The Radius of Metric Subregularity

There is a basic paradigm, called here the radius of well-posedness, which quantifies the "distance" from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.Comment: 20 page

A unified approach to shape and topological sensitivity analysis of discretized optimal design problems

We introduce a unified sensitivity concept for shape and topological perturbations and perform the sensitivity analysis for a discretized PDE-constrained design optimization problem in two space dimensions. We assume that the design is represented by a piecewise linear and globally continuous level set function on a fixed finite element mesh and relate perturbations of the level set function to perturbations of the shape or topology of the corresponding design. We illustrate the sensitivity analysis for a problem that is constrained by a reaction-diffusion equation and draw connections between our discrete sensitivities and the well-established continuous concepts of shape and topological derivatives. Finally, we verify our sensitivities and illustrate their application in a level-set-based design optimization algorithm where no distinction between shape and topological updates has to be made

On the role of semismoothness in nonsmooth numerical analysis: Theory

For the numerical solution of nonsmooth problems, sometimes it is not necessary that an exact subgradient/generalized Jacobian is at our disposal, but that a certain semismoothness property is fulfilled. In this paper we consider not only semismoothness of nonsmooth real- and vector-valued mappings, but also its interplay with the semismoothness$^*$ property for multifunctions. In particular, we are interested in the semismoothness of solution maps to parametric semismooth$^*$ inclusions. Our results are expressed in terms of suitable generalized derivatives of the set-valued part, i.e., by limiting coderivatives or by SC (subspace containing) derivatives. As a byproduct we identify a class of multifunctions having the remarkable property that they are strictly proto-differentiable almost everywhere (with respect to some Hausdorff measure) on their graph

On the SCD semismooth* Newton method for generalized equations with application to a class of static contact problems with Coulomb friction

In the paper, a variant of the semismooth* Newton method is developed for the numerical solution of generalized equations, in which the multi-valued part is a so-called SCD (subspace containing derivative) mapping. Under a rather mild regularity requirement, the method exhibits (locally) superlinear convergence behavior. From the main conceptual algorithm, two implementable variants are derived whose efficiency is tested via a generalized equation modeling a discretized static contact problem with Coulomb friction

Teaching mechanics with individual exercise assignments and automated correction

Solving exercise problems by yourself is a vital part of developing a mechanical understanding. Yet, most mechanics lectures have more than 200 participants, so the workload for manually creating and correcting assignments limits the number of exercises. The resulting example pool is usually much smaller than the number of participants, making verifying whether students can solve problems themselves considerably harder. At the same time, unreflected copying of tasks already solved does not foster the understanding of the subject and leads to a false self-assessment. We address these issues by providing a scalable approach for creating, distributing, and correcting exercise assignments for problems related to statics, strength of materials, dynamics, and hydrostatics. The overall concept allows us to provide individual exercise assignments for each student. A quantitative survey among students of our recent statics lecture assesses the acceptance of our teaching tool. The feedback indicates a clear added value for the lecture, which fosters self-directed and reflective learning

A novel section-section potential for short-range interactions between plane beams

We derive a novel formulation for the interaction potential between deformable fibers due to short-range fields arising from intermolecular forces. The formulation improves the existing section-section interaction potential law for in-plane beams by considering an offset between interacting cross sections. The new law is asymptotically consistent, which is particularly beneficial for computationally demanding scenarios involving short-range interactions like van der Waals and steric forces. The formulation is implemented within a framework of rotation-free Bernoulli-Euler beams utilizing the isogeometric paradigm. The improved accuracy of the novel law is confirmed through thorough numerical studies. We apply the developed formulation to investigate the complex behavior observed during peeling and pull-off of elastic fibers interacting via the Lennard-Jones potential

Advances in structure elucidation of small molecules using mass spectrometry

The structural elucidation of small molecules using mass spectrometry plays an important role in modern life sciences and bioanalytical approaches. This review covers different soft and hard ionization techniques and figures of merit for modern mass spectrometers, such as mass resolving power, mass accuracy, isotopic abundance accuracy, accurate mass multiple-stage MS(n) capability, as well as hybrid mass spectrometric and orthogonal chromatographic approaches. The latter part discusses mass spectral data handling strategies, which includes background and noise subtraction, adduct formation and detection, charge state determination, accurate mass measurements, elemental composition determinations, and complex data-dependent setups with ion maps and ion trees. The importance of mass spectral library search algorithms for tandem mass spectra and multiple-stage MS(n) mass spectra as well as mass spectral tree libraries that combine multiple-stage mass spectra are outlined. The successive chapter discusses mass spectral fragmentation pathways, biotransformation reactions and drug metabolism studies, the mass spectral simulation and generation of in silico mass spectra, expert systems for mass spectral interpretation, and the use of computational chemistry to explain gas-phase phenomena. A single chapter discusses data handling for hyphenated approaches including mass spectral deconvolution for clean mass spectra, cheminformatics approaches and structure retention relationships, and retention index predictions for gas and liquid chromatography. The last section reviews the current state of electronic data sharing of mass spectra and discusses the importance of software development for the advancement of structure elucidation of small molecules

On a globally convergent semismooth* Newton method in nonsmooth nonconvex optimzation

In this paper we present GSSN, a globalized SCD semismooth* Newton method for solving nonsmooth nonconvex optimization problems. The global convergence properties of the method are ensured by the proximal gradient method, whereas locally superlinear convergence is established via the SCD semismooth* Newton method under quite weak assumptions. The Newton direction is based on the SC (subspace containing) derivative of the subdifferential mapping and can be computed by the (approximate) solution of an equality-constrained quadratic program. Special attention is given to the efficient numerical implementation of the overall method