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Like most of the numerical libraries for .NET, NMath is little more than a wrapper over an Intel MKL embedded in the .NET assembly, probably by linking with C++/CLI to create a mixed assembly. You've probably just benchmarked those bits that are not actually written in .NET.
Even more surprisingly, the performance discrepancies were sometimes huge. The most expensive commercial numerical library we tested (IMSL) was over 500× slower than the free FFTW library at the FFT benchmark and none of the libraries made any use of multiple cores at the time.
The performance advantage is actually provided by Intel MKL, which offers implementations extremely optimized for many CPUs. From my point of view, this is the crucial point. Using straight-forward, naiv C/C++ code wont necessarily give you superior performance over C#/.NET, it's sometimes even worse. However C++/CLI allows you to exploit all the "dirty" optimization options.
Today it's industry standard to make mixed .Net/native libraries in order to take advantages of both platforms for performance optimization. Not only NMath, many commercial and free libraries with .net interface working like this. For example: Math.NET Numerics, dnAnalytics, Extreme Optimization, FinMath and many others. Integration with MKL is extremely popular for .net numerical libraries, and most of them just use Managed C++ assembly as an intermediate level. But this solution has a number of drawbacks:
There is a common misbelief that arbitrary precision computations are very slow. Indeed, mainstream numerical software packages are largely responsible for this false perception. Their codes were written decades ago using textbook algorithms, without proper optimization nor updates for the latest hardware.
We are determined to change the situation by developing high-performance numerical libraries for computations with arbitrary precision, tuned for modern CPU architectures, multi-core parallelism and relying on recent state-of-the-art algorithms. All combined makes our toolbox order(s) of magnitude faster compared to famous competitors:
There is a need for scientists and engineers to have a numerical librarythat:is free (in the sense of freedom, not in the sense of gratis; see theGNU General Public License), so that people can use that library,redistribute it, modify it ...is written in C using modern coding conventions, calling conventions,scoping ...is clearly and pedagogically documented; preferably with TeXinfo, so asto allow online info, WWW and TeX output.uses top quality state-of-the-art algorithms.is portable and configurable using autoconf and automake.basically, is GNUlitically correct.There are strengths and weaknesses with existing libraries:
If the project has a philosophy it is to "Think in C". Since we areworking in C we should only do what is natural in C, rather than tryingto simulate features of other languages. If there is something which isunnatural in C and has to be simulated then we avoid using it. If thismeans leaving something out of the library, or only offering a limitedversion then so be it. It is not worthwhile making the libraryover-complicated. There are numerical libraries in other languages, andif people need the features of those languages it would be sensible forthem to use the corresponding libraries, rather than coercing a Clibrary into doing that job.
NEC Numeric Library Collection - collection of mathematical libraries that powerfully supports the development of numerical simulation programs in a wide range of fields. The libraries are available on Vector Engine. (C)
afnl - Fortran 90 numerical library with modules for numerical types, constants, error handling, integration, optimization, linear algebra, sorting and search, special functions, statistics, polynomials, root-finding, Fourier transforms, and dates and times, by Alberto Ramos (OS: GPL-2.0 License)
NumericalHUB - Set of modern Fortran numerical libraries covering: zeroes of Systems of Equations, Lagrange Interpolation, Finite Differences, Cauchy Problems, Boundary Value Problems, and Initial Boundary Value Problems, associated with book How to learn Applied Mathematics through modern FORTRAN
Maintenance management in wind energy industry has great impact on overall wind power cost. Optimizing maintenance strategies can substantially reduces the cost and makes wind energy more competitive among the energy resources. Due to the extreme conditions of remote or offshore sites where the wind turbines are installed, corrective maintenance and time-based preventive maintenance are the most adopted strategies in the wind industry in recent years. However, there is need to further reduce wind power cost via maintenance strategy improvement to increase its competitiveness. Industry and research community have been focusing on various maintenance strategies to save the maintenance cost.This thesis is devoted to developing cost-effective maintenance strategies for wind farms, focusing on conventional time-based maintenance optimization, and prognostics and condition-based maintenance (CBM) optimization within the CBM strategy framework.Studies are performed on improving corrective maintenance and time-based preventive maintenance strategies, which are currently widely adopted in wind industry. Opportunistic maintenance methods are proposed, which take advantage of economic dependencies existing among the wind turbines, and corrective maintenance chances, to implement preventive maintenance simultaneously. Imperfect preventive maintenance actions are considered as well. The methods demonstrate the immediate benefits of saving the overall maintenance cost for a wind farm.In the more advanced CBM strategy, the health conditions of components are monitored and predicted, based on which maintenance actions are scheduled to prevent unexpectedfailures while reducing the maintenance costs. Prognostic techniques are essential in CBM. In particular, the wind direction and speed around wind turbines are changing overtime, which leads to instantaneously time-varying load applied to the wind turbines rotors. With focus on gearbox failure due to the gear tooth crack, an integrated prognostics method is developed considering instantaneously varying load condition. The numerical examples demonstrate that the gearbox remaining useful life prediction considering time-varying load is more accurate compared to existing methods under constant-load assumption. In a subsequent extended study, uncertainty in gear tooth crack initiation time is further considered for wind turbine gearbox prognostics method development. The method provides more accurate gearbox remaining useful life prediction compared to the results without considering time-varying load condition.This thesis also proposes a CBM method considering different turbine types and lead times, as well as the production loss during the shutdown time. The capability to accurately estimate the average maintenance cost for a wind farm with diverse turbines is a key contribution of the proposed method. In addition, this thesis accounts for the inaccuracy in the simulation-based algorithms that most complex problems are solved with. A numerical method for CBM optimization of wind farms is developed to avoid the variations in CBM cost evaluation, which leads to a smooth cost function surface andbenefits the optimization process.The research in this thesis provides innovative methods for maintenance management in the wind power industry. The developed methods will help to significantly reduce the overall maintenance cost within either conventional maintenance or CBM strategies that the wind farm owners may apply. It will improve the competitive advantage of the wind energy, and promote a clean and sustainable energy future for the society in Canada and worldwide.
Abstract: Inaccuracies in the models of the Earth system, i.e., structural and parametric model errors, lead to inaccurate climate change projections.Errors in the model can originate from unresolved phenomena due to a low numerical resolution, as well as misrepresentations of physical phenomena or boundaries (e.g., orography). Therefore, such models lead to inaccurate short--term forecasts of weather and extreme events, and more importantly, long term climate projections. While calibration methods have been introduced to address for parametric uncertainties, e.g., by better estimation of system parameters from observations, addressing structural uncertainties, especially in an interpretable manner, remains a major challenge.Therefore, with increases in both the amount and frequency of observations of the Earth system, algorithmic innovations are required to identify interpretable representations of the model errors from observations. We introduce a flexible, general-purpose framework to discover interpretable model errors, and show its performance on a canonical prototype of geophysical turbulence, the two--level quasi--geostrophic system. Accordingly, a Bayesian sparsity--promoting regression framework is proposed, that uses a library of kernels for discovery of model errors. As calculating the library from noisy and sparse data (e.g., from observations) using convectional techniques leads to interpolation errors, here we use a coordinate-based multi--layer embedding to impute the sparse observations. We demonstrate the importance of alleviating spectral bias, and propose a random Fourier feature layer to reduce it in the proposed embeddings, and subsequently enable an accurate discovery. Our framework is demonstrated to successfully identify structural model errors due to linear and nonlinear processes (e.g., radiation, surface friction, advection), as well as misrepresented orography.
Abstract: Carbon capture and storage (CCS) is one of the most promising technologies for reducing greenhouse gas emissions and relies on numerical reservoir simulations for identifying and monitoring CO2 storage sites. In many commercial settings however, numerical reservoir simulations are too computationally expensive for important downstream application such as optimization or uncertainty quantification. Deep learning-based surrogate models offer the possibility to solve PDEs many orders of magnitudes faster than conventional simulators, but they are difficult to scale to industrial-scale problem settings. Using model-parallel deep learning, we train the largest CO2 surrogate model to date on a 3D simulation grid with two million grid points. To train the 3D simulator, we generate a new training dataset based on a real-world CCS simulation benchmark. Once trained, each simulation with the network is five orders of magnitude faster than a numerical reservoir simulator and 4,500 times cheaper. This paves the way to applications that require thousands of (sequential) simulations, such as optimizing the location of CO2 injection wells to maximize storage capacity and minimize risk of leakage. 2b1af7f3a8