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The paper considers grad-div stabilized equal-order finite elements (FE) methods for the linearized Navier-Stokes equations. A block triangular preconditioner for the resulting system of algebraic equations is proposed which is closely related to the Augmented Lagrangian (AL) preconditioner. A field-of-values analysis of a preconditioned Krylov subspace method shows convergence bounds that are independent of the mesh parameter variation. Numerical studies support the theory and demonstrate the robustness of the approach also with respect to the viscosity parameter variation, as is typical for AL preconditioners when applied to inf-sup stable FE pairs. The numerical experiments also address the accuracy of grad-div stabilized equal-order FE method for the steady state Navier-Stokes equations.

This paper delves into the equivalence problem of Smith forms for multivariate polynomial matrices. Generally speaking, multivariate ($n \geq 2$) polynomial matrices and their Smith forms may not be equivalent. However, under certain specific condition, we derive the necessary and sufficient condition for their equivalence. Let $F\in K[x_1,\ldots,x_n]^{l\times m}$ be of rank $r$, $d_r(F)\in K[x_1]$ be the greatest common divisor of all the $r\times r$ minors of $F$, where $K$ is a field, $x_1,\ldots,x_n$ are variables and $1 \leq r \leq \min\{l,m\}$. Our key findings reveal the result: $F$ is equivalent to its Smith form if and only if all the $i\times i$ reduced minors of $F$ generate $K[x_1,\ldots,x_n]$ for $i=1,\ldots,r$.

In this paper we give local error estimates in Sobolev norms for the Galerkin method applied to strongly elliptic pseudodifferential equations on a polygon. By using the K-operator, an operator which averages the values of the Galerkin solution, we construct improved approximations.

This paper leans on two similar areas so far detached from each other. On the one hand, Dung's pioneering contributions to abstract argumentation, almost thirty years ago, gave rise to a plethora of successors, including abstract dialectical frameworks (ADFs). On the other hand, Boolean networks (BNs), devised as models of gene regulation, have been successful for studying the behavior of molecular processes within cells. ADFs and BNs are similar to each other: both can be viewed as functions from vectors of bits to vectors of bits. As soon as similarities emerge between these two formalisms, however, differences appear. For example, conflict-freedom is prominent in argumentation (where we are interested in a self-consistent, i.e., conflict-free, set of beliefs) but absent in BNs. By contrast, asynchrony (where only one gene is updated at a time) is conspicuous in BNs and lacking in argumentation. Finally, while a monotonicity-based notion occurs in signed reasoning of both argumentation and gene regulation, a different, derivative-based notion only appears in the BN literature. To identify common mathematical structure between both formalisms, these differences need clarification. This contribution is a partial review of both these areas, where we cover enough ground to exhibit their more evident similarities, to then reconcile some of their apparent differences. We highlight a range of avenues of research resulting from ironing out discrepancies between these two fields. Unveiling their common concerns should enable these two areas to cross-fertilize so as to transfer ideas and results between each other.

This paper deals with the numerical solution of conservation laws in the two dimensional case using a novel compact implicit time discretization that enable applications of fast algebraic solvers. We present details for the second order accurate parametric scheme based on the finite volume method including simple variants of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) approximations. We present numerical experiments for representative linear and nonlinear problems.

In this paper, conditional stability estimates are derived for unique continuation and Cauchy problems associated to the Poisson equation in ultra-weak variational form. Numerical approximations are obtained as minima of regularized least squares functionals. The arising dual norms are replaced by discretized dual norms, which leads to a mixed formulation in terms of trial- and test-spaces. For stable pairs of such spaces, and a proper choice of the regularization parameter, the $L_2$-error on a subdomain in the obtained numerical approximation can be bounded by the best possible fractional power of the sum of the data error and the error of best approximation. Compared to the use of a standard variational formulation, the latter two errors are measured in weaker norms. To avoid the use of $C^1$-finite element test spaces, nonconforming finite element test spaces can be applied as well. They either lead to the qualitatively same error bound, or in a simplified version, to such an error bound modulo an additional data oscillation term. Numerical results illustrate our theoretical findings.

This paper deals with a novel nonlinear coupled nonlocal reaction-diffusion system proposed for image restoration, characterized by the advantages of preserving low gray level features and textures.The gray level indicator in the proposed model is regularized using a new method based on porous media type equations, which is suitable for recovering noisy blurred images. The well-posedness, regularity, and other properties of the model are investigated, addressing the lack of theoretical analysis in those existing similar types of models. Numerical experiments conducted on texture and satellite images demonstrate the effectiveness of the proposed model in denoising and deblurring tasks.

This paper addresses second-order stochastic optimization for estimating the minimizer of a convex function written as an expectation. A direct recursive estimation technique for the inverse Hessian matrix using a Robbins-Monro procedure is introduced. This approach enables to drastically reduces computational complexity. Above all, it allows to develop universal stochastic Newton methods and investigate the asymptotic efficiency of the proposed approach. This work so expands the application scope of secondorder algorithms in stochastic optimization.

This paper discusses lowest-order nonstandard finite element methods for space discretization and explicit and implicit schemes for time discretization of the biharmonic wave equation with clamped boundary conditions. A modified Ritz projection operator defined on $H^2_0(\Omega)$ ensures error estimates under appropriate regularity assumptions on the solution. Stability results and error estimates of optimal order are established in suitable norms for the semidiscrete and explicit/implicit fully-discrete versions of the proposed schemes. Finally, we report on numerical experiments using explicit and implicit schemes for time discretization and Morley, discontinuous Galerkin, and {C$^0$ interior} penalty schemes for space discretization, that validate the theoretical error estimates.