For control problems with control constraints, a local convergence rate is established for an hp-method based on collocation at the Radau quadrature points in each mesh interval of the discretization. If the continuous problem has a sufficiently smooth solution and the Hamiltonian satisfies a strong convexity condition, then the discrete problem possesses a local minimizer in a neighborhood of the continuous solution, and as either the number of collocation points or the number of mesh intervals increase, the discrete solution convergences to the continuous solution in the sup-norm. The convergence is exponentially fast with respect to the degree of the polynomials on each mesh interval, while the error is bounded by a polynomial in the mesh spacing. An advantage of the hp-scheme over global polynomials is that there is a convergence guarantee when the mesh is sufficiently small, while the convergence result for global polynomials requires that a norm of the linearized dynamics is sufficiently small. Numerical examples explore the convergence theory.

A local convergence rate is established for a Gauss orthogonal collocation method applied to optimal control problems with control constraints. If the Hamiltonian possesses a strong convexity property, then the theory yields convergence for problems whose optimal state and costate possess two square integrable derivatives. The convergence theory is based on a stability result for the sup-norm change in the solution of a variational inequality relative to a 2-norm perturbation, and on a Sobolev space bound for the error in interpolation at the Gauss quadrature points and the additional point −1. The tightness of the convergence theory is examined using a numerical example.

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The scale at which MS‐ and NMR‐based platforms generate metabolomics datasets for both research, core, and clinical facilities to address challenges in the various sciences—ranging from biomedical to agricultural—is underappreciated. Thus, metabolomics efforts spanning microbe, environment, plant, animal, and human systems have led to continual and concomitant growth of in silico resources for analysis and interpretation of these datasets. These software tools, resources, and databases drive the field forward to help keep pace with the amount of data being generated and the sophisticated and diverse analytical platforms that are being used to generate these metabolomics datasets. To address challenges in data preprocessing, metabolite annotation, statistical interrogation, visualization, interpretation, and integration, the metabolomics and informatics research community comes up with hundreds of tools every year. The purpose of the present review is to provide a brief and useful summary of more than 95 metabolomics tools, software, and databases that were either developed or significantly improved during 2017–2018. We hope to see this review help readers, developers, and researchers to obtain informed access to these thorough lists of resources for further improvisation, implementation, and application in due course of time.

A framework for the global analysis of multi-speed analytical ultracentrifugation sedimentation velocity experiments is presented. We discuss extensions to the adaptive space-time finite element fitting methods implemented in UltraScan-III to model sedimentation velocity experiments where a single run is performed at multiple rotor speeds, and describe extensions in the optimization routines used for fitting experimental data collected at arbitrary multi-speed profiles. Our implementation considers factors such as speed dependent rotor stretching, the resulting radial shifting of the finite element solution's boundary conditions, and changes in the associated time-invariant noise. We also address the calculation of acceleration rates and acceleration zones from existing radial acceleration and time records, as well as utilization of the time state object available at high temporal resolution from the new Beckman Optima AUC instrument. Analysis methods in UltraScan-III support unconstrained models that extract reliable information for both the sedimentation and the diffusion coefficients. These methods do not rely on any assumptions and allow for arbitrary variations in both sedimentation and diffusion transport. We have adapted these routines for the multi-speed case, and developed optimized and general grid based fitting methods to handle changes in the information content of the simulation matrix for different speed steps. New graphical simulation tools are presented that assist the investigator to estimate suitable grid metrics and evaluate information content based on edit profiles for individual experiments.

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