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We are interested in the question of stability in the field of shape optimization, with focus on the strategy using second order shape derivative. More precisely, we identify structural hypotheses on the hessian of the considered shape…
We study optimal design problems involving variational inequalities with unilateral conditions in the domain and pointwise boundary observation. We use regularizing and penalization tehniques in the setting of the Hamiltonian approach to…
We provide a framework for the numerical approximation of distributed optimal control problems, based on least-squares finite element methods. Our proposed method simultaneously solves the state and adjoint equations and is $\inf$--$\sup$…
In this paper we prove the existence of an optimal domain $\Omega_{opt}$ for the shape optimization problem $$\max\Big\{\lambda_q(\Omega)\ :\ \Omega\subset D,\ \lambda_p(\Omega)=1\Big\},$$ where $q<p$ and $D$ is a prescribed bounded subset…
Recently the authors have studied a state and control constrained optimal control problem with fractional elliptic PDE as constraints. The goal of this paper is to continue that program forward and introduce an algorithm to solve such…
In this thesis we develop a functional analytic framework for shape optimization with elliptic partial differential equation (PDE) constraints in classical function spaces (H\"older spaces). This approach is motivated by shape optimization…
In this article we shall study the following elliptic system with coefficients: \begin{equation}\notag \left\{\begin{aligned} -\epsilon^2\Delta u +c(x)u=b(x)|v|^{q-1}v, &\text{ and } -\epsilon^2\Delta v +c(x)v=a(x) |u|^{p-1}u &&\text{in }…
This work studies a variational formulation and numerical solution of a regularized morphoelasticity problem of shape evolution. The foundation of our analysis is based on the governing equations of linear elasticity, extended to account…
We consider optimization problems constrained by partial differential equations (PDEs) with additional constraints placed on the solution of the PDEs. We develop a general and versatile framework using infinite-valued penalization functions…
We study the problem of maximizing the geometric mean of $d$ low-degree non-negative forms on the real or complex sphere in $n$ variables. We show that this highly non-convex problem is NP-hard even when the forms are quadratic and is…
Visualizing graphs using virtual physical models is probably the most heavily used technique for drawing graphs in practice. There are many algorithms that are efficient and produce high-quality layouts. If one requires that the layout also…
In this article, we present two different approaches for obtaining quantitative inequalities in the context of parabolic optimal control problems. Our model consists of a linearly controlled heat equation with Dirichlet boundary condition…
We consider the following eigenvalue optimization in the composite membrane problem with fractional Laplacian: given a bounded domain $\Omega\subset \mathbb{R}^n$, $\alpha>0$ and $0<A<|\Omega|$, find a subset $D\subset \Omega$ of area $A$…
This paper is concerned with error estimates for the numerical approximation for affine optimal control problems subject to semilinear elliptic PDEs. To investigate the error estimates, we focus on local minimizers that satisfy certain…
In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) $\min \{x^* C x…
We consider the existence of optimal shapes in the context of the thermomechanical system of partial differential equations (PDE) using the recent approach based on elliptic regularity theory. We give an extended and improved definition of…
One of the most common problems of scientific applications is computation of the derivative of a function specified by possibly noisy or imprecise experimental data. Application of conventional techniques for numerically calculating…
We consider the unit ball $\Omega\subset \mathbb{R}^N$ ($N\ge2$) filled with two materials with different conductivities. We perform shape derivatives up to the second order to find out precise information about locally optimal…
We generalize the shape optimization problem for the existence of stable equilibrium configurations of nematic and cholesteric liquid crystal drops surrounded by an isotropic solution to include a broader family of admissible domains with…
In this paper we study the regularity of the optimal sets for the shape optimization problem \[ \min\Big\{\lambda_1(\Omega)+\dots+\lambda_k(\Omega)\ :\ \Omega\subset\mathbb{R}^d,\ \text{open}\ ,\ |\Omega|=1\Big\}, \] where…