Related papers: Second-order Shape Optimization for Geometric Inve…
In this paper we present a shape optimization scheme which utilizes the alternating direction method of multipliers (ADMM) to approximate a direction of steepest descent in $W^{1,\infty}$. The followed strategy is a combination of the…
This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic…
Many data-fitting applications require the solution of an optimization problem involving a sum of large number of functions of high dimensional parameter. Here, we consider the problem of minimizing a sum of $n$ functions over a convex…
Optimization under structural constraints is typically analyzed through projection or penalty methods, obscuring the geometric mechanism by which constraints shape admissible dynamics. We propose an operator-theoretic formulation in which…
We consider the inverse obstacle scattering problem of determining both the shape and the "equivalent impedance" from far field measurements at a fixed frequency. In this work, the surface impedance is represented by a second order surface…
Most problems involving simultaneous localization and mapping can nowadays be solved using one of two fundamentally different approaches. The traditional approach is given by a least-squares objective, which minimizes many local photometric…
For the numerical solution of shape optimization problems, particularly those constrained by partial differential equations (PDEs), the quality of the underlying mesh is of utmost importance. Particularly when investigating complex…
We propose a robust numerical framework for PDE-constrained shape optimization and Willmore-driven surface hole filling. To address two central challenges -- slow progress in flat energy landscapes, which can trigger premature stagnation at…
Shape constraints, such as non-negativity, monotonicity, convexity or supermodularity, play a key role in various applications of machine learning and statistics. However, incorporating this side information into predictive models in a hard…
An analogue of the total variation prior for the normal vector field along the boundary of piecewise flat shapes in 3D is introduced. A major class of examples are triangulated surfaces as they occur for instance in finite element…
We present an extension of an algorithm for the classical scalar $p$-Laplace Dirichlet problem to the vector-valued $p$-Laplacian with mixed boundary conditions in order to solve problems occurring in shape optimization using a $p$-harmonic…
We consider shape and topology optimization for fluids which are governed by the Navier--Stokes equations. Shapes are modelled with the help of a phase field approach and the solid body is relaxed to be a porous medium. The phase field…
We present a new approach to discretizing shape optimization problems that generalizes standard moving mesh methods to higher-order mesh deformations and that is naturally compatible with higher-order finite element discretizations of…
This paper describes a class of shape optimization problems for optical metamaterials comprised of periodic microscale inclusions composed of a dielectric, low-dimensional material suspended in a non-magnetic bulk dielectric. The shape…
We give curvature-dependant convergence rates for the optimization of weakly convex functions defined on a manifold of 1-bounded geometry via Riemannian gradient descent and via the dynamic trivialization algorithm. In order to do this, we…
We prove gradient estimates for hypersurfaces in the hyperbolic space $\mathbb{H}^{n+1},$ expanding by negative powers of a certain class of homogeneous curvature functions. We obtain optimal gradient estimates for hypersurfaces evolving by…
We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like…
This paper investigates a category of constrained fractional optimization problems that emerge in various practical applications. The objective function for this category is characterized by the ratio of a numerator and denominator, both…
We present a mathematical and algorithmic scheme for learning the principal geometric elements in an image or 3D object. We build on recent work that convexifies the basic problem of finding a combination of a small number shapes that…
We present a geometric multilevel optimization approach that smoothly incorporates box constraints. Given a box constrained optimization problem, we consider a hierarchy of models with varying discretization levels. Finer models are…