arXiv subjects

Aikaterini Aretaki

Publications and source records attributed to Aikaterini Aretaki.

6 recordsLinked to original sources

arXiv2022

Random geometries for optimal control PDE problems based on fictitious domain FEMS and cut elements

This work investigates an elliptic optimal control problem defined on uncertain domains and discretized by a fictitious domain finite element method and cut elements. Key ingredients of the study are to manage cases considering the usually computationally "forbidden" combination of poorly conditioned equation system matrices due to challenging geometries, optimal control searches with iterative methods, slow convergence to system solutions on deterministic and non--deterministic level, and expensive remeshing due to geometrical changes. We overcome all these difficulties, utilizing the advantages of proper preconditioners adapted to unfitted mesh methods, improved types of Monte Carlo methods, and mainly employing the advantages of embedded FEMs, based on a fixed background mesh computed once even if geometrical changes are taking place. The sensitivity of the control problem is introduced in terms of random domains, employing a Quasi-Monte Carlo method and emphasizing a deterministic target state. The variational discretization concept is adopted, optimal error estimates for the state, adjoint state, and control are derived that confirm the efficiency of the cut finite element method in challenging geometries. The performance of a multigrid scheme especially developed for unfitted finite element discretizations adapted to the optimal control problem is also tested. Some fundamental preconditioners are applied to the arising sparse linear systems coming from the discretization of the state and adjoint state variational forms in the spatial domain. The corresponding convergence rates along with the quality of the prescribed preconditioners are verified by numerical examples.

math.NA↗

arXiv2022

Equal higher order analysis of an unfitted discontinuous Galerkin method for Stokes flow systems

In this work, we analyze an unfitted discontinuous Galerkin discretization for the numerical solution of the Stokes system based on equal higher-order discontinuous velocities and pressures. This approach combines the best from both worlds, firstly the advantages of a piece-wise discontinuous high-order accurate approximation and secondly the advantages of an unfitted to the true geometry grid around possibly complex objects and/or geometrical deformations. Utilizing a fictitious domain framework, the physical domain of interest is embedded in an unfitted background mesh and the geometrically unfitted discretization is built upon symmetric interior penalty discontinuous Galerkin formulation. To enhance stability we enrich the discrete variational formulation with a pressure stabilization term. Moreover, the present contribution adopts high order ghost penalty strategies to address the ill conditioning of the system matrix caused by small truncated elements with respect to the unfitted boundary. Motivated by continuous unfitted FEM [21,74,75] along with other unfitted mesh surveys grounded on discontinuous spaces [10,44,45,73], we use proper velocity and pressure ghost penalties defined on faces of cut cells to establish a robust high-order method, in spite of the cell agglomeration technique usually applied on dG methods. The current presentation should prove valuable in engineering applications where special emphasis is placed on the optimal effective approximation attaining much smaller relative errors in coarser meshes. Inf-sup stability, the optimal order of convergence, and the condition number sensitivity with respect to cut configuration are investigated. Numerical examples verify the theoretical results.

math.NA↗

arXiv2012

The k-rank numerical radii

The $k$-rank numerical range $Λ_{k}(A)$ is expressed via an intersection of a countable family of numerical ranges $\{F(M^{*}_νAM_ν)\}_{ν\in\mathbb{N}}$ with respect to $n\times (n-k+1)$ isometries $M_ν$. This implication for $Λ_{k}(A)$ provides further elaboration of the $k$-rank numerical radii of $A$.

math.FA↗

arXiv2011

The higher rank numerical range of nonnegative matrices

In this article the well known "Perron-Frobenius theory" is investigated involving the higher rank numerical range $Λ_{k}(A)$ of an irreducible and entrywise nonnegative matrix $A$ and extending the notion of elements of maximum modulus in $Λ_{k}(A)$. Further, an application of this theory to the $Λ_{k}(L(λ))$ of a Perron polynomial $L(λ)$ is elaborated via its companion matrix $C_{L}$.

math.RA↗

arXiv2011

The higher rank numerical range of matrix polynomials

The notion of the higher rank numerical range $Λ_{k}(L(λ))$ for matrix polynomials $L(λ)=A_{m}λ^{m}+...+A_{1}λ+A_{0}$ is introduced here and some fundamental geometrical properties are investigated. Further, the sharp points of $Λ_{k}(L(λ))$ are defined and their relation to the numerical range $w(L(λ))$ is presented. A connection of $Λ_{k}(L(λ))$ with the vector-valued higher rank numerical range $Λ_{k}(A_{0},..., A_{m})$ is also discussed.

math.RA↗

arXiv2009

Investigating the Numerical Range of Non Square Matrices

A presentation of numerical range for rectangular matrices is undertaken in this paper, introducing two different definitions and elaborating basic properties. Then we are extended to the treatment of rank-k numerical range.

math.FA↗