State and parameter estimation for model-based retinal laser treatment

Parameter estimation and model reduction for retinal laser treatment

“…In this section, we briefly describe the PDE-model for heat absorption used for retinal laser treatment as well as its time and space discretization. For more details, we refer to the previous works [15,16]. The computational domain, denoted by Ω ⊂ R 3 , is depicted in Figure 2 and consists of a cylinder with radius R that encloses the irradiated area, given by a smaller cylinder with radius R I .…”

“…To this end, we investigate and compare two parametric model order reduction (MOR) approaches in our particular application in this section. More precisely, we first extend an approach based on Taylor expansion [15,16] and combine it with the interpolation-based parametric model order reduction method (pMOR) from [3]. Second, we consider a global basis (gb) approach [4, Section 4.1] and pair it with a discrete empirical interpolation method (DEIM) [7].…”

“…The general framework of parametric model order reduction is as follows. Consider the dynamical control system with input space R m , output space R l and state space R n depending on a parameter α ∈ D ⊂ R q , ẋ = Ax + B(α)u, y = C(α)x (16) with matrix A ∈ R n×n , vector-valued functions B : D → R n×m , C : D → R ℓ×n and a control input u ∈ R m . Suppose that suitable projection matrices V, W ∈ R n×d , d ≪ n with full rank are given.…”

“…Thus, we presented in [15] an approach adapting the parametric model reduction method from [3] to obtain a low-dimensional representation of the heat equation which retains the absorption dependence and is suitable for applications in the kHz-range. Further, we presented first results regarding observers for the reduced models states and the absorption coefficient, such as an extended Kalman filter and a moving horizon estimator in [16]. In [18], we proposed a Model Predictive Control (MPC) scheme to ensure a safe and effective treatment.…”

“…In this paper, we thoroughly compare two parametric model reduction techniques in our particular application for the consideration of either one or two independent absorption coefficients in the eye. First, we extend the approach of [15,16] to the case of two parameters, where we combine a Taylor series expansion of the nonlinear parameter dependency and with the interpolation based parametric model order reduction method (pMOR) from [3]. Second, we consider a global basis (gb) approach [4, Section 4.1], where we construct a basis from system snapshots sampled at different parameters.…”