Freie Universität Berlin
Repository: Freie Universität Berlin (FU), Math Department (fu_mi_publications)Not a member yet
2251 research outputs found
Sort by
Moist available potential energy of the mean state of the atmosphere and the thermodynamic potential for warm conveyor belts and convection
Get PDFMuch of our understanding of atmospheric circulation
comes from relationships between aspects of the circulation
and the mean state of the atmosphere. In particular,
the concept of mean available potential energy (MAPE) has
been used previously to relate the strength of the extratropical
storm tracks to the zonal-mean temperature and humidity
distributions. Here, we calculate for the first time the MAPE
of the zonally varying (i.e., three-dimensional) time-mean
state of the atmosphere including the effects of latent heating.
We further calculate a local MAPE by restricting the domain
to an assumed eddy size, and we partition this local MAPE
into convective and nonconvective components. Local convective
MAPE maximizes in the subtropics and midlatitudes,
in many cases in regions of the world that are known to have
intense convection. Local nonconvective MAPE has a spatial
pattern similar to the Eady growth rate, although local
nonconvective MAPE has the advantage that it takes into account
latent heating. Furthermore, the maximum potential ascent
associated with local nonconvective MAPE is related to
the frequency of warm conveyor belts (WCBs), which are
ascending airstreams in extratropical cyclones with large impacts
on weather. This maximum potential ascent can be calculated
based only on mean temperature and humidity, and
WCBs tend to start in regions of high maximum potential ascent
on a given day. These advances in the use of MAPE are
expected to be helpful to connect changes in the mean state
of the atmosphere, such as under global warming, to changes
in important aspects of extratropical circulation
Towards Optimal Sobolev Norm Rates for the Vector-Valued Regularized Least-Squares Algorithm
No full textWe present the first optimal rates for infinite-dimensional vector-valued ridge regression on a continuous scale of norms that interpolate between L2 and the hypothesis space, which we consider as a vector-valued reproducing kernel Hilbert space. These rates allow to treat the misspecified case in which the true regression function is not contained in the hypothesis space. We combine standard assumptions on the capacity of the hypothesis space with a novel tensor product construction of vector-valued interpolation spaces in order to characterize the smoothness of the regression function. Our upper bound not only attains the same rate as real-valued kernel ridge regression, but also removes the assumption that the target regression function is bounded. For the lower bound, we reduce the problem to the scalar setting using a projection argument. We show that these rates are optimal in most cases and independent of the dimension of the output space. We illustrate our results for the special case of vector-valued Sobolev space
On the ensemble Kalman inversion under inequality constraints
No full textThe ensemble Kalman inversion (EKI), a recently introduced optimisation method for solving inverse problems, is widely employed for the efficient and derivative-free estimation of unknown parameters. Specifically in cases involving ill-posed inverse problems and high-dimensional parameter spaces, the scheme has shown promising success. However, in its general form, the EKI does not take constraints into account, which are essential and often stem from physical limitations or specific requirements. Based on a log-barrier approach, we suggest adapting the continuous-time formulation of EKI to incorporate convex inequality constraints. We underpin this adaptation with a theoretical analysis that provides lower and upper bounds on the ensemble collapse, as well as convergence to the constraint optimum for general nonlinear forward models. Finally, we showcase our results through two examples involving partial differential equations (PDEs)
Happy People--Image Synthesis as Black-Box Optimization Problem in the Discrete Latent Space of Deep Generative Models
No full textIn recent years, optimization in the learned latent space of deep generative models has been successfully applied to black-box optimization problems such as drug design, image generation or neural architecture search. Existing models thereby leverage the ability of neural models to learn the data distribution from a limited amount of samples such that new samples from the distribution can be drawn. In this work, we propose a novel image generative approach that optimizes the generated sample with respect to a continuously quantifiable property. While we anticipate absolutely no practically meaningful application for the proposed framework, it is theoretically principled and allows to quickly propose samples at the mere boundary of the training data distribution. Specifically, we propose to use tree-based ensemble models as mathematical programs over the discrete latent space of vector quantized VAEs, which can be globally solved. Subsequent weighted retraining on these queries allows to induce a distribution shift. In lack of a practically relevant problem, we consider a visually appealing application: the generation of happily smiling faces (where the training distribution only contains less happy people) - and show the principled behavior of our approach in terms of improved FID and higher smile degree over baseline approaches
Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era
Get PDFThis study presents a collection of purely data-driven workflows for constructing reduced-order models (ROMs) for distributed dynamical systems. The ROMs we focus on, are data-assisted models inspired by, and templated upon, the theory of Approximate Inertial Manifolds (AIMs); the particular motivation is the so-called post-processing Galerkin method of Garcia-Archilla, Novo and Titi. Its applicability can be extended: the need for accurate truncated Galerkin projections and for deriving closed-formed corrections can be circumvented using machine learning tools. When the right latent variables are not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a manifold learning scheme) can be used to discover good sets of latent variables and test their explainability. The proposed methodology can express the ROMs in terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both Black-Box and (theoretically-informed and data-corrected) Gray-Box models are described; the necessity for the latter arises when truncated Galerkin projections are so inaccurate as to not be amenable to post-processing. We use the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative partial differential equations to illustrate and successfully test the overall framework
From continuous-time formulations to discretization schemes: tensor trains and robust regression for BSDEs and parabolic PDEs
No full texthe numerical approximation of partial differential equations (PDEs) poses formidable challenges in high dimensions since classical grid-based methods suffer from the so-called curse of dimensionality. Recent attempts rely on a combination of Monte Carlo methods and variational formulations, using neural networks for function approximation. Extending previous work (Richter et al., 2021), we argue that tensor trains provide an appealing framework for parabolic PDEs: The combination of reformulations in terms of backward stochastic differential equations and regression-type methods holds the promise of leveraging latent low-rank structures, enabling both compression and efficient computation. Emphasizing a continuous-time viewpoint, we develop iterative schemes, which differ in terms of computational efficiency and robustness. We demonstrate both theoretically and numerically that our methods can achieve a favorable trade-off between accuracy and computational efficiency. While previous methods have been either accurate or fast, we have identified a novel numerical strategy that can often combine both of these aspects
Navigating protein landscapes with a machine-learned transferable coarse-grained model
No full textThe most popular and universally predictive protein simulation models employ all-atom molecular dynamics (MD), but they come at extreme computational cost. The development of a universal, computationally efficient coarse-grained (CG) model with similar prediction performance has been a long-standing challenge. By combining recent deep learning methods with a large and diverse training set of all-atom protein simulations, we here develop a bottom-up CG force field with chemical transferability, which can be used for extrapolative molecular dynamics on new sequences not used during model parametrization. We demonstrate that the model successfully predicts folded structures, intermediates, metastable folded and unfolded basins, and the fluctuations of intrinsically disordered proteins while it is several orders of magnitude faster than an all-atom model. This showcases the feasibility of a universal and computationally efficient machine-learned CG model for proteins
Utilizing alignment-free methods to enable quantitative gene expression analysis of large collections of sequencing data
No full textDue to advances in sequencing technologies, the amount of sequencing data is continuously increasing and has reached an amount that calls for new data management methods, to actually utilize the sequencing data. In the last years, a number of different indices haven been developed to simplify the data, thereby reducing the amount of space needed and enabling analysis on large collections of sequencing data. In this thesis, the index Needle will be introduced, which allows (semi-)quantitative analyses on large data sets and outperforms other existing solutions with regards to both space and speed. Needle, like other indices, is based on alignment-free methods because in this way the costly step of classical sequence analyses, the alignment, can be omitted. Alignment-free methods are based on short subsequences of the actual sequence data. There are multiple different methods to determine these subsequences and this thesis provides a detailed analysis and comparison to determine the best method for such indices. Moreover, the benchmarking application minions is introduced, which will make comparisons between these methods easier as adding future new methods is simple. Needle is capable of utilizing large collections of sequencing data and determining their gene expressions. Three analyses are performed, which act as a proof of concept for how Needle can be utilized for large collections of sequencing data. Therefore, Needle is applied in this thesis to find cancer signatures, a newly annotated mouse transcript and tissue specific differentially expressed genes for different large data sets. In summary, indices like Needle are needed to actually take advantage of the data wealth currently present in the biological and medical research field
Ensemble Kalman Inversion for Image Guided Guide Wire Navigation in Vascular Systems
No full textThis paper addresses the challenging task of guide wire navigation in cardiovascular interventions, focusing on the parameter estimation of a guide wire system using Ensemble Kalman Inversion (EKI) with a subsampling technique. The EKI uses an ensemble of particles to estimate the unknown quantities. However since the data misfit has to be computed for each particle in each iteration, the EKI may become computationally infeasible in the case of high-dimensional data, e.g. high-resolution images. This issue can been addressed by randomised algorithms that utilize only a random subset of the data in each iteration. We introduce and analyse a subsampling technique for the EKI, which is based on a continuous-time representation of stochastic gradient methods and apply it to on the parameter estimation of our guide wire system. Numerical experiments with real data from a simplified test setting demonstrate the potential of the method
A mathematical programming approach for resource allocation of data analysis workflows on heterogeneous clusters
No full textScientific communities are motivated to schedule their large-scale data analysis workflows in heterogeneous cluster environments because of privacy and financial issues. In such environments containing considerably diverse resources, efficient resource allocation approaches are essential for reaching high performance. Accordingly, this research addresses the scheduling problem of workflows with bag-of-task form to minimize total runtime (makespan). To this aim, we develop a mixed-integer linear programming model (MILP). The proposed model contains binary decision variables determining which tasks should be assigned to which nodes. Also, it contains linear constraints to fulfill the tasks requirements such as memory and scheduling policy. Comparative results show that our approach outperforms related approaches in most cases. As part of the post-optimality analysis, some secondary preferences are imposed on the proposed model to obtain the most preferred optimal solution. We analyze the relaxation of the makespan in the hope of significantly reducing the number of consumed nodes