On the algorithmic solution of optimization problems subject to probabilistic/robust (probust) constraints

We present an adaptive grid refinement algorithm to solve probabilistic optimization problems with infinitely many random constraints. Using a bilevel approach, we iteratively aggregate inequalities that provide most information not in a geometric but in a probabilistic sense. This conceptual idea, for which a convergence proof is provided, is then adapted to an implementable algorithm. The efficiency of our approach when compared to naive methods based on uniform grid refinement is illustrated for a numerical test example as well as for a water reservoir problem with joint probabilistic filling level constraints.

Adaptive Mesh Refinement Strategies for a Novel Modelof Immiscible Fluid Flow in Fractures

In this paper, we consider two Adaptive Mesh Refinement (AMR) methods to simulate flow through fractures using a novel multiphase model. The approach represents the fluid using a two-dimensional parallel-plate model that employs techniques adapted from lattice-Boltzmann simulations to track the fluid interface. Here, we discuss different mesh refinement strategies for the model and compare their performance to that of a uniform grid. Results from the simulations are demonstrated showing excellent agreement between the model and analytical solutions for both unrefined and refined meshes. We also present results from the study that illustrate the behavior of the AMR front-tracking method. The AMR model is able to accurately track the interfacial properties in cases where uniform fine meshes would significantly increase the simulation cost.The ability of the model to dynamically refine the domain is demonstrated by presenting the results from an example with evolving interfaces.

Assessment of Globularity of Protein Structures via Minimum Volume Ellipsoids and Voxel-Based Atom Representation

A computer algorithm for assessment of globularity of protein structures is presented. By enclosing the input protein in a minimum volume ellipsoid (MVEE) and calculating a profile measuring how voxelized space within this shape (cubes on a uniform grid) is occupied by atoms, it is possible to estimate how well the molecule resembles a globule. For any protein to satisfy the proposed globularity criterion, its ellipsoid profile (EP) should first confirm that atoms adequately fill the ellipsoid’s center. This property should then propagate towards the surface of the ellipsoid, although with diminishing importance. It is not required to compute the molecular surface. Globular status (full or partial) is assigned to proteins with values of their ellipsoid profiles, called here the ellipsoid indexes (EI), above certain levels. Due to structural outliers which may considerably distort the measurements, a companion method for their detection and reduction of their influence is also introduced. It is based on kernel density estimation and is shown to work well as an optional input preparation step for MVEE. Finally, the complete workflow is applied to over two thousand representatives of SCOP 2.08 domain superfamilies, surveying the landscape of tertiary structure of proteins from the Protein Data Bank.

Elements of future snowpack modeling – Part 2: A modular and extendable Eulerian–Lagrangian numerical scheme for coupled transport, phase changes and settling processes

A coupled treatment of transport processes, phase changes and mechanical settling is the core of any detailed snowpack model. A key concept underlying the majority of these models is the notion of layers as deforming material elements that carry the information on their physical state. Thereby an explicit numerical solution of the ice mass continuity equation can be circumvented, although with the downside of virtual no flexibility in implementing different coupling schemes for densification, phase changes and transport. As a remedy we consistently recast the numerical core of a snowpack model into an extendable Eulerian–Lagrangian framework for solving the coupled non-linear processes. In the proposed scheme, we explicitly solve the most general form of the ice mass balance using the method of characteristics, a Lagrangian method. The underlying coordinate transformation is employed to state a finite-difference formulation for the superimposed (vapor and heat) transport equations which are treated in their Eulerian form on a moving, spatially non-uniform grid that includes the snow surface as a free upper boundary. This formulation allows us to unify the different existing viewpoints of densification in snow or firn models in a flexible way and yields a stable coupling of the advection-dominated mechanical settling with the remaining equations. The flexibility of the scheme is demonstrated within several numerical experiments using a modular solver strategy. We focus on emerging heterogeneities in (two-layer) snowpacks, the coupling of (solid–vapor) phase changes with settling at layer interfaces and the impact of switching to a non-linear mechanical constitutive law. Lastly, we discuss the potential of the scheme for extensions like a dynamical equation for the surface mass balance or the coupling to liquid water flow.

Some aspects in Kelvin-Helmholtz instability with and without Boussinesq approximation

The topic of this paper is the Kelvin-Helmholtz instability, a phenomenon which occurs on the interface of a stratified fluid, in the presence of a parallel shear flow, when there is a velocity and density difference across the interface of two adjacent layers. This paper focuses on a numerical simulation modelled by the Taylor-Goldstein equation, which represents a more realistic case compared to the basic Kelvin-Helmholtz shear flow. The Euler system is solved with new modelled smooth velocity and density profiles at the interface. The flux at cell boundaries is reconstructed by implementing a third order WENO (Weighted Essentially Non-Oscillatory) method. Next, a Riemann solver builds the fluxes at cell interfaces. The use of both Rusanov and HLLC solvers is investigated. Temporal discretization is done by applying the second order TVD (total variation diminishing) Runge-Kutta method on a uniform grid. Numerical simulations are performed comparatively for both Kelvin-Helmholtz and Taylor-Goldstein instabilities, on the same simulation domains. We find that increasing the number of grid points leads to a better accuracy in shear layer vortices visualization. Thus, we can conclude that applying the Taylor-Goldstein equation improves the realism in the general fluid instability modelling.

Adaptive method for the solution of 1D and 2D advection–diffusion equations used in environmental engineering

The paper concerns the numerical solution of one-dimensional (1D) and two-dimensional (2D) advection–diffusion equations. For the numerical solution of the 1D advection–diffusion equation, a method, originally proposed for the solution of the 1D pure advection equation, has been developed. A modified equation analysis carried out for the proposed method allowed increasing of the resulting solution accuracy and, consequently, to reduce the numerical dissipation and dispersion. This is achieved by proper choice of the involved weighting parameter being a function of the Courant number and the diffusive number. The method is adaptive because for uniform grid point and for uniform flow velocity, the weighting parameter takes a constant value, whereas for non-uniform grid and for varying flow velocity, its value varies in the region of solution. For the solution of the 2D transport equation, the dimensional decomposition in the form of Strang splitting technique is used. Consequently, this equation is reduced to a series of the 1D equations with regard to x- and y-directions which next are solved using the aforementioned method. The results of computational experiments compared with the exact solutions confirmed that the proposed approaches ensure high solution accuracy of the transport equations.

Some aspects of extrapolation based on interpolation polynomials

The problem of extrapolation on the basis of interpolation polynomials is considered in the paper. A simple computational procedure is proposed to find the predicted value for a polynomial of any degree under conditions of a uniform grid. An algorithm for determining the best polynomial for extrapolation is proposed. To construction of integral transformation for operator of equation of convective diffusion under mixed boundary conditions.

Grasp Stability Prediction for a Dexterous Robotic Hand Combining Depth Vision and Haptic Bayesian Exploration

Grasp stability prediction of unknown objects is crucial to enable autonomous robotic manipulation in an unstructured environment. Even if prior information about the object is available, real-time local exploration might be necessary to mitigate object modelling inaccuracies. This paper presents an approach to predict safe grasps of unknown objects using depth vision and a dexterous robot hand equipped with tactile feedback. Our approach does not assume any prior knowledge about the objects. First, an object pose estimation is obtained from RGB-D sensing; then, the object is explored haptically to maximise a given grasp metric. We compare two probabilistic methods (i.e. standard and unscented Bayesian Optimisation) against random exploration (i.e. uniform grid search). Our experimental results demonstrate that these probabilistic methods can provide confident predictions after a limited number of exploratory observations, and that unscented Bayesian Optimisation can find safer grasps, taking into account the uncertainty in robot sensing and grasp execution.