Scalable shape optimization methods for structured inverse modeling in 3D diffusive processes

Abstract

In this work we consider inverse modeling of the shape of cells in the outermost layer of human skin. We propose a novel algorithm that combines mathematical shape optimization with high-performance computing. Our aim is to fit a parabolic model for drug diffusion through the skin to data measurements. The degree of freedom is not the permeability itself, but the shape that distinguishes regions of high and low diffusivity. These are the cells and the space in between. The key part of the method is the computation of shape gradients, which are then applied as deformations to the finite element mesh, in order to minimize a tracking type objective function. Fine structures in the skin require a very high resolution in the computational model. We therefor investigate the scalability of our algorithm up to millions of discretization elements.

Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization

“…Shape optimization is a challenging field with many interesting applications. As examples, we mention aerodynamic shape optimization [15], acoustic shape optimization [21] or optimization of interfaces in transmission problems [7,11,12]. The general structure of such a a PDE constrained shape optimization problem is of the form where Ω ⊂ D is an open subset of a hold-all domain D ⊂ R d and J is a real valued functional.…”

Efficient PDE Constrained Shape Optimization Based on Steklov--Poincaré-Type Metrics

“…This changes for highly parallel application on supercomputers as investigated in [17]. Operations, which are only performed on surfaces, can drastically affect the scalability of the overall algorithm if the computational load is not balanced also with respect to surface elements.…”

“…Shape optimization is of interest in many fields of application -in particular in the context of partial differential equations (PDE). As examples, we mention aerodynamic shape optimization [22], acoustic shape optimization [30] or optimization of interfaces in transmission problems [10,18,20] and in electrostatics [4]. In industry, shapes are often represented within a finite dimensional design space.…”

Mesh Quality Preserving Shape Optimization Using Nonlinear Extension Operators

“…Possible variants are that the outer shape of is to be determined, e.g., when represents a solid body, or interior interfaces, which separate spatially discontinuous coefficients such as material properties. Shape optimization in general is nowadays an active field of research with applications ranging from magnetostatics [8], interface identification in transmission processes [12,22,27], fluid dynamics [2,9,25], acoustics [31], image restoration and segmentation [14] and composite material identification [23,28] to nano-optics [15].…”