Strong convergence rates of probabilistic integrators for ordinary differential equations

Probabilistic integration of a continuous dynamical system is a way of systematically introducing model error, at scales no larger than errors introduced by standard numerical discretisation, in order to enable thorough exploration of possible responses of the system to inputs. It is thus a potentially useful approach in a number of applications such as forward uncertainty quantification, inverse problems, and data assimilation. We extend the convergence analysis of probabilistic integrators for deterministic ordinary differential equations, as proposed by Conrad et al.\ (\textit{Stat.\ Comput.}, 2017), to establish mean-square convergence in the uniform norm on discrete- or continuous-time solutions under relaxed regularity assumptions on the driving vector fields and their induced flows. Specifically, we show that randomised high-order integrators for globally Lipschitz flows and randomised Euler integrators for dissipative vector fields with polynomially-bounded local Lipschitz constants all have the same mean-square convergence rate as their deterministic counterparts, provided that the variance of the integration noise is not of higher order than the corresponding deterministic integrator. These and similar results are proven for probabilistic integrators where the random perturbations may be state-dependent, non-Gaussian, or non-centred random variables.Comment: 25 page

The Optimal Uncertainty Algorithm in the Mystic Framework

We have recently proposed a rigorous framework for Uncertainty Quantification (UQ) in which UQ objectives and assumption/information set are brought into the forefront, providing a framework for the communication and comparison of UQ results. In particular, this framework does not implicitly impose inappropriate assumptions nor does it repudiate relevant information. This framework, which we call Optimal Uncertainty Quantification (OUQ), is based on the observation that given a set of assumptions and information, there exist bounds on uncertainties obtained as values of optimization problems and that these bounds are optimal. It provides a uniform environment for the optimal solution of the problems of validation, certification, experimental design, reduced order modeling, prediction, extrapolation, all under aleatoric and epistemic uncertainties. OUQ optimization problems are extremely large, and even though under general conditions they have finite-dimensional reductions, they must often be solved numerically. This general algorithmic framework for OUQ has been implemented in the mystic optimization framework. We describe this implementation, and demonstrate its use in the context of the Caltech surrogate model for hypervelocity impact

Optimal Uncertainty Quantification

We propose a rigorous framework for Uncertainty Quantification (UQ) in which the UQ objectives and the assumptions/information set are brought to the forefront. This framework, which we call Optimal Uncertainty Quantification (OUQ), is based on the observation that, given a set of assumptions and information about the problem, there exist optimal bounds on uncertainties: these are obtained as extreme values of well-defined optimization problems corresponding to extremizing probabilities of failure, or of deviations, subject to the constraints imposed by the scenarios compatible with the assumptions and information. In particular, this framework does not implicitly impose inappropriate assumptions, nor does it repudiate relevant information. Although OUQ optimization problems are extremely large, we show that under general conditions, they have finite-dimensional reductions. As an application, we develop Optimal Concentration Inequalities (OCI) of Hoeffding and McDiarmid type. Surprisingly, contrary to the classical sensitivity analysis paradigm, these results show that uncertainties in input parameters do not necessarily propagate to output uncertainties. In addition, a general algorithmic framework is developed for OUQ and is tested on the Caltech surrogate model for hypervelocity impact, suggesting the feasibility of the framework for important complex systems

Regulation of arginine transport by GCN2 eIF2 kinase is important for replication of the intracellular parasite Toxoplasma gondii

Toxoplasma gondii is a prevalent protozoan parasite that can infect any nucleated cell but cannot replicate outside of its host cell. Toxoplasma is auxotrophic for several nutrients including arginine, tryptophan, and purines, which it must acquire from its host cell. The demands of parasite replication rapidly deplete the host cell of these essential nutrients, yet Toxoplasma successfully manages to proliferate until it lyses the host cell. In eukaryotic cells, nutrient starvation can induce the integrated stress response (ISR) through phosphorylation of an essential translation factor eIF2. Phosphorylation of eIF2 lowers global protein synthesis coincident with preferential translation of gene transcripts involved in stress adaptation, such as that encoding the transcription factor ATF4 (CREB2), which activates genes that modulate amino acid metabolism and uptake. Here, we discovered that the ISR is induced in host cells infected with Toxoplasma. Our results show that as Toxoplasma depletes host cell arginine, the host cell phosphorylates eIF2 via protein kinase GCN2 (EIF2AK4), leading to induced ATF4. Increased ATF4 then enhances expression of the cationic amino acid transporter CAT1 (SLC7A1), resulting in increased uptake of arginine in Toxoplasma-infected cells. Deletion of host GCN2, or its downstream effectors ATF4 and CAT1, lowers arginine levels in the host, impairing proliferation of the parasite. Our findings establish that Toxoplasma usurps the host cell ISR to help secure nutrients that it needs for parasite replication

Criticality for the Gehring link problem

In 1974, Gehring posed the problem of minimizing the length of two linked curves separated by unit distance. This constraint can be viewed as a measure of thickness for links, and the ratio of length over thickness as the ropelength. In this paper we refine Gehring's problem to deal with links in a fixed link-homotopy class: we prove ropelength minimizers exist and introduce a theory of ropelength criticality. Our balance criterion is a set of necessary and sufficient conditions for criticality, based on a strengthened, infinite-dimensional version of the Kuhn--Tucker theorem. We use this to prove that every critical link is C^1 with finite total curvature. The balance criterion also allows us to explicitly describe critical configurations (and presumed minimizers) for many links including the Borromean rings. We also exhibit a surprising critical configuration for two clasped ropes: near their tips the curvature is unbounded and a small gap appears between the two components. These examples reveal the depth and richness hidden in Gehring's problem and our natural extension.Comment: This is the version published by Geometry & Topology on 14 November 200

Aerothermal modeling program. Phase 2, element B: Flow interaction experiment

NASA has instituted an extensive effort to improve the design process and data base for the hot section components of gas turbine engines. The purpose of element B is to establish a benchmark quality data set that consists of measurements of the interaction of circular jets with swirling flow. Such flows are typical of those that occur in the primary zone of modern annular combustion liners. Extensive computations of the swirling flows are to be compared with the measurements for the purpose of assessing the accuracy of current physical models used to predict such flows

NASTRAN as an analytical research tool for composite mechanics and composite structures

Selected examples are described in which NASTRAN is used as an analysis research tool for composite mechanics and for composite structural components. The examples were selected to illustrate the importance of using NASTRAN as an analysis tool in this rapidly advancing field

Variable load automatically tests dc power supplies

Continuously variable load automatically tests dc power supplies over an extended current range. External meters monitor current and voltage, and multipliers at the outputs facilitate plotting the power curve of the unit

Quantum Invariants of Templates

We define invariants for templates that appear in certain dynamical systems. Invariants are derived from certain bialgebras. Diagrammatic relations between projections of templates and the algebraic structures are used to define invariants. We also construct 3-manifolds via framed links associated to tamplate diagrams, so that any 3-manifold invariant can be used as a template invariant