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This paper proposes and analyzes a novel efficient high-order finite volume method for the ideal magnetohydrodynamics (MHD). As a distinctive feature, the method simultaneously preserves two critical physical constraints: a discretely divergence-free (DDF) constraint on the magnetic field and the positivity-preserving (PP) property, which ensures the positivity of density, pressure, and internal energy. To enforce the DDF condition in each cell, we design a new discrete projection approach that projects the reconstructed point values at the cell interface into a DDF space, without using any approximation polynomials. This projection method is highly efficient, easy to implement, and particularly suitable for the high-order finite volume methods that return only the point values (no explicit approximation polynomials) in the reconstruction. Moreover, we also develop a new finite volume framework for constructing provably PP schemes for the ideal MHD system. The framework comprises the discrete projection technique, a suitable approximation to the Godunov--Powell source terms, and a simple PP limiter. We provide rigorous analysis of the PP property of the proposed finite volume method, demonstrating that the DDF condition and the proper approximation to the source terms eliminate the impact of magnetic divergence terms on the PP property. The analysis is challenging due to the internal energy function's nonlinearity and the intricate relationship between the DDF and PP properties. To address these challenges, we adopt the recently developed geometric quasilinearization approach [K. Wu and C.-W. Shu, SIAM Rev., 65 (2023), pp. 1031--1073], which transforms a nonlinear constraint into a family of linear constraints. Finally, we validate the effectiveness of the proposed method through several benchmark and demanding numerical examples. The results demonstrate that the proposed method is robust, accurate, and highly effective, confirming the significance of the proposed DDF projection and PP techniques. [ABSTRACT FROM AUTHOR]

A typo in the paper [M. Hladík, SIAM J. Matrix Anal. Appl., 44 (2023), pp. 175-195] is corrected. [ABSTRACT FROM AUTHOR]

This paper is the first in a series of two articles, aiming to prove the convergence of the HX preconditioner originally proposed by Hiptmair and Xu [SIAM J. Numer. Anal., 45 (2007), pp. 2483-2509] for Maxwell's equations with jump coefficients. In this paper, we establish several extensions of the discrete regular decomposition for edge finite element functions defined in threedimensional domains. The functions defined by the new discrete regular decompositions can inherit zero degrees of freedom of the considered edge finite element function on some faces and edges of polyhedral domains as well as of some non-Lipschitz domains and possess nearly optimal stability with only a logarithmic factor. [ABSTRACT FROM AUTHOR]

In the numerical simulation of ideal magnetohydrodynamics (MHD), keeping the pressure and density always positive is essential for both physical considerations and numerical stability. This is however a challenging task, due to the underlying relation between such a positivity-preserving (PP) property and the magnetic divergence-free (DF) constraint as well as the strong nonlinearity of the MHD equations. In this paper, we present the first rigorous PP analysis of the central discontinuous Galerkin (CDG) methods and construct arbitrarily high-order provably PP CDG schemes for ideal MHD. By the recently developed geometric quasilinearization (GQL) approach, our analysis reveals that the PP property of standard CDG methods is closely related to a discrete magnetic DF condition, whose form was unknown prior to our analysis and differs from that for the noncentral discontinuous Galerkin (DG) and finite volume methods in [K. Wu, SIAM J. Numer. Anal., 56 (2018), pp. 2124-2147]. The discovery of this relation lays the foundation for the design of our PP CDG schemes. In the one-dimensional (1D) case, the discrete DF condition is naturally satisfied, and we rigorously prove that the standard CDG method is PP under a condition that can be enforced easily with an existing PP limiter. However, in the multidimensional cases, the corresponding discrete DF condition is highly nontrivial, yet critical, and we analytically prove that the standard CDG method, even with the PP limiter, is not PP in general, as it generally fails to meet the discrete DF condition. We address this issue by carefully analyzing the structure of the discrete divergence terms and then constructing new locally DF CDG schemes for Godunov's modified MHD equations with an additional source term. The key point is to find out the suitable discretization of the source term such that it exactly cancels out all the terms in the discovered discrete DF condition. Based on the GQL approach, we prove in theory the PP property of the new multidimensional CDG schemes under a CFL condition. The robustness and accuracy of the proposed PP CDG schemes are further validated by several demanding 1D, two-dimensional, and three-dimensional numerical MHD examples, including the high-speed jets and blast problems with very low plasma beta. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the backward problem for nonlinear parabolic equations involving a pseudo-differential operator in the n-dimensional space. We prove that the problem is ill-posed in the sense of Hadamard, i.e., the solution, if it exists, does not depend continuously on the data. To regularize the problem, we propose two modified versions of the so-called optimal filtering method of Seidman [T.I. Seidman, Optimal filtering for the backward heat equation, SIAM J. Numer. Anal., 33 (1996), 162–170]. According to different a priori assumptions on the regularity of the exact solution, we obtain some sharp optimal estimates of the Hölder-Logarithmic type in the Sobolev space Hq(Rn). [ABSTRACT FROM AUTHOR]

In this paper, we extend the divergence-free VEM of [L. Beirão da Veiga, C. Lovadina and G. Vacca, Virtual elements for the Navier–Stokes problem on polygonal meshes, SIAM J. Numer. Anal.56 (2018) 1210–1242] to the Oseen problem, including a suitable stabilization procedure that guarantees robustness in the convection-dominated case without disrupting the divergence-free property. The stabilization is inspired from [N. Ahmed, G. R. Barrenechea, E. Burman, J. Guzman, A. Linke and C. Merdon, A pressure-robust discretization of Oseen's equation using stabilization in the vorticity equation, SIAM J. Numer. Anal.59 (2021) 2746–2774] and includes local SUPG-like terms of the vorticity equation, internal jump terms for the velocity gradients, and an additional VEM stabilization. We derive theoretical convergence results that underline the robustness of the scheme in different regimes, including the convection-dominated case. Furthermore, as in the non-stabilized case, the influence of the pressure on the velocity error is moderate, as it appears only through higher-order terms. [ABSTRACT FROM AUTHOR]

The classical porous medium equation is widely used to model different natural phenomena related to diffusion, filtration and heat propagation. In this short communication, we prove that the solution of porous medium equation can be locally approximated by the solution of a class of nonlocal dispersal equation. Our work is a counterpart to the important works (see Berestycki et al. [J. Funct. Anal. 271 (2016), pp. 2701–2751; J. Math. Biol. 72 (2016), pp. 1693–1745]; Dipierro et al. [J. Eur. Math. Soc. (JEMS) 19 (2017), pp. 957–966; J. Geom. Anal. 29 (2019), pp. 1428–1455]; Hansen and Netuka [Potential Anal. 2 (1993), pp. 67–71]; Ignat and Rossi [J. Funct. Anal. 251 (2007), pp. 399–437]; Shen and Xie [J. Differential Equations 259 (2015), pp. 7375–7405]; Sprekels and Valdinoci [SIAM J. Control Optim. 55 (2017), pp. 70–93]). [ABSTRACT FROM AUTHOR]

This erratum clarifies that equations (4.21) and (4.23) in Böttcher and Porter [SIAM J. Appl. Math., 81 (2021), pp. 2704-2724] assume that the underlying evolution operator has an orthonormal basis. The results and conclusions of the paper are unaffected by this clarification. [ABSTRACT FROM AUTHOR]

In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank–Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type [Van Kan, SIAM J. Sci. Statist. Comput.7 (1986) 870–891] in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete L∞(0, T; L2) norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper presents methods of spectral dichotomy of a matrix which compute spectral projectors on the subspace associated with the eigenvalues external to the parabolas described by a general equation. These methods are modifications of the one proposed in [A. N. Malyshev and M. Sadkane, SIAM J. MATRIX ANAL. APPL. 18 (2), 265-278, 1997] which uses the spectral dichotomy Theoretical and method of a matrix with respect to the imaginary axis. algorithmic aspects of the methods are developed. Numerical results obtained by applying methods presented on matrices are reported. [ABSTRACT FROM AUTHOR]

We analyze the recently introduced family of preconditioners in [M. M. Rana et al., SIAM J. Sci. Comput., 43 (2021), pp. S475-S495] for the stage equations of implicit Runge--Kutta methods for s-stage methods. We simplify the formulas for the eigenvalues and eigenvectors of the preconditioned systems for a general s-stage method and use these to obtain convergence rate estimates for preconditioned GMRES for some common choices of the implicit Runge-Kutta methods. This analysis is based on understanding the inherent matrix structure of these problems and exploiting it to qualitatively predict and explain the main observed features of the GMRES convergence behavior, using tools from approximation and potential theory based on Schwarz--Christoffel maps for curves and close, connected domains in the complex plane. We illustrate our analysis with numerical experiments showing very close correspondence of the estimates and the observed behavior, suggesting the analysis reliably captures the essence of these preconditioners. [ABSTRACT FROM AUTHOR]

We consider the linear degenerate wave equation, on the interval (0, 1) wtt - (x\α wx)x = p(t)m&$955;(x)w, with bilinear control p and Neumann boundary conditions. We study the controllability of this nonlinear control system, locally around a constant reference tra jectory, the "ground state." Under some classical and generic assumptions on \mu, we prove that there exists a threshold value for time, T\α = 2 1 \α, such that the reachable set is a neighborhood of the ground state. If T < T\α it is contained in a C1-submanifold of infinite codimension. Finally, it T = T\α and \α \in [0,1) it is a C1-submanifold of codimension 1 and if \α\in (1,2) the reachable set is a neighborhood of the ground state. The case \α = 1 remains open. This extends to the degenerate case the work [K. Beauchard, J. Differential Equations, 250 (2011), pp. 2064--2098] and adapts to bilinear controls the work [F. Alabau-Boussouira, P. Cannarsa, and G. Leugering, SIAM J. Control Optim., 55 (2017), pp. 2052--2087]. Our proofs are based on a careful analysis of the spectral problem and on Ingham type results, which are extensions of Kadec's 1/4 theorem. [ABSTRACT FROM AUTHOR]

Anomalous diffusion is often modelled in terms of the subdiffusion equation, which can involve a weakly singular source term. For this case, many predominant time-stepping methods, including the correction of high-order backward differentiation formula (BDF) schemes [B. Jin, B. Y. Li, and Z. Zhou, SIAM J. Sci. Comput., 39 (2017), pp. A3129-A3152], may suffer from a severe order reduction. To fill in this gap, we propose a smoothing method for time-stepping schemes, where the singular term is regularized by using an m-fold integral-differential calculus and the equation is discretized by the k-step BDF convolution quadrature, called the IDm-BDFk method. We prove that the desired kth-order convergence can be recovered even if the source term is weakly singular and the initial data is not compatible. Numerical experiments illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

We study the exploratory Hamilton-Jacobi-Bellman (HJB) equation arising from the entropy-regularized exploratory control problem, which was formulated by Wang, Zariphopoulou, and Zhou (J. Mach. Learn. Res., 21 (2020), 198) in the context of reinforcement learning in continuous time and space. We establish the well-posedness and regularity of the viscosity solution to the equation, as well as the convergence of the exploratory control problem to the classical stochastic control problem when the level of exploration decays to zero. We then apply the general results obtained to the exploratory temperature control problem, which was introduced by Gao, Xu, and Zhou (SIAM J. Control Optim., 60 (2022), pp. 1250-1268) to design an endogenous temperature schedule for simulated annealing in the context of nonconvex optimization. We derive an explicit rate of convergence for this problem as exploration diminishes to zero, and find that the stationary distribution of the optimally controlled process exists, which is however neither a Dirac mass on the global optimum nor a Gibbs measure. [ABSTRACT FROM AUTHOR]

Objective: The Global Lung Initiative (GLI)-2012 equation has now been used as the reference for the interpretation of the spirometry worldwide. The equations have been derived using modern statistical methods called Generalized Additive Models for Location, Scale and Shape (GAMLSS) and Spline. A part of the spirometric data from which the equations were derived is the data collected from the Thai population. The present study aimed to create the reference equations using only the data from the Thai population retrieved from the previous cross-sectional study and compared them to the GLI-2012’s. Materials and Methods: The previous data of each spirometric parameters including FVC, FEV₁, FEV₁/FVC, and FEF25-75% in 1,616 male and 2,280 female subjects were retrieved for the statistical modeling by GAMLSS and Spline packages in R Statistics. Lambda-Mu-Sigma (LMS) models were used to establish the best fitted models defined by the smallest Schwarz’s Bayesian Criterion (SBC). The authors also used a set of spirometric data of healthy elderly subjects to validate the equations and compare with the GLI-2012. Results: The GAMLSS-Thai equations were then derived to establish the predicted (Mu) and the lower limit of normal (LLN) values for each parameter. For 140 male and 651 female subjects aged 60 to 80 years, the predicted performance of GAMLSS-Thai is better than GLI-2012 equations as shown by various statistical tools including z-score, mean absolute percentage error (MAPE), and root mean square deviation (RMSD) in all spirometric parameters. Conclusion: The GAMLSS-Thai equation can be alternatively used as reference spirometric values in subjects aged 10 to 80 years in Thai population. It is more appropriate for subjects aged 60 to 80 years than the GLI-2012 equation. [ABSTRACT FROM AUTHOR]

This is a corrigendum to the article ``Error estimates for Galerkin approximations of the Serre equations"" by D. C. Antonopoulos, V. A. Dougalis, and D. E. Mitsotakis, SIAM J. Numer. Anal., 55 (2017), pp. 841--868. [ABSTRACT FROM AUTHOR]