scholarly journals A projected super-penalty method for the $$C^1$$-coupling of multi-patch isogeometric Kirchhoff plates

2021 ◽  
Vol 67 (4) ◽  
pp. 1133-1153
Author(s):  
Luca Coradello ◽  
Gabriele Loli ◽  
Annalisa Buffa
Keyword(s):  
Rates Of Convergence ◽  
Saddle Point Problem ◽  
Point Problem ◽  
B Splines ◽  
Kirchhoff Plates ◽  
Optimal Accuracy ◽  
Conforming Meshes ◽  
Coarse Meshes ◽  
Coupling Algorithm ◽  
Penalty Strategy

AbstractThis work focuses on the development of a super-penalty strategy based on the $$L^2$$ L 2 -projection of suitable coupling terms to achieve $$C^1$$ C 1 -continuity between non-conforming multi-patch isogeometric Kirchhoff plates. In particular, the choice of penalty parameters is driven by the underlying perturbed saddle point problem from which the Lagrange multipliers are eliminated and is performed to guarantee the optimal accuracy of the method. Moreover, by construction, the method does not suffer from boundary locking, especially on very coarse meshes. We demonstrate the applicability of the proposed coupling algorithm to Kirchhoff plates by studying several benchmark examples discretized by non-conforming meshes. In all cases, we recover the optimal rates of convergence achievable by B-splines where we achieve a substantial gain in accuracy per degree-of-freedom compared to other choices of the penalty parameters.

scholarly journals Computational aspects of the weak micro‐periodicity saddle point problem

PAMM ◽  
2021 ◽  
Vol 20 (1) ◽  
Author(s):  
Ritukesh Bharali ◽  
Fredrik Larsson ◽  
Ralf Jänicke
Keyword(s):  
Saddle Point ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Computational Aspects

Shape and parameter reconstruction for the Robin transmission inverse problem

2016 ◽  
Vol 24 (6) ◽  
Author(s):  
Antoine Laurain ◽  
Houcine Meftahi
Keyword(s):  
Inverse Problem ◽  
Shape Optimization ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Optimization Approach ◽  
Reconstruction Method ◽  
Piecewise Constant ◽  
Piecewise Constant Conductivity ◽  
Level Set Approach ◽  
Parameter Reconstruction

AbstractIn this paper we consider the inverse problem of simultaneously reconstructing the interface where the jump of the conductivity occurs and the Robin parameter for a transmission problem with piecewise constant conductivity and Robin-type transmission conditions on the interface. We propose a reconstruction method based on a shape optimization approach and compare the results obtained using two different types of shape functionals. The reformulation of the shape optimization problem as a suitable saddle point problem allows us to obtain the optimality conditions by using differentiability properties of the min-sup combined with a function space parameterization technique. The reconstruction is then performed by means of an iterative algorithm based on a conjugate shape gradient method combined with a level set approach. To conclude we give and discuss several numerical examples.

Accelerated Methods for Saddle-Point Problem

2020 ◽  
Vol 60 (11) ◽  
pp. 1787-1809
Author(s):  
M. S. Alkousa ◽  
A. V. Gasnikov ◽  
D. M. Dvinskikh ◽  
D. A. Kovalev ◽  
F. S. Stonyakin
Keyword(s):  
Saddle Point ◽  
Saddle Point Problem ◽  
Point Problem ◽  
Accelerated Methods

scholarly journals Numerical solutions of symmetric saddle point problem by direct methods

2013 ◽  
Vol 46 (3) ◽  
Author(s):  
Alicja Smoktunowicz ◽  
Felicja Okulicka-Dłużewska
Keyword(s):  
Saddle Point ◽  
Numerical Solutions ◽  
Direct Methods ◽  
Qr Decomposition ◽  
Saddle Point Problem ◽  
Singular Value ◽  
Point Problem ◽  
Numerical Comparison ◽  
Augmented System ◽  
Value Decomposition

AbstractNumerical stability of two main direct methods for solving the symmetric saddle point problem are analyzed. The first one is a generalization of Golub’s method for the augmented system formulation (ASF) and uses the Householder QR decomposition. The second method is supported by the singular value decomposition (SVD). Numerical comparison of some direct methods are given.

On first-order algorithms for l1/nuclear norm minimization

Acta Numerica ◽  
2013 ◽  
Vol 22 ◽  
pp. 509-575 ◽  
Author(s):  
Yurii Nesterov ◽  
Arkadi Nemirovski
Keyword(s):  
Saddle Point ◽  
Gradient Methods ◽  
Saddle Point Problem ◽  
Optimization Techniques ◽  
Nuclear Norm ◽  
Point Problem ◽  
Nuclear Norm Minimization ◽  
Dantzig Selector ◽  
First Order ◽  
Norm Minimization

In the past decade, problems related to l1/nuclear norm minimization have attracted much attention in the signal processing, machine learning and optimization communities. In this paper, devoted to l1/nuclear norm minimization as ‘optimization beasts’, we give a detailed description of two attractive first-order optimization techniques for solving problems of this type. The first one, aimed primarily at lasso-type problems, comprises fast gradient methods applied to composite minimization formulations. The second approach, aimed at Dantzig-selector-type problems, utilizes saddle-point first-order algorithms and reformulation of the problem of interest as a generalized bilinear saddle-point problem. For both approaches, we give complete and detailed complexity analyses and discuss the application domains.

scholarly journals A change-point problem and inference for segment signals

2018 ◽  
Vol 22 ◽  
pp. 210-235
Author(s):  
Victor-Emmanuel Brunel
Keyword(s):  
Change Point ◽  
Rates Of Convergence ◽  
Extensive Study ◽  
Change Points ◽  
Point Problem ◽  
Minimal Size ◽  
Detection And Estimation ◽  
Change Point Problem ◽  
Minimax Rates ◽  
The Impact

We address the problem of detection and estimation of one or two change-points in the mean of a series of random variables. We use the formalism of set estimation in regression: to each point of a design is attached a binary label that indicates whether that point belongs to an unknown segment and this label is contaminated with noise. The endpoints of the unknown segment are the change-points. We study the minimal size of the segment which allows statistical detection in different scenarios, including when the endpoints are separated from the boundary of the domain of the design, or when they are separated from one another. We compare this minimal size with the minimax rates of convergence for estimation of the segment under the same scenarios. The aim of this extensive study of a simple yet fundamental version of the change-point problem is two-fold: understanding the impact of the location and the separation of the change points on detection and estimation and bringing insights about the estimation and detection of convex bodies in higher dimensions.

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