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Global well-posedness for the primitive equations coupled to nonlinear moisture dynamics with phase changes
Get PDFIn this work we study the global solvability of the primitive equations for the atmosphere coupled to moisture dynamics with phase changes for warm clouds, where water is present in the form of water vapor and in the liquid state as cloud water and rain water. This moisture model contains closures for the phase changes condensation and evaporation, as well as the processes of auto conversion of cloud water into rainwater and the collection of cloud water by the falling rain droplets. It has been used by Klein and Majda in [17] and corresponds to a basic form of the bulk microphysics closure in the spirit of Kessler [16] and Grabowski and Smolarkiewicz [12]. The moisture balances are strongly coupled to the thermodynamic equation via the latent heat associated to the phase changes. In [14] we assumed the velocity field to be given and proved rigorously the global existence and uniqueness of uniformly bounded solutions of the moisture balances coupled to the thermodynamic equation. In this paper we present the solvability of a full moist atmospheric flow model, where the moisture model is coupled to the primitive equations of atmospherical dynamics governing the velocity field. For the derivation of a priori estimates for the velocity field we thereby use the ideas of Cao and Titi [6], who succeeded in proving the global solvability of the primitive equations
The fractional p-Laplacian emerging from homogenization of the random conductance model with degenerate ergodic weights and unbounded-range jumps
Get PDFWe study a general class of discrete p-Laplace operators in the random conductance
model with long-range jumps and ergodic weights. Using a variational formulation of the
problem, we show that under the assumption of bounded �rst moments and a suitable
lower moment condition on the weights, the homogenized limit operator is a fractional
p-Laplace operator.
Under strengthened lower moment conditions, we can apply our insights also to the
spectral homogenization of the discrete Laplace operator to the continuous fractional
Laplace operator
Ensemble Kalman filter for neural network-based one-shot inversion
No full textWe studythe useofnoveltechniquesarisinginmachinelearningforinverse problems. Our approach replaces the complex forward model by a neural network, which is trained simultaneously in a one-shot sense when estimating the unknown parametersfromdata, ie, theneuralnetworkistrainedonlyfortheunknownparameter. By establishing a link to the Bayesian approach to inverse problems we develop an algorithmic framework that ensures the feasibility of the parameter estimate with respect to the forward model. We propose an efficient, derivative-free optimization method based on variants of the ensemble Kalman inversion. Numerical experiments show that the ensemble Kalman filter for neural network-based one-shot inversion is a promising direction combining optimization and machine learning techniques for inverse problems
Kernel Conditional Density Operators
No full textWe introduce a novel conditional density estimation model termed the conditional density operator (CDO). It naturally captures multivariate, multimodal output densities and shows performance that is competitive with recent neural conditional density models and Gaussian processes. The proposed model is based on a novel approach to the reconstruction of probability densities from their kernel mean embeddings by drawing connections to estimation of Radon-Nikodym derivatives in the reproducing kernel Hilbert space (RKHS). We prove finite sample bounds for the estimation error in a standard density reconstruction scenario, independent of problem dimensionality. Interestingly, when a kernel is used that is also a probability density, the CDO allows us to both evaluate and sample the output density efficiently. We demonstrate the versatility and performance of the proposed model on both synthetic and real-world data
Fast Optical Absorption Spectra Calculations for Periodic Solid State Systems
No full textWe present a method to construct an efficient approximation to the bare exchange andscreened direct interaction kernels of the Bethe-Salpeter Hamiltonian for periodic solid state systemsvia the interpolative separable density fitting technique. We show that the cost of constructing the approximate Bethe-Salpeter Hamiltonian scales nearly optimally asO(Nk) with respect to thenumber of samples in the Brillouin zoneNk. In addition, we show that the cost for applying the Bethe-Salpeter Hamiltonian to a vector scales asO(NklogNk). Therefore the optical absorption spectrum,as well as selected excitation energies can be efficiently computed via iterative methods such as the Lanczos method. This is a significant reduction from theO(N2k) andO(N3k) scaling associated with a brute force approach for constructing the Hamiltonian and diagonalizing the Hamiltonian respectively. We demonstrate the efficiency and accuracy of this approach with both one-dimensionalmodel problems and three-dimensional real materials (graphene and diamond). For the diamond system withNk= 2197, it takes 6 hours to assemble the Bethe-Salpeter Hamiltonian and 4 hours to fully diagonalize the Hamiltonian using 169 cores when the brute force approach is used. The new method takes less than 3 minutes to set up the Hamiltonian and 24 minutes to compute the absorption spectrum on a single core
A Mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation
Get PDFIn this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes
Low-cost scalable discretization, prediction and feature selection for complex systems
No full textFinding reliable discrete approximations of complex systems is a key prerequisite when applying many of the most popular modeling tools. Common discretization approaches (e.g., the very popular K-means clustering) are crucially limited in terms of quality, parallelizability, and cost. We introduce a low-cost improved quality scalable probabilistic approximation (SPA) algorithm, allowing for simultaneous data-driven optimal discretization, feature selection, and prediction. We prove its optimality, parallel efficiency, and a linear scalability of iteration cost. Cross-validated applications of SPA to a range of large realistic data classification and prediction problems reveal marked cost and performance improvements. For example, SPA allows the data-driven next-day predictions of resimulated surface temperatures for Europe with the mean prediction error of 0.75°C on a common PC (being around 40% better in terms of errors and five to six orders of magnitude cheaper than with common computational instruments used by the weather services)
The added value of large-eddy and storm-resolving models for simulating clouds and precipitation
No full textMore than one hundred days were simulated over very large domains with fine (0.156 km to 2.5 km) grid spacing for realistic conditions to test the hypothesis that storm (kilometer) and large-eddy (hectometer) resolving simulations would provide an improved representation of clouds and precipitation in atmospheric simulations. At scales that resolve convective storms (storm-resolving for short), the vertical velocity variance becomes resolved and a better physical basis is achieved for representing clouds and precipitation. Similarly to past studies we found an improved representation of precipitation at kilometer scales, as compared to models with parameterized convection. The main precipitation features (location, diurnal cycle and spatial propagation) are well captured already at kilometer scales, and refining resolution to hectometer scales does not substantially change the simulations in these respects. It does, however, lead to a reduction in the precipitation on the time-scales considered – most notably over the ocean in the tropics. Changes in the distribution of precipitation, with less frequent extremes are also found in simulations incorporating hectometer scales. Hectometer scales appear to be more important for the representation of clouds, and make it possible to capture many important aspects of the cloud field, from the vertical distribution of cloud cover, to the distribution of cloud sizes, and to the diel (daily) cycle. Qualitative improvements, particularly in the ability to differentiate cumulus from stratiform clouds, are seen when one reduces the grid spacing from kilometer to hectometer scales. At the hectometer scale new challenges arise, but the similarity of observed and simulated scales, and the more direct connection between the circulation and the unconstrained degrees of freedom make these challenges less daunting. This quality, combined with already improved simulation as compared to more parameterized models, underpins our conviction that the use and further development of storm-resolving models offers exciting opportunities for advancing understanding of climate and climate change
Theory and simulation of open systems out of equilibrium
Get PDFWe consider the theoretical model of Bergmann and Lebowitz for open systems out of equilibrium and translate its principles in the adaptive resolution simulation molecular dynamics technique. We simulate Lennard-Jones fluids with open boundaries in a thermal gradient and find excellent agreement of the stationary responses with the results obtained from the simulation of a larger locally forced closed system. The encouraging results pave the way for a computational treatment of open systems far from equilibrium framed in a well-established theoretical model that avoids possible numerical artifacts and physical misinterpretations
Graph Kernel Koopman Embedding for Human Microbiome Analysis
No full textMore and more diseases have been found to be strongly correlated with disturbances in the microbiome constitution, e.g., obesity, diabetes, or some cancer types. Thanks to modern high-throughput omics technologies, it becomes possible to directly analyze human microbiome and its influence on the health status. Microbial communities are monitored over long periods of time and the associations between their members are explored. These relationships can be described by a time-evolving graph. In order to understand responses of the microbial community members to a distinct range of perturbations such as antibiotics exposure or diseases and general dynamical properties, the time-evolving graph of the human microbial communities has to be analyzed. This becomes especially challenging due to dozens of complex interactions among microbes and metastable dynamics. The key to solving this problem is the representation of the time-evolving graphs as fixed-length feature vectors preserving the original dynamics. We propose a method for learning the embedding of the time-evolving graph that is based on the spectral analysis of transfer operators and graph kernels. We demonstrate that our method can capture temporary changes in the time-evolving graph on both synthetic data and real-world data. Our experiments demonstrate the efficacy of the method. Furthermore, we show that our method can be applied to human microbiome data to study dynamic processes