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Subsampling in ensemble Kalman inversion
No full textWe consider the Ensemble Kalman Inversion which has been recently introduced as an efficient, gradient-free optimisation method to estimate unknown parameters in an inverse setting. In the case of large data sets, the Ensemble Kalman Inversion becomes computationally infeasible as the data misfit needs to be evaluated for each particle in each iteration. Here, randomised algorithms like stochastic gradient descent have been demonstrated to successfully overcome this issue by using only a random subset of the data in each iteration, so-called subsampling techniques. Based on a recent analysis of a continuous-time representation of stochastic gradient methods, we propose, analyse, and apply subsampling-techniques within Ensemble Kalman Inversion. Indeed, we propose two different subsampling techniques: either every particle observes the same data subset (single subsampling) or every particle observes a different data subset (batch subsampling)
Workshop Report 21w5167 Optimization under Uncertainty: Learning and Decision Making in 2021
No full textThis workshop brought together researchers in optimization under uncertainty and uncertainty quantification, whose work stands to benefit from cutting-edge machine learning techniques for data-driven differential equations. The workshop focused on mathematical challenges at the interface of applied mathematics, optimization, probability and statistics, and machine learning. Through eleven scientific talks, three panel discussions, and unstructured interactions, the workshop facilitated an exchange of ideas towards new, powerful methods for optimization under uncertainty. In what follows, we give an overview of the presentations, discussions, and outcomes
Future changes in North Atlantic winter cyclones in CESM-LE – Part 1: Cyclone intensity, potential vorticity anomalies, and horizontal wind speed
Get PDFStrong low-level winds associated with extratropical
cyclones can have substantial impacts on society. The
wind intensity and the spatial distribution of wind maxima
may change in a warming climate; however, the involved
changes in cyclone structure and dynamics are not entirely
clear. Here, such structural changes of strong North Atlantic
cyclones in a warmer climate close to the end of the current
century are investigated with storm-relative composites
based on Community Earth System Model Large Ensemble
(CESM-LE) simulations. Furthermore, a piecewise potential
vorticity inversion is applied to associate such changes
in low-level winds to changes in potential vorticity (PV)
anomalies at different levels. Projected changes in cyclone
intensity are generally rather small. However, using cyclonerelative
composites, we identify an extended wind footprint
southeast of the center of strong cyclones, where the wind
speed tends to intensify in a warmer climate. Both an amplified
low-level PV anomaly driven by enhanced diabatic
heating and a dipole change in upper-level PV anomalies
contribute to this wind intensification. On the contrary, wind
changes associated with lower- and upper-level PV anomalies
mostly compensate for each other upstream of the cyclone
center. Wind changes at upper levels are dominated
by changes in upper-level PV anomalies and the background
flow. Altogether, our results indicate that a complex interaction
of enhanced diabatic heating and altered non-linear
upper-tropospheric wave dynamics shape future changes in
near-surface winds in North Atlantic cyclones
Instability of the isothermal, hydrostatic equatorial atmosphere at rest under the full Coriolis acceleration
No full textThe traditional approximation neglects the cosine components of the Coriolis acceleration, and this approximation has been widely used in the study of geophysical phenomena. However, the justification of the traditional approximation is questionable under a few circumstances. In particular, dynamics with substantial vertical velocities or geophysical phenomena in the tropics have non-negligible cosine Coriolis terms. Such cases warrant investigations with the non-traditional setting, i.e., the full Coriolis acceleration. In this manuscript, we study the effect of the non-traditional setting on an isothermal, hydrostatic and compressible atmosphere assuming a meridionally homogeneous flow. Employing linear stability analysis, we show that, given appropriate boundary conditions, i.e. free-slip boundary at the surface and non-reflecting boundary at the top, the equatorial atmosphere at rest becomes unstable. Numerical experiments were conducted, and Rayleigh damping is used as a numerical approximation for the non-reflecting top boundary. Our two main results are as follows. 1) Experiments involving the full Coriolis terms exhibit exponentially growing instability while experiments subject to the same initial condition and the traditional approximation remain stable. 2) The experimental instability growth rate is close to the theoretical value. Despite the limitations of our investigations wherein only studies on the f-plane are conducted, and effects from the β-plane approximation are ignored, the presence of this instability may have physical and experimental implications for the non-traditional setting. A discussion of the limitations and implications of our study concludes our investigations. Specifically, the influence on numerical deep-atmosphere models and the validity of assuming meridionally homogeneous flow are discussed.
Ray , Mark Schlutow, Rupert Klei
A probabilistic framework for particle-based reaction–diffusion dynamics using classical Fock space representations
No full textThe modeling and simulation of stochastic reaction–diffusion processes is a topic of steady interest that is approached with a wide range of methods. At the level of particle-resolved descriptions, where chemical reactions are coupled to the spatial diffusion of individual particles, there exist comprehensive numerical simulation schemes, while the corresponding mathematical formalization is relatively underdeveloped. The aim of this paper is to provide a framework to systematically formulate the probabilistic evolution equation, termed chemical diffusion master equation (CDME), that governs particle-based stochastic reaction–diffusion processes. To account for the non-conserved and unbounded particle number of this type of open systems, we employ a classical analogue of the quantum mechanical Fock space that contains the symmetrized probability densities of the many-particle configurations in space. Following field-theoretical ideas of second quantization, we introduce creation and annihilation operators that act on single-particle densities and provide natural representations of symmetrized probability densities as well as of reaction and diffusion operators. These operators allow us to consistently and systematically formulate the CDME for arbitrary reaction schemes. The resulting form of the CDME further serves as the foundation to derive more coarse-grained descriptions of reaction–diffusion dynamics. In this regard, we show that a discretization of the evolution equation by projection onto a Fock subspace generated by a finite set of single-particle densities leads to a generalized form of the well-known reaction–diffusion master equation, which supports non-local reactions between grid cells and which converges properly in the continuum limit
Numerical and Experimental Evaluation of Shock Dividers
No full textMitigation of pressure pulsations in the exhaust of a pulse detonation combustor is crucial for operation with a downstream turbine. For this purpose, a device termed the shock divider is designed and investigated. The intention of the divider is to split the leading shock wave into two weaker waves that propagate along separated ducts with different cross sections, allowing the shock waves to travel with different velocities along different paths. The separated shock waves redistribute the energy of the incident shock wave. The shock dynamics inside the divider are investigated using numerical simulations. A second-order dimensional split finite volume MUSCL-scheme is used to solve the compressible Euler equations. Furthermore, low-cost simulations are performed using geometrical shock dynamics to predict the shock wave propagation inside the divider. The numerical simulations are compared to high-speed schlieren images and time-resolved total pressure recording. For the latter, a high-frequency pressure probe is placed at the divider outlet, which is shown to resolve the transient total pressure during the shock passage. Moreover, the separation of the shock waves is investigated and found to grow as the divider duct width ratio increases. The numerical and experimental results allow for a better understanding of the dynamic evolution of the flow inside the divider and inform its capability to reduce the pressure pulsations at the exhaust of the pulse detonation combustor
Boundary Conditions on Conical Hydrophobic Inclusions in Lipid Membranes
Get PDFUsing all-atom Molecular Dynamics simulations including explicit water, we consider hydrophobic
inclusions of conical shape that are inserted into phospholipid bilayer membranes with the goal to
determine the boundary conditions at the inclusion-membrane interface. We determine the mean
membrane shape around the inclusion and from that extract the membrane vertical-displacement,
thickness and bending-angle pro�les as possible order parameters in a coarse-grained description
of the membrane shape. Via comparison with solutions of membrane-shape equations obtained
by Landau theory, we investigate the appropriate boundary condition at the inclusion-membrane
interface. We �nd that in the considered cone opening angle range, the boundary values of the
di�erent order parameters that describe the membrane deformation are rather constant, which
re
ects an inherently preferred shape of membranes at conical inclusions
Robust SDE-Based Variational Formulations for Solving Linear PDEs via Deep Learning
No full textThe combination of Monte Carlo methods and deep learning has recently led to efficient algorithms for solving partial differential equations (PDEs) in high dimensions. Related learning problems are often stated as variational formulations based on associated stochastic differential equations (SDEs), which allow the minimization of corresponding losses using gradient-based optimization methods. In respective numerical implementations it is therefore crucial to rely on adequate gradient estimators that exhibit low variance in order to reach convergence accurately and swiftly. In this article, we rigorously investigate corresponding numerical aspects that appear in the context of linear Kolmogorov PDEs. In particular, we systematically compare existing deep learning approaches and provide theoretical explanations for their performances. Subsequently, we suggest novel methods that can be shown to be more robust both theoretically and numerically, leading to substantial performance improvements
Solving the time-independent Schrödinger equation for chains of coupled excitons and phonons using tensor trains
No full textWe demonstrate how to apply the tensor-train format to solve the time-independent Schrödinger equation for quasi one-dimensional excitonic chain systems with and without periodic boundary conditions. The coupled excitons and phonons are modeled by Fröhlich-Holstein type Hamiltonians with on-site and nearest-neighbor interactions only. We reduce the memory consumption as well as the computational costs significantly by employing efficient decompositions to construct low rank tensor-train representations, thus mitigating the curse of dimensionality. In order to compute also higher quantum states, we introduce an approach which directly incorporates the Wielandt deflation technique into the alternating linear scheme for the solution of eigenproblems. Besides systems with coupled excitons and phonons, we also investigate uncoupled problems for which (semi-)analytical results exist. There, we find that in case of homogeneous systems the tensor-train ranks of state vectors only marginally depend on the chain length which results in a linear growth of the storage consumption. However, the CPU time increases slightly faster with the chain length than the storage consumption because the alternating linear scheme adopted in our work requires more iterations to achieve convergence for longer chains and a given rank. Finally, we demonstrate that the tensor-train approach to the quantum treatment of coupled excitons and phonons makes it possible to directly tackle the phenomenon of mutual self-trapping. We are able to confirm the main results of the Davydov theory, i.e., the dependence of the wave packet width and the corresponding stabilization energy on the exciton-phonon coupling strength, though only for a certain range of that parameter. In future work, our approach will allow calculations also beyond the validity regime of that theory and/or beyond the restrictions of the Fröhlich-Holstein type Hamiltonians
Time-dependent friction effects on vibrational infrared frequencies and line shapes of liquid water
No full textFrom ab initio simulations of liquid water, the time-dependent friction functions and time-averaged
non-linear effective bond potentials for the OH stretch and HOH bend vibrations are extracted. The
obtained friction exhibits adiabatic contributions at and below the vibrational time scales, but also
much slower non-adiabatic contributions, reflecting homogeneous and inhomogeneous line broadening
mechanisms, respectively. Compared to the gas phase, hydration softens both stretch and bend
potentials, which by itself would lead to a red-shift of the corresponding vibrational bands. In
contrast, non-adiabatic friction contributions cause a spectral blue shift. For the stretch mode, the
potential effect dominates and thus a significant red shift when going from gas to the liquid phase
results. For the bend mode, potential and non-adiabatic friction effects are of comparable magnitude,
so that a slight blue shift results, in agreement with well-known but puzzling experimental findings.
The observed line broadening is shown to be roughly equally caused by adiabatic and non-adiabatic
friction contributions for both, the stretch and bend modes in liquid water. Thus, the understanding
of infrared vibrational frequencies and line shapes is considerably advanced by the quantitative
analysis of the time-dependent friction that acts on vibrational modes in liquid