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Efficient numerical methods for large-scale scientific computations: Introduction
Accepted Manuscript Efficient numerical methods for large-scale scientific computations: Introduction Zahari Zlatev, Ivan Dimov, Ivan Lirkov PII: DOI: Reference:
S0377-0427(15)00274-5 http://dx.doi.org/10.1016/j.cam.2015.05.001 CAM 10162
To appear in:
Journal of Computational and Applied Mathematics
Please cite this article as: Z. Zlatev, I. Dimov, I. Lirkov, Efficient numerical methods for large-scale scientific computations: Introduction, Journal of Computational and Applied Mathematics (2015), http://dx.doi.org/10.1016/j.cam.2015.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Efficient Numerical Methods for Large-scale Scientific Computations: Introduction Zahari Zlatev 1) , Ivan Dimov 2 ) and Ivan Lirkov 2 ) 1)
Department of Environmental Science, Aarhus University, Roskilde, Denmark, [email protected]
2)
Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria, [email protected], [email protected]
The mathematical models are important tools in various research studies related to different areas of science and engineering. These models are very often described by systems of partial differential equations (PDEs), which in most of the cases are non-linear and time-dependent. It is necessary to apply different numerical methods both to discretize the systems of PDEs and to handle the arising after the discretization sub-problems (in many of the cases these sub-problems are systems of non-linear algebraic equations, which are usually huge in size; containing many millions of equations). The requirements for achieving sufficiently accurate model results lead very often to the necessity to discretize the models on fine grids. It is clear that the resulting sub-problems are indeed becoming very large in such a case and the use of efficient numerical methods is crucial in this situation. In fact, it becomes possible to handle successfully many of the advanced mathematical models only when efficient numerical methods are selected and correctly implemented during their treatment on the available high-speed computers. However, the selection of efficient numerical methods is by no means an easy task. It should be emphasized here that even when two large-scale models are described mathematically by equations containing the same formulae, the use of different numerical methods in their treatment might be appropriate when the operators involved in the models have different properties in spite of the fact that they are described by terms involving the same derivatives. For example, it will in general be better to treat the diffusion of pollutants in the atmosphere and the diffusion of pollutants in water by different numerical methods in spite of the fact that the mathematical models are described by the same type of equations. The fact that there arise many difficulties related to the choice of efficient numerical methods was well understood from the very beginning of the development of the numerical analysis and the computational science. Moreover, it is also well known that it is difficult to find the most efficient numerical method and to justify fully its application in a particular large-scale scientific and/or engineering application. Finding the most efficient numerical method is practically impossible in many of the complicated situations that have often to be handled in practice and this fact was also well known from the very beginning of the development of the numerical analysis. This is why R. W. Hamming wrote in one of the first books on numerical methods, [4], that the choice of a numerical method is in nearly all cases a question of finding a good compromise.
Many numerical methods are described and tested in connection with different large-scale scientific problems in the papers contained in CAM Special Issue on “Efficient Numerical Methods for Large-scale Scientific Computations”. There are
(a) various methods based on the implementation of finite element and finite difference methods, (b) different optimization techniques, (c) algorithms for treatment of Stokes and Navier-Stokes equations, (d) some modern parallel algorithms designed for powerful high-speed computers, (e) numerical methods based on the use of various Monte Carlo techniques, (f) some methods of Runge-Kutta type and (g) devises based on operator splitting procedures. Even more important is the fact that the listed above types of advanced numerical methods were actually designed for and used in many scientific and engineering areas: (1) Computational Biology, (2) Atomic Physics, (3) Quantum Chemistry, (4) Elastic Structures, (5) Epidemic Propagation, (6) Tumor Growth and Tumor Invasion, (7) Drop-coalescence, (8) Regional Air Pollution Problems, (9) Effect of Climatic Changes on the Humans Living in Big European Cities,
(10) Ensemble Modelling Related to Air Pollution Studies, (11) Treatment of Maxwell Equations and (12) Problems Arising in Finance Mathematics. This means that the readers will find in our Special Issue on “Efficient Numerical Methods for Large-scale Scientific Computations” descriptions of many numerical methods implemented in different application areas and will be able, when they need it, to select if not the best algorithms then some good ones for their specific needs. R. W. Hamming is very famous also for another statement in his book [4]. He wrote there that the purpose of the computations is insight not the numbers. In fact, in nearly all large conferences on numerical methods held in the last three decades of the twentieth century this statement was quoted at least once. However, the truth is that this statement became much more relevant, although it not quoted so often anymore, in the last ten-fifteen years, because the modern highspeed computer are producing enormous files of output data containing millions and millions of numbers. It is clear that taking random numbers from the output files has no meaning. It is necessary to study carefully the output data, by applying some very comprehensive devices, in order to be able to show clearly the different trends and the important relationships hidden in the enormous output data files. It will be illustrative to give an example in order to explain better the statements from the above paragraph. Numerical results obtained in computations carried out by using the Unified Danish Eulerian Model (UNI-DEM), fully described in [5] and [7], are used in this example. Actually, impacts of the climatic change on several critical pollution levels in different parts of Europe were systematically studied by using high-speed computers ([1], see also [5] and [7]) and the runs were carried out on a time-period of sixteen consecutive years with fourteen different scenarios, [2], [6], [8], [9] and [10]. Many different problems had to be resolved during this very comprehensive study. One of these problems was related to the correct use of the huge output data files in order to illustrate different trends and relationships. The total size of these files was greater than 200 Mega-Bytes. The computational difficulties during the runs of the model and the preparation of the output files were enormous, but these difficulties will not be discussed here (some details are given in [3]; see also [5] and [7]). The output data had to be used for many purposes. By selecting only the relevant data, one can illustrate and make easily understandable different important situations. This is demonstrated in the figures given below. All of these figures are related to the distribution of bad days in space and time. Consider an arbitrary point in the spatial domain of the model. If for some day the eight-hours averaged ozone concentration exceeds at least once 60 ppb, then the day under consideration is called a bad day. The number of bad days should not exceed 25, which is legislated in a directive of the parliament of the European Union (the exceedance of the critical level of 25 days could cause health problems for people suffering from asthmatic disease). Relevant results are shown and some relationships and trends are clearly seen in the figures given below:
(A) The distribution in years 1989 and 2003 of the numbers of bad days in Europe and the increases (in percent) caused by climatic changes are shown in Fig. 1. It is seen that (a) the numbers of bad days may vary very much from one year to another (cold and warm summers can cause rather considerable differences) and (b) the climatic changes are causing increases of the numbers of bad days in many parts of Europe. (B) The distribution of the numbers of bad days in Balkan Peninsula for 2004 is presented in Fig. 2. It is immediately seen that the critical level is exceeded in all countries of this peninsula. (C) The impact of the climatic changes on the numbers of bad days in the Balkan Peninsula for 2004 is shown in Fig. 3. It is seen that the climatic changes will lead to some increases of the numbers of bad days in nearly all countries in the Balkan Peninsula. (D) The variations of the numbers of bad days in several European cities during the studied period of 16 years are presented in Fig. 4. It is clearly seen that (a) the critical level is very often exceeded and (b) there is a trend for decreasing of the numbers of bad days at the end of the studied period of sixteen year (which is caused by the fact that the emissions in Europe were gradually decreased after year 1989). It was necessary to develop a special package of programs for the representations shown in the four figures. It should be emphasized here that the calculated by the model data and the graphical package can also be used to illustrate many other interesting trends and relationships, see [1], [2], [5] – [10]. The authors of the papers published in this Special Issue have taken care to illustrate the results obtained by the developed by them efficient numerical methods in appropriate figures and to explain different trends and relationships. Many of the papers in this Special Issue were presented at the Eighth International Conference on Numerical Methods and Applications held in August 20-24, 2014 in Borovets (Bulgaria). Some details about this conference can be found in the web-site: http://parallel.bas.bg/dpa/NMA_2014/. The conferences on Numerical Methods and Applications are regularly held in every fourth year. The next conference will be held in August 2018. Borovets is a very nice and quiet small town in the Southern part of Bulgaria and more precisely in the Rila Mountain (the highest mountain in Bulgaria and the Balkan Peninsula). It is primarily a well-known winter resort, but also the summer days are beautiful there. Normally, more than hundred participants from many countries attended the previous conferences on Numerical Methods and Applications, presented their results and had numerous interesting discussions during their stay.
We advise the readers of this Special Issue to consider the possibility of attending the next conference on Numerical Methods and Applications.
Figure 1 Distribution of the bad days in Europe in 1989 (the upper left-hand-side plot) and in 2003 (the upper right-hand-side plot) and increases caused by climatic changes (given in percent) of the numbers of bad days in 1989 (the lower left-hand-side plot) and in 2003 (the lower right-hand-side plot).
Figure 2 Numbers of bad days in the countries in the Balkan Peninsula for 2004.
We, the guest-editors of this Special Issue, should like to thank very much the authors of all papers for accepting our invitation to submit their papers, reading carefully the comments of the referees
and taking into account all recommendations during the preparation of the revised papers and re-submitting the final versions in time.
Figure 3 Increases of the number of bad days (in percent) when one of the Climatic Scenario is used.
We should like also to thank the referees of the papers of this Special Issue (including also the referees of the papers, which were not accepted for publication) for preparing carefully and in time their reviews and for the constructive criticism, which resulted in considerable improvements of the quality of the accepted papers.
We should like to thank very much the Editorial Board of the Journal of “Computational and Applied Mathematics” for the kind permission to prepare this Special Issue on “Efficient Numerical Methods for Large-scale Scientific Computations” and for helping us permanently during the whole process of preparation of this special issue.
Figure 4 Comparison of the numbers of bad days in six European cities.
Finally, people from the Publishing Company, Elsevier, helped us many times when we had difficulties with EES (Elsevier Editing System) during our work related to the preparation of this Special Issue. We should like to thank all of them very much.
References
[1] V. Alexandrov, W. Owczarz, P. G. Thomsen and Z. Zlatev: “Parallel runs of large air pollution models on a grid of SUN computers”, Mathematics and Computers in Simulations, Vol. 65 (2004), pp. 557-577. [2] I. Dimov, G. Geernaert and Z. Zlatev: “Impact of future climate changes on high pollution levels”, International Journal of Environment and Pollution Vol. 32, Issue 2 (2008), pp. 200230. [3] I. Faragó, Á. Havasi and Z. Zlatev: “Efficient algorithms for large scale scientific applications: Introduction”. Computers and Mathematics with Applications, Vol. 67 (2014), pp. 2085-2087. [4] R. W. Hamming: “Numerical methods for scientists and engineers”, McGraw-Hill Book Company, New York, San Francisco, Toronto, London, 1962. [5] Z. Zlatev: “Computer Treatment of Large Air Pollution Models”, Kluwer Academic Publishers, Dordrecht, 1995 (at present this book is distributed by the Springer-Verlag, BerlinHeidelberg). [6] Z. Zlatev: “Impact of future climate changes on high ozone levels in European suburban areas”. Climatic Change, Vol. 101 (2010), pp. 447-483. [7] Z. Zlatev and I. Dimov: “Computational and Numerical Challenges in Environmental Modelling”, Elsevier, Amsterdam, 2006. [8] Z. Zlatev, K. Georgiev and I. Dimov: “Influence of climatic changes on air pollution levels in the Balkan Peninsula”. Computers and Mathematics with Applications, Vol. 65 (2013), pp. 544562. [9] Z. Zlatev, Á. Havasi and I. Faragó: “Influence of climatic changes on pollution levels in Hungary and its surrounding countries”. Atmosphere, Vol. 2 (2011), pp. 201-221. [10] Z. Zlatev and L. Moseholm: Impact of climate changes on pollution levels in Denmark, Environmental Modelling, Vol. 217 (2008), pp. 305-319.
S0377-0427(15)00274-5 http://dx.doi.org/10.1016/j.cam.2015.05.001 CAM 10162
To appear in:
Journal of Computational and Applied Mathematics
Please cite this article as: Z. Zlatev, I. Dimov, I. Lirkov, Efficient numerical methods for large-scale scientific computations: Introduction, Journal of Computational and Applied Mathematics (2015), http://dx.doi.org/10.1016/j.cam.2015.05.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Efficient Numerical Methods for Large-scale Scientific Computations: Introduction Zahari Zlatev 1) , Ivan Dimov 2 ) and Ivan Lirkov 2 ) 1)
Department of Environmental Science, Aarhus University, Roskilde, Denmark, [email protected]
2)
Institute of Information and Communication Technologies, Bulgarian Academy of Sciences, Sofia, Bulgaria, [email protected], [email protected]
The mathematical models are important tools in various research studies related to different areas of science and engineering. These models are very often described by systems of partial differential equations (PDEs), which in most of the cases are non-linear and time-dependent. It is necessary to apply different numerical methods both to discretize the systems of PDEs and to handle the arising after the discretization sub-problems (in many of the cases these sub-problems are systems of non-linear algebraic equations, which are usually huge in size; containing many millions of equations). The requirements for achieving sufficiently accurate model results lead very often to the necessity to discretize the models on fine grids. It is clear that the resulting sub-problems are indeed becoming very large in such a case and the use of efficient numerical methods is crucial in this situation. In fact, it becomes possible to handle successfully many of the advanced mathematical models only when efficient numerical methods are selected and correctly implemented during their treatment on the available high-speed computers. However, the selection of efficient numerical methods is by no means an easy task. It should be emphasized here that even when two large-scale models are described mathematically by equations containing the same formulae, the use of different numerical methods in their treatment might be appropriate when the operators involved in the models have different properties in spite of the fact that they are described by terms involving the same derivatives. For example, it will in general be better to treat the diffusion of pollutants in the atmosphere and the diffusion of pollutants in water by different numerical methods in spite of the fact that the mathematical models are described by the same type of equations. The fact that there arise many difficulties related to the choice of efficient numerical methods was well understood from the very beginning of the development of the numerical analysis and the computational science. Moreover, it is also well known that it is difficult to find the most efficient numerical method and to justify fully its application in a particular large-scale scientific and/or engineering application. Finding the most efficient numerical method is practically impossible in many of the complicated situations that have often to be handled in practice and this fact was also well known from the very beginning of the development of the numerical analysis. This is why R. W. Hamming wrote in one of the first books on numerical methods, [4], that the choice of a numerical method is in nearly all cases a question of finding a good compromise.
Many numerical methods are described and tested in connection with different large-scale scientific problems in the papers contained in CAM Special Issue on “Efficient Numerical Methods for Large-scale Scientific Computations”. There are
(a) various methods based on the implementation of finite element and finite difference methods, (b) different optimization techniques, (c) algorithms for treatment of Stokes and Navier-Stokes equations, (d) some modern parallel algorithms designed for powerful high-speed computers, (e) numerical methods based on the use of various Monte Carlo techniques, (f) some methods of Runge-Kutta type and (g) devises based on operator splitting procedures. Even more important is the fact that the listed above types of advanced numerical methods were actually designed for and used in many scientific and engineering areas: (1) Computational Biology, (2) Atomic Physics, (3) Quantum Chemistry, (4) Elastic Structures, (5) Epidemic Propagation, (6) Tumor Growth and Tumor Invasion, (7) Drop-coalescence, (8) Regional Air Pollution Problems, (9) Effect of Climatic Changes on the Humans Living in Big European Cities,
(10) Ensemble Modelling Related to Air Pollution Studies, (11) Treatment of Maxwell Equations and (12) Problems Arising in Finance Mathematics. This means that the readers will find in our Special Issue on “Efficient Numerical Methods for Large-scale Scientific Computations” descriptions of many numerical methods implemented in different application areas and will be able, when they need it, to select if not the best algorithms then some good ones for their specific needs. R. W. Hamming is very famous also for another statement in his book [4]. He wrote there that the purpose of the computations is insight not the numbers. In fact, in nearly all large conferences on numerical methods held in the last three decades of the twentieth century this statement was quoted at least once. However, the truth is that this statement became much more relevant, although it not quoted so often anymore, in the last ten-fifteen years, because the modern highspeed computer are producing enormous files of output data containing millions and millions of numbers. It is clear that taking random numbers from the output files has no meaning. It is necessary to study carefully the output data, by applying some very comprehensive devices, in order to be able to show clearly the different trends and the important relationships hidden in the enormous output data files. It will be illustrative to give an example in order to explain better the statements from the above paragraph. Numerical results obtained in computations carried out by using the Unified Danish Eulerian Model (UNI-DEM), fully described in [5] and [7], are used in this example. Actually, impacts of the climatic change on several critical pollution levels in different parts of Europe were systematically studied by using high-speed computers ([1], see also [5] and [7]) and the runs were carried out on a time-period of sixteen consecutive years with fourteen different scenarios, [2], [6], [8], [9] and [10]. Many different problems had to be resolved during this very comprehensive study. One of these problems was related to the correct use of the huge output data files in order to illustrate different trends and relationships. The total size of these files was greater than 200 Mega-Bytes. The computational difficulties during the runs of the model and the preparation of the output files were enormous, but these difficulties will not be discussed here (some details are given in [3]; see also [5] and [7]). The output data had to be used for many purposes. By selecting only the relevant data, one can illustrate and make easily understandable different important situations. This is demonstrated in the figures given below. All of these figures are related to the distribution of bad days in space and time. Consider an arbitrary point in the spatial domain of the model. If for some day the eight-hours averaged ozone concentration exceeds at least once 60 ppb, then the day under consideration is called a bad day. The number of bad days should not exceed 25, which is legislated in a directive of the parliament of the European Union (the exceedance of the critical level of 25 days could cause health problems for people suffering from asthmatic disease). Relevant results are shown and some relationships and trends are clearly seen in the figures given below:
(A) The distribution in years 1989 and 2003 of the numbers of bad days in Europe and the increases (in percent) caused by climatic changes are shown in Fig. 1. It is seen that (a) the numbers of bad days may vary very much from one year to another (cold and warm summers can cause rather considerable differences) and (b) the climatic changes are causing increases of the numbers of bad days in many parts of Europe. (B) The distribution of the numbers of bad days in Balkan Peninsula for 2004 is presented in Fig. 2. It is immediately seen that the critical level is exceeded in all countries of this peninsula. (C) The impact of the climatic changes on the numbers of bad days in the Balkan Peninsula for 2004 is shown in Fig. 3. It is seen that the climatic changes will lead to some increases of the numbers of bad days in nearly all countries in the Balkan Peninsula. (D) The variations of the numbers of bad days in several European cities during the studied period of 16 years are presented in Fig. 4. It is clearly seen that (a) the critical level is very often exceeded and (b) there is a trend for decreasing of the numbers of bad days at the end of the studied period of sixteen year (which is caused by the fact that the emissions in Europe were gradually decreased after year 1989). It was necessary to develop a special package of programs for the representations shown in the four figures. It should be emphasized here that the calculated by the model data and the graphical package can also be used to illustrate many other interesting trends and relationships, see [1], [2], [5] – [10]. The authors of the papers published in this Special Issue have taken care to illustrate the results obtained by the developed by them efficient numerical methods in appropriate figures and to explain different trends and relationships. Many of the papers in this Special Issue were presented at the Eighth International Conference on Numerical Methods and Applications held in August 20-24, 2014 in Borovets (Bulgaria). Some details about this conference can be found in the web-site: http://parallel.bas.bg/dpa/NMA_2014/. The conferences on Numerical Methods and Applications are regularly held in every fourth year. The next conference will be held in August 2018. Borovets is a very nice and quiet small town in the Southern part of Bulgaria and more precisely in the Rila Mountain (the highest mountain in Bulgaria and the Balkan Peninsula). It is primarily a well-known winter resort, but also the summer days are beautiful there. Normally, more than hundred participants from many countries attended the previous conferences on Numerical Methods and Applications, presented their results and had numerous interesting discussions during their stay.
We advise the readers of this Special Issue to consider the possibility of attending the next conference on Numerical Methods and Applications.
Figure 1 Distribution of the bad days in Europe in 1989 (the upper left-hand-side plot) and in 2003 (the upper right-hand-side plot) and increases caused by climatic changes (given in percent) of the numbers of bad days in 1989 (the lower left-hand-side plot) and in 2003 (the lower right-hand-side plot).
Figure 2 Numbers of bad days in the countries in the Balkan Peninsula for 2004.
We, the guest-editors of this Special Issue, should like to thank very much the authors of all papers for accepting our invitation to submit their papers, reading carefully the comments of the referees
and taking into account all recommendations during the preparation of the revised papers and re-submitting the final versions in time.
Figure 3 Increases of the number of bad days (in percent) when one of the Climatic Scenario is used.
We should like also to thank the referees of the papers of this Special Issue (including also the referees of the papers, which were not accepted for publication) for preparing carefully and in time their reviews and for the constructive criticism, which resulted in considerable improvements of the quality of the accepted papers.
We should like to thank very much the Editorial Board of the Journal of “Computational and Applied Mathematics” for the kind permission to prepare this Special Issue on “Efficient Numerical Methods for Large-scale Scientific Computations” and for helping us permanently during the whole process of preparation of this special issue.
Figure 4 Comparison of the numbers of bad days in six European cities.
Finally, people from the Publishing Company, Elsevier, helped us many times when we had difficulties with EES (Elsevier Editing System) during our work related to the preparation of this Special Issue. We should like to thank all of them very much.
References
[1] V. Alexandrov, W. Owczarz, P. G. Thomsen and Z. Zlatev: “Parallel runs of large air pollution models on a grid of SUN computers”, Mathematics and Computers in Simulations, Vol. 65 (2004), pp. 557-577. [2] I. Dimov, G. Geernaert and Z. Zlatev: “Impact of future climate changes on high pollution levels”, International Journal of Environment and Pollution Vol. 32, Issue 2 (2008), pp. 200230. [3] I. Faragó, Á. Havasi and Z. Zlatev: “Efficient algorithms for large scale scientific applications: Introduction”. Computers and Mathematics with Applications, Vol. 67 (2014), pp. 2085-2087. [4] R. W. Hamming: “Numerical methods for scientists and engineers”, McGraw-Hill Book Company, New York, San Francisco, Toronto, London, 1962. [5] Z. Zlatev: “Computer Treatment of Large Air Pollution Models”, Kluwer Academic Publishers, Dordrecht, 1995 (at present this book is distributed by the Springer-Verlag, BerlinHeidelberg). [6] Z. Zlatev: “Impact of future climate changes on high ozone levels in European suburban areas”. Climatic Change, Vol. 101 (2010), pp. 447-483. [7] Z. Zlatev and I. Dimov: “Computational and Numerical Challenges in Environmental Modelling”, Elsevier, Amsterdam, 2006. [8] Z. Zlatev, K. Georgiev and I. Dimov: “Influence of climatic changes on air pollution levels in the Balkan Peninsula”. Computers and Mathematics with Applications, Vol. 65 (2013), pp. 544562. [9] Z. Zlatev, Á. Havasi and I. Faragó: “Influence of climatic changes on pollution levels in Hungary and its surrounding countries”. Atmosphere, Vol. 2 (2011), pp. 201-221. [10] Z. Zlatev and L. Moseholm: Impact of climate changes on pollution levels in Denmark, Environmental Modelling, Vol. 217 (2008), pp. 305-319.