Near-circularity for the rational Zolotarev problem in the complex plane

AbstractWe consider the rational Zolotarev problemminrϵRHmaxzϵE|r(Z)|minzϵF|r(Z)| for compact sets E,F⊈C, where Rll denotes the set of all rational functions of degree ⩽l. This problem is of importance, e.g., for the determination of optimal parameters for the method of alternating directions (ADI method) which is used for the iterative solution of large linear systems. For E and F being real intervals, the solution of this problem was given explicitly in terms of elliptic functions by Zolotarev in the last century. For complex domains, however, little is known as yet about this problem. In this paper, after reviewing some results on the asymptotic behavior, we prove a result which is similar to the near-circularity criterion as it is well known in connection to classical approximation by polynomials or rational functions. If we assume that both sets E and F are bounded by Jordan curves, this gives us a lower bound for the minimal value in the rational Zolotarev problem. Moreover, we derive upper bounds for the modulus of the doubly connected region D: Ĉ\(E∪:F) and show how the near-circularity criterion can be used for the construction of the rational minimal solutions for small degrees

Shape Optimization by Constrained First-Order Least Mean Approximation

In this work, the problem of shape optimization, subject to PDE constraints, is reformulated as an $L^p$ best approximation problem under divergence constraints to the shape tensor introduced in Laurain and Sturm: ESAIM Math. Model. Numer. Anal. 50 (2016). More precisely, the main result of this paper states that the $L^p$ distance of the above approximation problem is equal to the dual norm of the shape derivative considered as a functional on $W^{1,p^\ast}$ (where $1/p + 1/p^\ast = 1$). This implies that for any given shape, one can evaluate its distance from being a stationary one with respect to the shape derivative by simply solving the associated $L^p$-type least mean approximation problem. Moreover, the Lagrange multiplier for the divergence constraint turns out to be the shape deformation of steepest descent. This provides a way, as an alternative to the approach by Deckelnick, Herbert and Hinze: ESAIM Control Optim. Calc. Var. 28 (2022), for computing shape gradients in $W^{1,p^\ast}$ for $p^\ast \in ( 2 , \infty )$. The discretization of the least mean approximation problem is done with (lowest-order) matrix-valued Raviart-Thomas finite element spaces leading to piecewise constant approximations of the shape deformation acting as Lagrange multiplier. Admissible deformations in $W^{1,p^\ast}$ to be used in a shape gradient iteration are reconstructed locally. Our computational results confirm that the $L^p$ distance of the best approximation does indeed measure the distance of the considered shape to optimality. Also confirmed by our computational tests are the observations that choosing $p^\ast$ (much) larger than 2 (which means that $p$ must be close to 1 in our best approximation problem) decreases the chance of encountering mesh degeneracy during the shape gradient iteration.Comment: 20 pages, 8 figure

Weakly symmetric stress equilibration for hyperelastic materialmodels

A stress equilibration procedure for hyperelastic material models is proposed andanalyzed in this paper. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs, in a vertex-patch-wise manner, an $H(div)$-conforming approximation to the first Piola-Kirchhoff stress. This is done in such a way that its associated Cauchy stress is weakly symmetric in the sense that its anti-symmetric part is zero tested against continuous piecewise linear functions. Our main result is the identification of the subspace of test functions perpendicular to the range of the local equilibration system on each patch which turn out to be rigid body modes associated with the current configuration. Momentum balance properties are investigated analytically and numerically and the resulting stress reconstruction is shown to provide improved results for surface traction forces by computational experiments

Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity

A stress equilibration procedure for linear elasticity is proposed and analyzed in this paper with emphasis on the behavior for (nearly) incompressible materials. Based on the displacement-pressure approximation computed with a stable finite element pair, it constructs an $H (\text{div})$-conforming, weakly symmetric stress reconstruction. Our focus is on the Taylor-Hood combination of continuous finite element spaces of polynomial degrees $k+1$ and $k$ for the displacement and the pressure, respectively. Our construction leads then to reconstructed stresses by Raviart-Thomas elements of degree $k$ which are weakly symmetric in the sense that its anti-symmetric part is zero tested against continuous piecewise polynomial functions of degree $k$. The computation is performed locally on a set of vertex patches covering the computational domain in the spirit of equilibration \cite{BraSch:08}. Due to the weak symmetry constraint, the local problems need to satisfy consistency conditions associated with all rigid body modes, in contrast to the case of Poisson's equation where only the constant modes are involved. The resulting error estimator is shown to constitute a guaranteed upper bound for the error with a constant that depends only on the shape regularity of the triangulation. Local efficiency, uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator

Multilevel preconditioning based on discrete symmetrization for convection-diffusion equations

AbstractThe subject of this paper is an additive multilevel preconditioning approach for convection-diffusion problems. Our particular interest is in the convergence behavior for convection-dominated problems which are discretized by the streamline diffusion method. The multilevel preconditioner is based on a transformation of the discrete problem which reduces the relative size of the skew-symmetric part of the operator. For the constant coefficient case, an analysis of the convergence properties of this multilevel preconditioner is given in terms of its dependence on the convection size. Moreover, the results of computational experiments for more general convection-diffusion problems are presented and our new preconditioner is compared to standard multilevel preconditioning

Redox regulation of ischemic limb neovascularization - What we have learned from animal studies

Mouse hindlimb ischemia has been widely used as a model to study peripheral artery disease. Genetic modulation of the enzymatic source of oxidants or components of the antioxidant system reveal that physiological levels of oxidants are essential to promote the process of arteriogenesis and angiogenesis after femoral artery occlusion, although mice with diabetes or atherosclerosis may have higher deleterious levels of oxidants. Therefore, fine control of oxidants is required to stimulate vascularization in the limb muscle. Oxidants transduce cellular signaling through oxidative modifications of redox sensitive cysteine thiols. Of particular importance, the reversible modification with abundant glutathione, called S-glutathionylation (or GSH adducts), is relatively stable and alters protein function including signaling, transcription, and cytoskeletal arrangement. Glutaredoxin-1 (Glrx) is an enzyme which catalyzes reversal of GSH adducts, and does not scavenge oxidants itself. Glrx may control redox signaling under fluctuation of oxidants levels. In ischemic muscle increased GSH adducts through Glrx deletion improves in vivo limb revascularization, indicating endogenous Glrx has anti-angiogenic roles. In accordance, Glrx overexpression attenuates VEGF signaling in vitro and ischemic vascularization in vivo. There are several Glrx targets including HIF-1α which may contribute to inhibition of vascularization by reducing GSH adducts. These animal studies provide a caution that excess antioxidants may be counter-productive for treatment of ischemic limbs, and highlights Glrx as a potential therapeutic target to improve ischemic limb vascularization