Bayesian experimental design for linear elasticity

Figure 1.  Test object $ \Omega $, i.e., a unit square with rounded corners. Left: $ \Omega $ embedded in $ \mathbb{R}^2 $. Right: Arclength parametrization of $ \Omega $ including the Neumann (blue) and Dirichlet (orange) boundaries

Figure 2.  The pressure field $ g_p $ for $ p = 1+\tfrac{\pi}{2}r $ and three values for $ \zeta $

Figure 3.  Subdomains of $ \Omega $ employed in an (approximately) piecewise constant parametrization for the Lamé parameter perturbation $ \eta $

Figure 4.  Exhaustive search on a $ 200\times 200\times 200 $ grid. Left: Optimal design of three pressure activations corresponding to the arclength parameter triplet $ p^* = (0.72, 1.28, 2.72) $. Middle: Progress of the search, with an "iteration" referring to the instances when the estimate for $ p^* $ was updated in the exhaustive search. Right: Evolution of the optimization target, with the final optimal value $ \Phi_{\rm A}(p^*) = 1.46 \cdot 10^{18} $

Figure 5.  Greedy sequential algorithm. Left: Optimized design for three pressure activations corresponding to the arclength parameter triplet $ p^* = (0.92, 2.34, 2.92) $. Middle: Progress of the algorithm, with each "iteration" referring to an introduction of a new pressure activation whose position has been optimized by a one-dimensional exhaustive search. Right: Evolution of the optimization target, with the final value $ \Phi_{\rm A}(p^*) = 1.52 \cdot 10^{18} $

Figure 6.  Sequential algorithm enhanced by gradient descent. Left: Optimized design for three pressure activations corresponding to the arclength parameter triplet $ p^* = (0.72, 1.29, 2.71) $. Middle: Progress of the algorithm, with an "iteration" referring to the introduction of a new pressure activation via a one-dimensional exhaustive search or a step of gradient descent for fine-tuning the design after such an introduction. Right: Evolution of the optimization target, with the final value $ \Phi_{\rm A}(p^*) = 1.46 \cdot 10^{18} $

Figure 7.  Exhaustive search on a $ 200\times 200\times 200 $ grid for a finer discretization of the Lamé parameters with 200 degrees of freedom. Left: Optimal design of three pressure activations corresponding to the arclength parameter triplet $ p^* = (0.74, 1.26, 3.14) $. Middle: Progress of the search, with an "iteration" referring to the instances when the estimate for $ p^* $ was updated in the exhaustive search. Right: Evolution of the optimization target, with the final optimal value $ \Phi_{\rm A}(p^*) = 6.62 \cdot 10^{18} $

Figure 8.  Greedy sequential algorithm. Left: Optimized design for ten pressure activations. Middle: Progress of the algorithm, with each "iteration" referring to an introduction of a new pressure activation whose position has been optimized by a one-dimensional exhaustive search. Right: Evolution of the optimization target, with the final value $ \Phi_{\rm A}(p^*) = 3.86 \cdot 10^{17} $

Figure 9.  Sequential algorithm enhanced by gradient descent. Left: Optimized design for ten pressure activations. Middle: Progress of the algorithm, with an "iteration" referring to the introduction of a new pressure activation via a one-dimensional exhaustive search or a step of gradient descent for fine-tuning the design after such an introduction. Right: Evolution of the optimization target, with the final value $ \Phi_{\rm A}(p^*) = 3.78 \cdot 10^{17} $

Figure 10.  Gradient descent with an equidistant initial guess for the pressure activations. Left: Optimized design of ten pressure activations. Middle: Progress of the algorithm. Right: Evolution of the optimization target, with the final value $ \Phi_{\rm A}(p^*) = 3.77 \cdot 10^{17} $

Figure 11.  Exhaustive search for a stiff material (gray cast iron). Left: Optimal design of three pressure activations corresponding to the arclength parameter triplet $ p^* = (1.10, 2.88, 3.31) $. Middle: Progress of the search, with an "iteration" referring to the instances when the estimate for $ p^* $ was updated in the exhaustive search. Right: Evolution of the optimization target, with the final optimal value $ \Phi_{\rm A}(p^*) = 5.42\cdot 10^{20} $

Figure 12.  Exhaustive search for soft material (rubber). Left: Optimal design of three pressure activations corresponding to the arclength parameter triplet $ p^* = (1.12, 2.72, 3.29) $. Middle: Progress of the search, with an "iteration" referring to the instances when the estimate for $ p^* $ was updated in the exhaustive search. Right: Evolution of the optimization target, with the final optimal value $ \Phi_{\rm A}(p^*) = 9.32\cdot 10^{16} $

Figure 13.  Exhaustive search on a $ 200\times 200\times 200 $ grid with the standard deviation of noise decreased to one tenth compared to Figure 4. Left: Optimal design of three pressure activations corresponding to the arclength parameter triplet $ p^* = (0.68, 1.16, 2.70) $. Middle: Progress of the search, with an "iteration" referring to the instances when the estimate for $ p^* $ was updated in the exhaustive search. Right: Evolution of the optimization target, with the final optimal value $ \Phi_{\rm A}(p^*) = 1.22 \cdot 10^{18} $