A deterministic approach to the adapted optode placement for illumination of highly scattering tissue

Abstract

A novel approach is presented for computing optode placements that are adapted to specific geometries and tissue characteristics, e.g., in optical tomography and photodynamic cancer therapy. The method is based on optimal control techniques together with a sparsity-promoting penalty that favors pointwise solutions, yielding both locations and magnitudes of light sources. In contrast to current discrete approaches, the need for specifying an initial set of candidate configurations as well as the exponential increase in complexity with the number of optodes are avoided. This is demonstrated with computational examples from photodynamic therapy.

A Priori Error Analysis for Discretization of Sparse Elliptic Optimal Control Problems in Measure Space

“…If Ω c and Ω o are disjoint, we can typically expect the optimal control to be a linear combination of Dirac delta functions; cf. [5] for an application, where this is explicitly desired.…”

Sparse Solutions in Optimal Control of PDEs with Uncertain Parameters: The Linear Case

“…However, since we interpret regions where optimal controls are non-zero as regions where we propose to place control devices, it is more meaningful to require that the stochastic controls share their sparsity structure. This can be achieved by adding a sparsity-enforcing term to the objective functional in (4): (5) Here, β > 0 and the outer integral in the sparsity-enforcing term is over the pointwise marginal distribution of the squared controls. Note that the sparsity term is well-defined and finite for u ∈ U ad .…”

“…While the optimal controls are stochastic, i.e., they depend on ω, the controller locations resulting from (5) are deterministic, i.e., they only depend on the probability space, but not the individual event ω ∈ Ω. A practical interpretation of this approach is that the optimal location of controllers is computed by solving (5) in an offline phase, while the optimal controls u(ω) are computed in an online phase corresponding to the particular realization of the random variable ω. Let us give two application examples for an optimal control formulation of the form (21).…”

“…In this paper, we aim at optimal control problems under uncertainty, where the control objective involves a sparsifying term and, as a consequence, distributed optimal controls vanish on parts of the domain. The areas where controls are nonzero are interpreted as locations where it is most efficient to employ control devices [5,7,8,10,12,13,21,35].…”

Finite element error analysis for measure-valued optimal control problems governed by a 1D wave equation with variable coefficients

“…Motivated by industrial applications as well as applications in the natural sciences, in which one is interested to place actuators in form of point sources in an optimal way, see, e.g., [4,9] or in the reconstruction of point sources from given measurements, see, e.g., [34,44], measure valued optimal control problems involving PDEs gained attention in the last years. These problems can be translated into optimization problems in terms of the coordinates and coefficients of the point sources.…”