亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

Adaptive finite elements combined with geometric multigrid solvers are one of the most efficient numerical methods for problems such as the instationary Navier-Stokes equations. Yet despite their efficiency, computations remain expensive and the simulation of, for example, complex flow problems can take many hours or days. GPUs provide an interesting avenue to speed up the calculations due to their very large theoretical peak performance. However, the large degree of parallelism and non-standard API make the use of GPUs in scientific computing challenging. In this work, we develop a GPU acceleration for the adaptive finite element library Gascoigne and study its effectiveness for different systems of partial differential equations. Through the systematic formulation of all computations as linear algebra operations, we can employ GPU-accelerated linear algebra libraries, which simplifies the implementation and ensures the maintainability of the code while achieving very efficient GPU utilizations. Our results for a transport-diffusion equation, linear elasticity, and the instationary Navier-Stokes equations show substantial speedups of up to 20X compared to multi-core CPU implementations.

The high dimensional nature of genomics data complicates feature selection, in particular in low sample size studies - not uncommon in clinical prediction settings. It is widely recognized that complementary data on the features, `co-data', may improve results. Examples are prior feature groups or p-values from a related study. Such co-data are ubiquitous in genomics settings due to the availability of public repositories. Yet, the uptake of learning methods that structurally use such co-data is limited. We review guided adaptive shrinkage methods: a class of regression-based learners that use co-data to adapt the shrinkage parameters, crucial for the performance of those learners. We discuss technical aspects, but also the applicability in terms of types of co-data that can be handled. This class of methods is contrasted with several others. In particular, group-adaptive shrinkage is compared with the better-known sparse group-lasso by evaluating feature selection. Finally, we demonstrate the versatility of the guided shrinkage methodology by showing how to `do-it-yourself': we integrate implementations of a co-data learner and the spike-and-slab prior for the purpose of improving feature selection in genetics studies.

Randomized algorithms have proven to perform well on a large class of numerical linear algebra problems. Their theoretical analysis is critical to provide guarantees on their behaviour, and in this sense, the stochastic analysis of the randomized low-rank approximation error plays a central role. Indeed, several randomized methods for the approximation of dominant eigen- or singular modes can be rewritten as low-rank approximation methods. However, despite the large variety of algorithms, the existing theoretical frameworks for their analysis rely on a specific structure for the covariance matrix that is not adapted to all the algorithms. We propose a general framework for the stochastic analysis of the low-rank approximation error in Frobenius norm for centered and non-standard Gaussian matrices. Under minimal assumptions on the covariance matrix, we derive accurate bounds both in expectation and probability. Our bounds have clear interpretations that enable us to derive properties and motivate practical choices for the covariance matrix resulting in efficient low-rank approximation algorithms. The most commonly used bounds in the literature have been demonstrated as a specific instance of the bounds proposed here, with the additional contribution of being tighter. Numerical experiments related to data assimilation further illustrate that exploiting the problem structure to select the covariance matrix improves the performance as suggested by our bounds.

Discretization of the uniform norm of functions from a given finite dimensional subspace of continuous functions is studied. Previous known results show that for any $N$-dimensional subspace of the space of continuous functions it is sufficient to use $e^{CN}$ sample points for an accurate upper bound for the uniform norm by the discrete norm and that one cannot improve on the exponential growth of the number of sampling points for a good discretization theorem in the uniform norm. In this paper we focus on two types of results, which allow us to obtain good discretization of the uniform norm with polynomial in $N$ number of points. In the first way we weaken the discretization inequality by allowing a bound of the uniform norm by the discrete norm multiplied by an extra factor, which may depend on $N$. In the second way we impose restrictions on the finite dimensional subspace under consideration. In particular, we prove a general result, which connects the upper bound on the number of sampling points in the discretization theorem for the uniform norm with the best $m$-term bilinear approximation of the Dirichlet kernel associated with the given subspace.

This paper discusses the error and cost aspects of ill-posed integral equations when given discrete noisy point evaluations on a fine grid. Standard solution methods usually employ discretization schemes that are directly induced by the measurement points. Thus, they may scale unfavorably with the number of evaluation points, which can result in computational inefficiency. To address this issue, we propose an algorithm that achieves the same level of accuracy while significantly reducing computational costs. Our approach involves an initial averaging procedure to sparsify the underlying grid. To keep the exposition simple, we focus only on one-dimensional ill-posed integral equations that have sufficient smoothness. However, the approach can be generalized to more complicated two- and three-dimensional problems with appropriate modifications.

In reinsurance, Poisson and Negative binomial distributions are employed for modeling frequency. However, the incomplete data regarding reported incurred claims above a priority level presents challenges in estimation. This paper focuses on frequency estimation using Schnieper's framework for claim numbering. We demonstrate that Schnieper's model is consistent with a Poisson distribution for the total number of claims above a priority at each year of development, providing a robust basis for parameter estimation. Additionally, we explain how to build an alternative assumption based on a Negative binomial distribution, which yields similar results. The study includes a bootstrap procedure to manage uncertainty in parameter estimation and a case study comparing assumptions and evaluating the impact of the bootstrap approach.

We consider the computation of statistical moments to operator equations with stochastic data. We remark that application of PINNs -- referred to as TPINNs -- allows to solve the induced tensor operator equations under minimal changes of existing PINNs code, and enabling handling of non-linear and time-dependent operators. We propose two types of architectures, referred to as vanilla and multi-output TPINNs, and investigate their benefits and limitations. Exhaustive numerical experiments are performed; demonstrating applicability and performance; raising a variety of new promising research avenues.

Agent-based models (ABMs) are simulation models used in economics to overcome some of the limitations of traditional frameworks based on general equilibrium assumptions. However, agents within an ABM follow predetermined, not fully rational, behavioural rules which can be cumbersome to design and difficult to justify. Here we leverage multi-agent reinforcement learning (RL) to expand the capabilities of ABMs with the introduction of fully rational agents that learn their policy by interacting with the environment and maximising a reward function. Specifically, we propose a 'Rational macro ABM' (R-MABM) framework by extending a paradigmatic macro ABM from the economic literature. We show that gradually substituting ABM firms in the model with RL agents, trained to maximise profits, allows for a thorough study of the impact of rationality on the economy. We find that RL agents spontaneously learn three distinct strategies for maximising profits, with the optimal strategy depending on the level of market competition and rationality. We also find that RL agents with independent policies, and without the ability to communicate with each other, spontaneously learn to segregate into different strategic groups, thus increasing market power and overall profits. Finally, we find that a higher degree of rationality in the economy always improves the macroeconomic environment as measured by total output, depending on the specific rational policy, this can come at the cost of higher instability. Our R-MABM framework is general, it allows for stable multi-agent learning, and represents a principled and robust direction to extend existing economic simulators.

Out-of-distribution (OOD) detection, crucial for reliable pattern classification, discerns whether a sample originates outside the training distribution. This paper concentrates on the high-dimensional features output by the final convolutional layer, which contain rich image features. Our key idea is to project these high-dimensional features into two specific feature subspaces, leveraging the dimensionality reduction capacity of the network's linear layers, trained with Predefined Evenly-Distribution Class Centroids (PEDCC)-Loss. This involves calculating the cosines of three projection angles and the norm values of features, thereby identifying distinctive information for in-distribution (ID) and OOD data, which assists in OOD detection. Building upon this, we have modified the batch normalization (BN) and ReLU layer preceding the fully connected layer, diminishing their impact on the output feature distributions and thereby widening the distribution gap between ID and OOD data features. Our method requires only the training of the classification network model, eschewing any need for input pre-processing or specific OOD data pre-tuning. Extensive experiments on several benchmark datasets demonstrates that our approach delivers state-of-the-art performance. Our code is available at //github.com/Hewell0/ProjOOD.

We derive information-theoretic generalization bounds for supervised learning algorithms based on the information contained in predictions rather than in the output of the training algorithm. These bounds improve over the existing information-theoretic bounds, are applicable to a wider range of algorithms, and solve two key challenges: (a) they give meaningful results for deterministic algorithms and (b) they are significantly easier to estimate. We show experimentally that the proposed bounds closely follow the generalization gap in practical scenarios for deep learning.