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Understanding fluid movement in multi-pored materials is vital for energy security and physiology. For instance, shale (a geological material) and bone (a biological material) exhibit multiple pore networks. Double porosity/permeability models provide a mechanics-based approach to describe hydrodynamics in aforesaid porous materials. However, current theoretical results primarily address state-state response, and their counterparts in the transient regime are still wanting. The primary aim of this paper is to fill this knowledge gap. We present three principal properties -- with rigorous mathematical arguments -- that the solutions under the double porosity/permeability model satisfy in the transient regime: backward-in-time uniqueness, reciprocity, and a variational principle. We employ the ``energy method'' -- by exploiting the physical total kinetic energy of the flowing fluid -- to establish the first property and Cauchy-Riemann convolutions to prove the next two. The results reported in this paper -- that qualitatively describe the dynamics of fluid flow in double-pored media -- have (a) theoretical significance, (b) practical applications, and (c) considerable pedagogical value. In particular, these results will benefit practitioners and computational scientists in checking the accuracy of numerical simulators. The backward-in-time uniqueness lays a firm theoretical foundation for pursuing inverse problems in which one predicts the prescribed initial conditions based on data available about the solution at a later instance.

We address the numerical treatment of source terms in algebraic flux correction schemes for steady convection-diffusion-reaction (CDR) equations. The proposed algorithm constrains a continuous piecewise-linear finite element approximation using a monolithic convex limiting (MCL) strategy. Failure to discretize the convective derivatives and source terms in a compatible manner produces spurious ripples, e.g., in regions where the coefficients of the continuous problem are constant and the exact solution is linear. We cure this deficiency by incorporating source term components into the fluxes and intermediate states of the MCL procedure. The design of our new limiter is motivated by the desire to preserve simple steady-state equilibria exactly, as in well-balanced schemes for the shallow water equations. The results of our numerical experiments for two-dimensional CDR problems illustrate potential benefits of well-balanced flux limiting in the scalar case.

The theory of generalized locally Toeplitz (GLT) sequences is a powerful apparatus for computing the asymptotic spectral distribution of matrices $A_n$ arising from numerical discretizations of differential equations. Indeed, when the mesh fineness parameter $n$ tends to infinity, these matrices $A_n$ give rise to a sequence $\{A_n\}_n$, which often turns out to be a GLT sequence. In this paper, we extend the theory of GLT sequences in several directions: we show that every GLT sequence enjoys a normal form, we identify the spectral symbol of every GLT sequence formed by normal matrices, and we prove that, for every GLT sequence $\{A_n\}_n$ formed by normal matrices and every continuous function $f:\mathbb C\to\mathbb C$, the sequence $\{f(A_n)\}_n$ is again a GLT sequence whose spectral symbol is $f(\kappa)$, where $\kappa$ is the spectral symbol of $\{A_n\}_n$. In addition, using the theory of GLT sequences, we prove a spectral distribution result for perturbed normal matrices.

One tuple of probability vectors is more informative than another tuple when there exists a single stochastic matrix transforming the probability vectors of the first tuple into the probability vectors of the other. This is called matrix majorization. Solving an open problem raised by Mu et al, we show that if certain monotones - namely multivariate extensions of R\'{e}nyi divergences - are strictly ordered between the two tuples, then for sufficiently large $n$, there exists a stochastic matrix taking the $n$-fold Kronecker power of each input distribution to the $n$-fold Kronecker power of the corresponding output distribution. The same conditions, with non-strict ordering for the monotones, are also necessary for such matrix majorization in large samples. Our result also gives conditions for the existence of a sequence of statistical maps that asymptotically (with vanishing error) convert a single copy of each input distribution to the corresponding output distribution with the help of a catalyst that is returned unchanged. Allowing for transformation with arbitrarily small error, we find conditions that are both necessary and sufficient for such catalytic matrix majorization. We derive our results by building on a general algebraic theory of preordered semirings recently developed by one of the authors. This also allows us to recover various existing results on majorization in large samples and in the catalytic regime as well as relative majorization in a unified manner.

In this paper we study the convergence of a second order finite volume approximation of the scalar conservation law. This scheme is based on the generalized Riemann problem (GRP) solver. We firstly investigate the stability of the GRP scheme and find that it might be entropy unstable when the shock wave is generated. By adding an artificial viscosity we propose a new stabilized GRP scheme. Under the assumption that numerical solutions are uniformly bounded, we prove consistency and convergence of this new GRP method.

In the present paper, we propose a block variant of the extended Hessenberg process for computing approximations of matrix functions and other problems producing large-scale matrices. Applications to the computation of a matrix function such as f(A)V, where A is an nxn large sparse matrix, V is an nxp block with p<<n, and f is a function are presented. Solving shifted linear systems with multiple right hand sides are also given. Computing approximations of these matrix problems appear in many scientific and engineering applications. Different numerical experiments are provided to show the effectiveness of the proposed method for these problems.

By using the stochastic particle method, the truncated Euler-Maruyama (TEM) method is proposed for numerically solving McKean-Vlasov stochastic differential equations (MV-SDEs), possibly with both drift and diffusion coefficients having super-linear growth in the state variable. Firstly, the result of the propagation of chaos in the L^q (q\geq 2) sense is obtained under general assumptions. Then, the standard 1/2-order strong convergence rate in the L^q sense of the proposed method corresponding to the particle system is derived by utilizing the stopping time analysis technique. Furthermore, long-time dynamical properties of MV-SDEs, including the moment boundedness, stability, and the existence and uniqueness of the invariant probability measure, can be numerically realized by the TEM method. Additionally, it is proven that the numerical invariant measure converges to the underlying one of MV-SDEs in the L^2-Wasserstein metric. Finally, the conclusions obtained in this paper are verified through examples and numerical simulations.

This work deals with an inverse source problem for the biharmonic wave equation. A two-stage numerical method is proposed to identify the unknown source from the multi-frequency phaseless data. In the first stage, we introduce some artificially auxiliary point sources to the inverse source system and establish a phase retrieval formula. Theoretically, we point out that the phase can be uniquely determined and estimate the stability of this phase retrieval approach. Once the phase information is retrieved, the Fourier method is adopted to reconstruct the source function from the phased multi-frequency data. The proposed method is easy-to-implement and there is no forward solver involved in the reconstruction. Numerical experiments are conducted to verify the performance of the proposed method.

Gradient-enhanced Kriging (GE-Kriging) is a well-established surrogate modelling technique for approximating expensive computational models. However, it tends to get impractical for high-dimensional problems due to the size of the inherent correlation matrix and the associated high-dimensional hyper-parameter tuning problem. To address these issues, a new method, called sliced GE-Kriging (SGE-Kriging), is developed in this paper for reducing both the size of the correlation matrix and the number of hyper-parameters. We first split the training sample set into multiple slices, and invoke Bayes' theorem to approximate the full likelihood function via a sliced likelihood function, in which multiple small correlation matrices are utilized to describe the correlation of the sample set rather than one large one. Then, we replace the original high-dimensional hyper-parameter tuning problem with a low-dimensional counterpart by learning the relationship between the hyper-parameters and the derivative-based global sensitivity indices. The performance of SGE-Kriging is finally validated by means of numerical experiments with several benchmarks and a high-dimensional aerodynamic modeling problem. The results show that the SGE-Kriging model features an accuracy and robustness that is comparable to the standard one but comes at much less training costs. The benefits are most evident for high-dimensional problems with tens of variables.

We introduce in this paper the numerical analysis of high order both in time and space Lagrange-Galerkin methods for the conservative formulation of the advection-diffusion equation. As time discretization scheme we consider the Backward Differentiation Formulas up to order $q=5$. The development and analysis of the methods are performed in the framework of time evolving finite elements presented in C. M. Elliot and T. Ranner, IMA Journal of Numerical Analysis \textbf{41}, 1696-1845 (2021). The error estimates show through their dependence on the parameters of the equation the existence of different regimes in the behavior of the numerical solution; namely, in the diffusive regime, that is, when the diffusion parameter $\mu$ is large, the error is $O(h^{k+1}+\Delta t^{q})$, whereas in the advective regime, $\mu \ll 1$, the convergence is $O(\min (h^{k},\frac{h^{k+1} }{\Delta t})+\Delta t^{q})$. It is worth remarking that the error constant does not have exponential $\mu ^{-1}$ dependence.