Bifurcation Analysis of Reaction Diffusion Systems on Arbitrary Surfaces

In this paper we present computational techniques to investigate the solutions of two-component, nonlinear reaction-diffusion (RD) systems on arbitrary surfaces. We build on standard techniques for linear and nonlinear analysis of RD systems, and extend them to operate on large-scale meshes for arbitrary surfaces. In particular, we use spectral techniques for a linear stability analysis to characterize and directly compose patterns emerging from homogeneities. We develop an implementation using surface finite element methods and a numerical eigenanalysis of the Laplace-Beltrami operator on surface meshes. In addition, we describe a technique to explore solutions of the nonlinear RD equations using numerical continuation. Here, we present a multiresolution approach that allows us to trace solution branches of the nonlinear equations efficiently even for large-scale meshes. Finally, we demonstrate the working of our framework for two RD systems with applications in biological pattern formation: a Brusselator model that has been used to model pattern development on growing plant tips, and a chemotactic model for the formation of skin pigmentation patterns. While these models have been used previously on simple geometries, our framework allows us to study the impact of arbitrary geometries on emerging patterns.Comment: This paper was submitted at the Journal of Mathematical Biology, Springer on 07th July 2015, in its current form (barring image references on the last page and cosmetic changes owning to rebuild for arXiv). The complete body of work presented here was included and defended as a part of my PhD thesis in Nov 2015 at the University of Ber

Numerical Continuation of Bound and Resonant States of the Two Channel Schr\"odinger Equation

Resonant solutions of the quantum Schr\"odinger equation occur at complex energies where the S-matrix becomes singular. Knowledge of such resonances is important in the study of the underlying physical system. Often the Schr\"odinger equation is dependent on some parameter and one is interested in following the path of the resonances in the complex energy plane as the parameter changes. This is particularly true in coupled channel systems where the resonant behavior is highly dependent on the strength of the channel coupling, the energy separation of the channels and other factors. In previous work it was shown that numerical continuation, a technique familiar in the study of dynamical systems, can be brought to bear on the problem of following the resonance path in one dimensional problems and multi-channel problems without energy separation between the channels. A regularization can be defined that eliminates coalescing poles and zeros that appear in the S-matrix at the origin due to symmetries. Following the zeros of this regularized function then traces the resonance path. In this work we show that this approach can be extended to channels with energy separation, albeit limited to two channels. The issue here is that the energy separation introduces branch cuts in the complex energy domain that need to be eliminated with a so-called uniformization. We demonstrate that the resulting approach is suitable for investigating resonances in two-channel systems and provide an extensive example

Computational Methods for Nonlinear Systems Analysis With Applications in Mathematics and Engineering

An investigation into current methods and new approaches for solving systems of nonlinear equations was performed. Nontraditional methods for implementing arc-length type solvers were developed in search of a more robust capability for solving general systems of nonlinear algebraic equations. Processes for construction of parameterized curves representing the many possible solutions to systems of equations versus finding single or point solutions were established. A procedure based on these methods was then developed to identify static equilibrium states for solutions to multi-body-dynamic systems. This methodology provided for a pictorial of the overall solution to a given system, which demonstrated the possibility of multiple candidate equilibrium states for which a procedure for selection of the proper state was proposed. Arc-length solvers were found to identify and more readily trace solution curves as compared to other solvers making such an approach practical. Comparison of proposed methods was made to existing methods found in the literature and commercial software with favorable results. Finally, means for parallel processing of the Jacobian matrix inherent to the arc-length and other nonlinear solvers were investigated, and an efficient approach for implementation was identified. Several case studies were performed to substantiate results. Commercial software was also used in some instances for additional results verification

Trifocal Relative Pose from Lines at Points and its Efficient Solution

We present a new minimal problem for relative pose estimation mixing point features with lines incident at points observed in three views and its efficient homotopy continuation solver. We demonstrate the generality of the approach by analyzing and solving an additional problem with mixed point and line correspondences in three views. The minimal problems include correspondences of (i) three points and one line and (ii) three points and two lines through two of the points which is reported and analyzed here for the first time. These are difficult to solve, as they have 216 and - as shown here - 312 solutions, but cover important practical situations when line and point features appear together, e.g., in urban scenes or when observing curves. We demonstrate that even such difficult problems can be solved robustly using a suitable homotopy continuation technique and we provide an implementation optimized for minimal problems that can be integrated into engineering applications. Our simulated and real experiments demonstrate our solvers in the camera geometry computation task in structure from motion. We show that new solvers allow for reconstructing challenging scenes where the standard two-view initialization of structure from motion fails.Comment: This material is based upon work supported by the National Science Foundation under Grant No. DMS-1439786 while most authors were in residence at Brown University's Institute for Computational and Experimental Research in Mathematics -- ICERM, in Providence, R

On Continuation Methods for Non-Linear Bi-Objective Optimization: Certified Interval-Based Approach

The global optimization of constrained Non-Linear Bi-Objective Optimization problems (MO) aims at covering their Pareto-optimal front which is in general a manifold in R^2. Continuation methods can help in this context as they can follow a continuous component of this front once an initial point on it is provided. They constitute somehow a generalization of the classical scalarizing framework which transforms the bi-objective problem into a parametric mono-objective problem. Recent works have shown that they can play a key role in global algorithms dedicated to bi-objective problems, e.g. population based algorithms, where they allow discovering large portions of locally Pareto optimal vectors, which turns out to strongly support diversification. In this paper, we provide a survey on continuation techniques in global optimization methods for MO, which allow discovering large portions of locally Pareto-optimal solutions. We also propose a rigorous active set management strategy on top of a previously proposed certified continuation method based on interval analysis, and illustrate it on a challenging bi-objective problem

Finite Strain Topology Optimization with Nonlinear Stability Constraints

This paper proposes a computational framework for the design optimization of stable structures under large deformations by incorporating nonlinear buckling constraints. A novel strategy for suppressing spurious buckling modes related to low-density elements is proposed. The strategy depends on constructing a pseudo-mass matrix that assigns small pseudo masses for DOFs surrounded by only low-density elements and degenerates to an identity matrix for the solid region. A novel optimization procedure is developed that can handle both simple and multiple eigenvalues wherein consistent sensitivities of simple eigenvalues and directional derivatives of multiple eigenvalues are derived and utilized in a gradient-based optimization algorithm - the method of moving asymptotes. An adaptive linear energy interpolation method is also incorporated in nonlinear analyses to handle the low-density elements distortion under large deformations. The numerical results demonstrate that, for systems with either low or high symmetries, the nonlinear stability constraints can ensure structural stability at the target load under large deformations. Post-analysis on the B-spline fitted designs shows that the safety margin, i.e., the gap between the target load and the 1st critical load, of the optimized structures can be well controlled by selecting different stability constraint values. Interesting structural behaviors such as mode switching and multiple bifurcations are also demonstrated.Comment: 77 pages, 44 Figure