Related papers: Fully and Semi-Automated Shape Differentiation in …
In the reduced order modeling (ROM) framework, the solution of a parametric partial differential equation is approximated by combining the high-fidelity solutions of the problem at hand for several properly chosen configurations. Examples…
In this paper, we consider a class of finite-sum convex optimization problems whose objective function is given by the summation of $m$ ($\ge 1$) smooth components together with some other relatively simple terms. We first introduce a…
Unfitted (also known as embedded or immersed) finite element approximations of partial differential equations are very attractive because they have much lower geometrical requirements than standard body-fitted formulations. These schemes do…
In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their…
Surrogate neural network-based partial differential equation (PDE) solvers have the potential to solve PDEs in an accelerated manner, but they are largely limited to systems featuring fixed domain sizes, geometric layouts, and boundary…
We propose efficient numerical schemes for implementing the natural gradient descent (NGD) for a broad range of metric spaces with applications to PDE-based optimization problems. Our technique represents the natural gradient direction as a…
In this paper, we consider both first- and second-order techniques to address continuous optimization problems arising in machine learning. In the first-order case, we propose a framework of transition from deterministic or…
Semi-Lagrangian methods are numerical methods designed to find approximate solutions to particular time-dependent partial differential equations (PDEs) that describe the advection process. We propose semi-Lagrangian one-step methods for…
Dual decomposition is a powerful technique for deriving decomposition schemes for convex optimization problems with separable structure. Although the Augmented Lagrangian is computationally more stable than the ordinary Lagrangian, the…
We present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology. We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape…
PDEs are central to scientific and engineering modeling, yet designing accurate numerical solvers typically requires substantial mathematical expertise and manual tuning. Recent neural network-based approaches improve flexibility but often…
We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is…
Recent works have shown that deep neural networks can be employed to solve partial differential equations, giving rise to the framework of physics informed neural networks. We introduce a generalization for these methods that manifests as a…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
Most of today's state-of-the-art methods for perspective shape from shading are modelled in terms of partial differential equations (PDEs) of Hamilton-Jacobi type. To improve the robustness of such methods w.r.t. noise and missing data,…
We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate…
This paper presents a quantum algorithm for the solution of prototypical second-order linear elliptic partial differential equations discretized by $d$-linear finite elements on Cartesian grids of a bounded $d$-dimensional domain. An…
In order to investigate correspondences between 3D shapes, many methods rely on a feature descriptor which is invariant under almost isometric transformations. An interesting class of models for such descriptors relies on partial…
We consider structured optimisation problems defined in terms of the sum of a smooth and convex function, and a proper, l.s.c., convex (typically non-smooth) one in reflexive variable exponent Lebesgue spaces $L_{p(\cdot)}(\Omega)$. Due to…
The inverse conductivity problem aims at determining the unknown conductivity inside a bounded domain from boundary measurements. In practical applications, algorithms based on minimizing a regularized residual functional subject to PDE…