On a continuation approach in Tikhonov regularization and its application in piecewise-constant parameter identification

We present a new approach to convexification of the Tikhonov regularization using a continuation method strategy. We embed the original minimization problem into a one-parameter family of minimization problems. Both the penalty term and the minimizer of the Tikhonov functional become dependent on a continuation parameter. In this way we can independently treat two main roles of the regularization term, which are stabilization of the ill-posed problem and introduction of the a priori knowledge. For zero continuation parameter we solve a relaxed regularization problem, which stabilizes the ill-posed problem in a weaker sense. The problem is recast to the original minimization by the continuation method and so the a priori knowledge is enforced. We apply this approach in the context of topology-to-shape geometry identification, where it allows to avoid the convergence of gradient-based methods to a local minima. We present illustrative results for magnetic induction tomography which is an example of PDE constrained inverse problem

Topology Optimization of Electric Machines based on Topological Sensitivity Analysis

Topological sensitivities are a very useful tool for determining optimal designs. The topological derivative of a domain-dependent functional represents the sensitivity with respect to the insertion of an infinitesimally small hole. In the gradient-based ON/OFF method, proposed by M. Ohtake, Y. Okamoto and N. Takahashi in 2005, sensitivities of the functional with respect to a local variation of the material coefficient are considered. We show that, in the case of a linear state equation, these two kinds of sensitivities coincide. For the sensitivities computed in the ON/OFF method, the generalization to the case of a nonlinear state equation is straightforward, whereas the computation of topological derivatives in the nonlinear case is ongoing work. We will show numerical results obtained by applying the ON/OFF method in the nonlinear case to the optimization of an electric motor.Comment: 20 pages, 7 figure

Variational approach to relaxed topological optimization: closed form solutions for structural problems in a sequential pseudo-time framework

The work explores a specific scenario for structural computational optimization based on the following elements: (a) a relaxed optimization setting considering the ersatz (bi-material) approximation, (b) a treatment based on a non-smoothed characteristic function field as a topological design variable, (c) the consistent derivation of a relaxed topological derivative whose determination is simple, general and efficient, (d) formulation of the overall increasing cost function topological sensitivity as a suitable optimality criterion, and (e) consideration of a pseudo-time framework for the problem solution, ruled by the problem constraint evolution. In this setting, it is shown that the optimization problem can be analytically solved in a variational framework, leading to, nonlinear, closed-form algebraic solutions for the characteristic function, which are then solved, in every time-step, via fixed point methods based on a pseudo-energy cutting algorithm combined with the exact fulfillment of the constraint, at every iteration of the non-linear algorithm, via a bisection method. The issue of the ill-posedness (mesh dependency) of the topological solution, is then easily solved via a Laplacian smoothing of that pseudo-energy. In the aforementioned context, a number of (3D) topological structural optimization benchmarks are solved, and the solutions obtained with the explored closed-form solution method, are analyzed, and compared, with their solution through an alternative level set method. Although the obtained results, in terms of the cost function and topology designs, are very similar in both methods, the associated computational cost is about five times smaller in the closed-form solution method this possibly being one of its advantages. Some comments, about the possible application of the method to other topological optimization problems, as well as envisaged modifications of the explored method to improve its performance close the workPeer ReviewedPostprint (author's final draft

Information Surfaces in Systems Biology and Applications to Engineering Sustainable Agriculture

Systems biology of plants offers myriad opportunities and many challenges in modeling. A number of technical challenges stem from paucity of computational methods for discovery of the most fundamental properties of complex dynamical systems. In systems engineering, eigen-mode analysis have proved to be a powerful approach. Following this philosophy, we introduce a new theory that has the benefits of eigen-mode analysis, while it allows investigation of complex dynamics prior to estimation of optimal scales and resolutions. Information Surfaces organizes the many intricate relationships among "eigen-modes" of gene networks at multiple scales and via an adaptable multi-resolution analytic approach that permits discovery of the appropriate scale and resolution for discovery of functions of genes in the model plant Arabidopsis. Applications are many, and some pertain developments of crops that sustainable agriculture requires.Comment: 24 Pages, DoCEIS 1

Level Set Jet Schemes for Stiff Advection Equations: The SemiJet Method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

Fast micro-differential evolution for topological active net optimization

This paper studies the optimization problem of topological active net (TAN), which is often seen in image segmentation and shape modeling. A TAN is a topological structure containing many nodes, whose positions must be optimized while a predefined topology needs to be maintained. TAN optimization is often time-consuming and even constructing a single solution is hard to do. Such a problem is usually approached by a ``best improvement local search'' (BILS) algorithm based on deterministic search (DS), which is inefficient because it spends too much efforts in nonpromising probing. In this paper, we propose the use of micro-differential evolution (DE) to replace DS in BILS for improved directional guidance. The resultant algorithm is termed deBILS. Its micro-population efficiently utilizes historical information for potentially promising search directions and hence improves efficiency in probing. Results show that deBILS can probe promising neighborhoods for each node of a TAN. Experimental tests verify that deBILS offers substantially higher search speed and solution quality not only than ordinary BILS, but also the genetic algorithm and scatter search algorithm