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A kernel-based approach to molecular conformation analysis
No full textWe present a novel machine learning approach to understanding conformation dynamics of biomolecules. The approach combines kernel-based techniques that are popular in the machine learning community with transfer operator theory for analyzing dynamical systems in order to identify conformation dynamics based on molecular dynamics simulation data. We show that many of the prominent methods like Markov State Models, EDMD, and TICA can be regarded as special cases of this approach and that new efficient algorithms can be constructed based on this derivation. The results of these new powerful methods will be illustrated with several examples, in particular the alanine dipeptide and the protein NTL9
A strongly convergent numerical scheme from ensemble Kalman inversion
No full textThe Ensemble Kalman methodology in an inverse problems setting can be viewed as an iterative scheme, which is a weakly tamed discretization scheme for a certain stochastic differential equation (SDE). Assuming a suitable approximation result, dynamical properties of the SDE can be rigorously pulled back via the discrete scheme to the original Ensemble Kalman inversion. The results of this paper make a step towards closing the gap of the missing approximation result by proving a strong convergence result in a simplified model of a scalar SDE. We focus here on a toy model with similar properties to the one arising in the context of an Ensemble Kalman filter. The proposed model can be interpreted as a single particle filter for a linear map and thus forms the basis for further analysis. The difficulty in the analysis arises from the formally derived limiting SDE with nonglobally Lipschitz continuous nonlinearities both in the drift and in the diffusion. Here the standard Euler--Maruyama scheme might fail to provide a strongly convergent numerical scheme and taming is necessary. In contrast to the strong taming usually used, the method presented here provides a weaker form of taming. We present a strong convergence analysis by first proving convergence on a domain of high probability by using a cutoff or localization, which then leads, combined with bounds on moments for both the SDE and the numerical scheme, by a bootstrapping argument to strong convergence
Importance sampling in path space for diffusion processes with slow-fast variables
Get PDFImportance sampling is a widely used technique to reduce the variance of a Monte Carlo estimator by an appropriate change of measure. In this work, we study importance sampling in the framework of diffusion process and consider the change of measure which is realized by adding a control force to the original dynamics. For certain exponential type expectation, the corresponding control force of the optimal change of measure leads to a zero-variance estimator and is related to the solution of a Hamilton-Jacobi-Bellmann equation. We focus on certain diffusions with both slow and fast variables, and the main result is that we obtain an upper bound of the relative error for the importance sampling estimators with control obtained from the limiting dynamics. We demonstrate our approximation strategy with a simple numerical example
A Moving Boundary Flux Stabilization Method for Cartesian Cut-Cell Grids using Directional Operator Splitting
No full textAn explicit moving boundary method for the numerical solution of time-dependent hyperbolic conservation laws on grids produced by the intersection of complex geometries with a regular Cartesian grid is presented. As it employs directional operator splitting, implementation of the scheme is rather straightforward. Extending the method for static walls from Klein et al., Phil. Trans. Roy. Soc., A367, no. 1907, 4559-4575 (2009), the scheme calculates fluxes needed for a conservative update of the near-wall cut-cells as linear combinations of standard fluxes from a one-dimensional extended stencil. Here the standard fluxes are those obtained without regard to the small sub-cell problem, and the linear combination weights involve detailed information regarding the cut-cell geometry. This linear combination of standard fluxes stabilizes the updates such that the time-step yielding marginal stability for arbitrarily small cut-cells is of the same order as that for regular cells. Moreover, it renders the approach compatible with a wide range of existing numerical flux-approximation methods. The scheme is extended here to time dependent rigid boundaries by reformulating the linear combination weights of the stabilizing flux stencil to account for the time dependence of cut-cell volume and interface area fractions. The two-dimensional tests discussed include advection in a channel oriented at an oblique angle to the Cartesian computational mesh, cylinders with circular and triangular cross-section passing through a stationary shock wave, a piston moving through an open-ended shock tube, and the flow around an oscillating NACA 0012 aerofoil profile
Asymptotics for moist deep convection I: Refined scalings and self-sustaining updrafts
Get PDFMoist processes are among the most important drivers of atmospheric dynamics, and scale analysis and asymptotics are cornerstones of theoretical meteorology. Accounting for moist processes in systematic scale analyses therefore seems of considerable importance for the field. Klein & Majda (TCFD, vol. 20, 525--552, (2006)) proposed a scaling regime for the incorporation of moist bulk microphysics closures in multi-scale asymptotic analyses of tropical deep convection. This regime is refined here to allow for mixtures of ideal gases and to establish consistency with a more general multiple scales modelling framework for atmospheric flows.
Deep narrow updrafts, so-called "hot towers", constitute principal building blocks of larger scale storm systems. They are analysed here in a sample application of the new scaling regime. A single quasi-onedimensional columnar cloud is considered on the vertical advective (or tower life cycle) time scale. The refined asymptotic scaling regime is essential for this example as it reveals a new mechanism for the self-sustainance of such updrafts. Even for strongly positive convectively available potential energy (CAPE), a vertical balance of buoyancy forces is found in the presence of precipitation. This balance induces a diagnostic equation for the vertical velocity and it is responsible for the generation of self-sustained balanced updrafts. The time dependent updraft
structure is encoded in a Hamilton-Jacobi equation for the precipitation mixing ratio. Numerical solutions of this equation suggest that the self-sustained updrafts may strongly enhance hot tower life cycles
A Semi-Implicit Multiscale Scheme for Shallow Water Flows at Low Froude Number
Get PDFA new large time step semi-implicit multiscale method is presented for the solution of low Froude-number shallow water flows. While on small scales which are under-resolved in time the impact of source terms on the divergence of the flow is essentially balanced, on large resolved scales the scheme propagates free gravity waves with minimized diffusion. The scheme features a scale decomposition based on multigrid ideas. Two different time integrators are blended at each scale depending on the scale-dependent Courant number for gravity wave propagation. The finite-volume discretization is based on a Cartesian grid and is second order accurate. The basic properties of the method are validated by numerical tests. This development is a further step in the development of asymptotically adaptive numerical methods for the computation of large scale atmospheric flows
Statistical Analysis of the First Passage Path Ensemble of Jump Processes
No full textThe transition mechanism of jump processes between two different subsets in state space reveals important dynamical information of the processes and therefore has attracted considerable attention in the past years. In this paper, we study the first passage path ensemble of both discrete-time and continuous-time jump processes on a finite state space. The main approach is to divide each first passage path into nonreactive and reactive segments and to study them separately. The analysis can be applied to jump processes which are non-ergodic, as well as continuous-time jump processes where the waiting time distributions are non-exponential. In the particular case that the jump processes are both Markovian and ergodic, our analysis elucidates the relations between the study of the first passage paths and the study of the transition paths in transition path theory. We provide algorithms to numerically compute statistics of the first passage path ensemble. The computational complexity of these algorithms scales with the complexity of solving a linear system, for which efficient methods are available. Several examples demonstrate the wide applicability of the derived results across research areas
Fuel injection control for a valve array in a Shockless Explosion Combustor
No full textApproximate constant volume combustion (aCVC) is a promising way to optimize the combustion process in a gas turbine, which would exceed the gain in efficiency resulting from optimizing other components significantly. This work deals with a recently proposed approach: shockless explosion combustion (SEC). Compared to already known concepts, such as pulsed detonation combustion (PDC), it overcomes several disadvantages, e.g., sharp pressure transitions and entropy generation due to shock waves. For an SEC, accurate fuel stratification is required to achieve a quasi-homogeneous auto-ignition. In an atmospheric test rig quasi-homogeneous ignitions were achieved previously in non-resonant operation. To achieve a resonant operation, which goes along with a higher firing frequency, lower ignition and injection times are required. For this purpose, an array of solenoid valves was designed to allow for highly dynamic operation within short filling time spans. Using a novel mixed-integer control approach, these solenoid valves were actuated such that a desired fuel profile was generated. In this paper, the mentioned test rig was used for non-reacting fuel measurements to compare the quality of the axial fuel stratification achieved by using the valve array with the one achieved by using a slower proportional valve. In the experimental investigation the actuation with the valve array proved to adjust the required fuel stratification with the same quality as the actuation with the proportional valve, which was already successfully applied to the reactive set-up. Hence, the mixed-integer controlled valve array is considered a useful concept for upcoming resonant reactive SEC investigations
Large deformation shape uncertainty quantification in acoustic scattering
No full textWe address shape uncertainty quantification for the two-dimensional Helmholtz transmission problem, where the shape of the scatterer is the only source of uncertainty. In the framework of the so-called deterministic approach, we provide a high-dimensional parametrization for the interface. Each domain configuration is mapped to a nominal configuration, obtaining a problem on a fixed domain with stochastic coefficients. To compute surrogate models and statistics of quantities of interest, we apply an adaptive, anisotropic Smolyak algorithm, which allows to attain high convergence rates that are independent of the number of dimensions activated in the parameter space. We also develop a regularity theory with respect to the spatial variable, with norm bounds that are independent of the parametric dimension. The techniques and theory presented in this paper can be easily generalized to any elliptic problem on a stochastic domain