maintaining the simplicity, flexibility and interpretability of PDE methods.