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Probability A Graduate Course

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Fundamentals Of Differential Geometry

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Design Sensitivity Analysis Of Structural Systems

If $- \infty < \alpha < \beta < \infty $ and $f \in C^{3} \left( [ \alpha , \beta ] \times {\bf R}^{2} , {\bf R} \right) $ is bounded, while $y \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ solves the typical one-dimensional problem of the calculus of variations to minimize the function $$F \left( y \right) = \int_{ \alpha }^{ \beta }f \left( x, y(x), y'(x) \right) dx,$$ then for any ${\phi } \in C^{2} \left( [ \alpha , \beta ] , {\bf R} \right) $ for which ${\phi }^{(k)}( \alpha ) = {\phi }^{(k)}( \beta ) = 0$ for every $k \in \{ 0, 1, 2 \} $, we prove that $\int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y^{2} } {\phi }^{2} - \frac{ {\partial }^{3}f }{ \partial y^{2} \partial y' } 2 {\phi }^{3} \right) dx$ $\geq \int_{\alpha }^{\beta } \left( \frac{ {\partial }^{2}f }{ \partial y \partial y' } 2 \phi \phi ' + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } 2 {\phi }^{2} \phi ' + \frac{ {\partial }^{2}f }{ {\partial y'}^{2} } \phi \phi " + \frac{ {\partial }^{3}f }{ \partial y {\partial y'}^{2} } \phi ' {\phi }^{2} + \frac{ {\partial }^{3}f }{ {\partial y'}^{3} } \phi {\phi '}^{2} \right) dx$, so either the above are variational inequalities of motion or the Lagrangian of motion is not $C^{3}$

Variational inequalities

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincar\'e type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.

Efficient PDE constrained shape optimization based on Steklov-Poincar\'e type metrics

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which instead require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov--Poincare-type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor involved in the derivation of shape derivatives.

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

Often, the unknown diffusivity in diffusive processes is structured by piecewise constant patches. This paper is devoted to efficient methods for the determination of such structured diffusion parameters by exploiting shape calculus. A novel shape gradient is derived in parabolic processes. Furthermore, quasi-Newton techniques are used in order to accelerate shape gradient based iterations in shape space. Numerical investigations support the theoretical results.

Structured Inverse Modeling in Parabolic Diffusion Problems

We examine the interaction of multigrid methods and shape optimization in appropriate shape spaces. The impact of discrete approximations of geometrical quantities, like the mean curvature, on a multigrid shape optimization algorithm with quasi-Newton updates is investigated. Both multigrid and quasi-Newton methods are necessary to achieve mesh-independent convergence and, thus, scalable algorithms for supercomputers. For the purpose of illustration, we consider a complex model for the identification of cellular structures in biology with minimal compliance in terms of elasticity and diffusion equations.

Algorithmic Aspects of Multigrid Methods for Optimization in Shape Spaces

Often, the unknown diffusivity in diffusive processes is structured by piecewise constant patches. This paper is devoted to efficient methods for the determination of such structured diffusion parameters by exploiting shape calculus. A novel shape gradient is derived in parabolic processes. Furthermore quasi-Newton techniques are used in order to accelerate shape gradient based iterations in shape space. Numerical investigations support the theoretical results.

Kathrin Welker

Papers

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

Efficient PDE constrained shape optimization based on Steklov-Poincar\'e type metrics

Structured Inverse Modeling in Parabolic Diffusion Problems

Algorithmic Aspects of Multigrid Methods for Optimization in Shape Spaces

Structured inverse modeling in parabolic diffusion processess