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Bayesian experimental design for linear elasticity

  • *Corresponding author: Nuutti Hyvönen

    *Corresponding author: Nuutti Hyvönen

The first author was supported by the German Research Foundation (DFG): project number 499303971. The second author was supported by the Academy of Finland: decisions 348503, 353081 and 359181

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  • This work considers Bayesian experimental design for the inverse boundary value problem of linear elasticity in a two-dimensional setting. The aim is to optimize the positions of compactly supported pressure activations on the boundary of the examined body in order to maximize the value of the resulting boundary deformations as data for the inverse problem of reconstructing the Lamé parameters inside the object. We resort to a linearized measurement model and adopt the framework of Bayesian experimental design, under the assumption that the prior and measurement noise distributions are mutually independent Gaussians. This enables the use of the standard Bayesian A-optimality criterion for deducing optimal positions for the pressure activations. The (second) derivatives of the boundary measurements with respect to the Lamé parameters and the positions of the boundary pressure activations are deduced to allow minimizing the corresponding objective function, i.e., the trace of the covariance matrix of the posterior distribution, by gradient-based optimization algorithms. Two-dimensional numerical experiments are performed to test the functionality of our approach: all introduced algorithms are able to improve experimental designs, but only exhaustive search reliably finds a global minimizer.

    Mathematics Subject Classification: Primary: 35Q74, 62K05, 62F15; Secondary: 35J25, 65N21, 74B05.

    Citation:

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  • 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} $

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