Alfredo Bellen and Marino Zennaro
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506546
- eISBN:
- 9780191709609
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506546.001.0001
- Subject:
- Mathematics, Numerical Analysis
The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary ...
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The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary differential equations (ODEs) are made using examples illustrating some unexpected and often surprising behaviours of the true and numerical solutions. The book briefly reviews the various approaches existing in the literature and develops an error and well-posedness analysis for general one-step and multistep methods. The continuous extensions of Runge-Kutta methods are presented in detail, which are useful for more general problems such as dense output and discontinuous equations. Some deeper insight into convergence and superconvergence is then carried out for DDEs with various kinds of delays. The stepsize control mechanism is developed on a firm mathematical basis. Classical results and an unconventional analysis of stability with respect to forcing term are reviewed for ODEs in view of the subsequent stability analysis for DDEs. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding investigations for the numerical methods are made. Reformulations of DDEs as partial differential equations and subsequent semi-discretization are described and compared with the classical approach. A list of available codes is provided.Less
The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary differential equations (ODEs) are made using examples illustrating some unexpected and often surprising behaviours of the true and numerical solutions. The book briefly reviews the various approaches existing in the literature and develops an error and well-posedness analysis for general one-step and multistep methods. The continuous extensions of Runge-Kutta methods are presented in detail, which are useful for more general problems such as dense output and discontinuous equations. Some deeper insight into convergence and superconvergence is then carried out for DDEs with various kinds of delays. The stepsize control mechanism is developed on a firm mathematical basis. Classical results and an unconventional analysis of stability with respect to forcing term are reviewed for ODEs in view of the subsequent stability analysis for DDEs. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding investigations for the numerical methods are made. Reformulations of DDEs as partial differential equations and subsequent semi-discretization are described and compared with the classical approach. A list of available codes is provided.
Alfredo Bellen and Marino Zennaro
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198506546
- eISBN:
- 9780191709609
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198506546.003.0004
- Subject:
- Mathematics, Numerical Analysis
This chapter focuses on the error and convergence analysis of the technique based on the use of a discrete method for ODEs endowed with some continuous extension, which is called the standard ...
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This chapter focuses on the error and convergence analysis of the technique based on the use of a discrete method for ODEs endowed with some continuous extension, which is called the standard approach. The technique is described for some of the main types of DDEs and neutral DDEs, ranging from constant or non-vanishing time dependent delays, to arbitrary time dependent delays, up to state dependent delays. It is shown that for time dependent and state dependent delays, the resulting method may become implicit even if the underlying ODE method is explicit and, even in this case, well-posedness is proven. The non-trivial problem of tracking the discontinuities is also considered. It is shown that despite the fact that any discrete method is, in principle, suitable for the standard approach, one-step methods (essentially Runge-Kutta methods) are preferable to multistep methods. A list of available codes is given.Less
This chapter focuses on the error and convergence analysis of the technique based on the use of a discrete method for ODEs endowed with some continuous extension, which is called the standard approach. The technique is described for some of the main types of DDEs and neutral DDEs, ranging from constant or non-vanishing time dependent delays, to arbitrary time dependent delays, up to state dependent delays. It is shown that for time dependent and state dependent delays, the resulting method may become implicit even if the underlying ODE method is explicit and, even in this case, well-posedness is proven. The non-trivial problem of tracking the discontinuities is also considered. It is shown that despite the fact that any discrete method is, in principle, suitable for the standard approach, one-step methods (essentially Runge-Kutta methods) are preferable to multistep methods. A list of available codes is given.
Stefan Schaefer
- Published in print:
- 2011
- Published Online:
- January 2012
- ISBN:
- 9780199691609
- eISBN:
- 9780191731792
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199691609.003.0007
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics
This tutorial gives a practical introduction to the Hybrid Monte Carlo algorithm and the analysis of Monte Carlo data. The method is exemplified at the ϕ 4 theory, for which all steps from the ...
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This tutorial gives a practical introduction to the Hybrid Monte Carlo algorithm and the analysis of Monte Carlo data. The method is exemplified at the ϕ 4 theory, for which all steps from the derivation of the relevant formulae to the actual implementation in a computer program are discussed in detail. It concludes with the analysis of Monte Carlo data, in particular their auto-correlations.Less
This tutorial gives a practical introduction to the Hybrid Monte Carlo algorithm and the analysis of Monte Carlo data. The method is exemplified at the ϕ 4 theory, for which all steps from the derivation of the relevant formulae to the actual implementation in a computer program are discussed in detail. It concludes with the analysis of Monte Carlo data, in particular their auto-correlations.
Jim Paul and Brendan Moynihan
- Published in print:
- 2013
- Published Online:
- November 2015
- ISBN:
- 9780231164689
- eISBN:
- 9780231535236
- Item type:
- book
- Publisher:
- Columbia University Press
- DOI:
- 10.7312/columbia/9780231164689.001.0001
- Subject:
- Business and Management, Finance, Accounting, and Banking
The author's meteoric rise took him from a small town in Northern Kentucky to governor of the Chicago Mercantile Exchange, yet he lost it all—his fortune, his reputation, and his job—in one fatal ...
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The author's meteoric rise took him from a small town in Northern Kentucky to governor of the Chicago Mercantile Exchange, yet he lost it all—his fortune, his reputation, and his job—in one fatal attack of excessive economic hubris. This honest, frank analysis revisits the events that led to the author's disastrous decision and examines the psychological factors behind bad financial practices in several economic sectors. The book begins with the unbroken string of successes that helped the author achieve a jet-setting lifestyle and land a key spot with the Chicago Mercantile Exchange. It then describes the circumstances leading up to his $1.6 million loss and the essential lessons he learned from it—primarily that, although there are as many ways to make money in the markets as there are people participating in them, all losses come from the same few sources. Investors lose money in the markets either because of errors in their analysis or because of psychological barriers preventing the application of analysis. While all analytical methods have some validity and make allowances for instances in which they do not work, psychological factors can keep an investor in a losing position, causing him to abandon one method for another in order to rationalize the decisions already made. This cautionary tale includes strategies for avoiding loss tied to a simple framework for understanding, accepting, and dodging the dangers of investing, trading, and speculating.Less
The author's meteoric rise took him from a small town in Northern Kentucky to governor of the Chicago Mercantile Exchange, yet he lost it all—his fortune, his reputation, and his job—in one fatal attack of excessive economic hubris. This honest, frank analysis revisits the events that led to the author's disastrous decision and examines the psychological factors behind bad financial practices in several economic sectors. The book begins with the unbroken string of successes that helped the author achieve a jet-setting lifestyle and land a key spot with the Chicago Mercantile Exchange. It then describes the circumstances leading up to his $1.6 million loss and the essential lessons he learned from it—primarily that, although there are as many ways to make money in the markets as there are people participating in them, all losses come from the same few sources. Investors lose money in the markets either because of errors in their analysis or because of psychological barriers preventing the application of analysis. While all analytical methods have some validity and make allowances for instances in which they do not work, psychological factors can keep an investor in a losing position, causing him to abandon one method for another in order to rationalize the decisions already made. This cautionary tale includes strategies for avoiding loss tied to a simple framework for understanding, accepting, and dodging the dangers of investing, trading, and speculating.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
- Published in print:
- 2014
- Published Online:
- September 2014
- ISBN:
- 9780199678792
- eISBN:
- 9780191780745
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199678792.003.0009
- Subject:
- Mathematics, Numerical Analysis, Computational Mathematics / Optimization
This chapter concerns the statement and properties of the steady Navier–Stokes equations and the corresponding weak formulation. This includes discussion of stability theory, bifurcation and ...
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This chapter concerns the statement and properties of the steady Navier–Stokes equations and the corresponding weak formulation. This includes discussion of stability theory, bifurcation and nonlinear iteration. This is followed by a description of finite element discretization and error analysis of discrete solutions.Less
This chapter concerns the statement and properties of the steady Navier–Stokes equations and the corresponding weak formulation. This includes discussion of stability theory, bifurcation and nonlinear iteration. This is followed by a description of finite element discretization and error analysis of discrete solutions.
Carey Witkov and Keith Zengel
- Published in print:
- 2019
- Published Online:
- November 2019
- ISBN:
- 9780198847144
- eISBN:
- 9780191882074
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198847144.001.0001
- Subject:
- Physics, Theoretical, Computational, and Statistical Physics, Particle Physics / Astrophysics / Cosmology
This book is the first to make chi-squared model testing, one of the data analysis methods used to discover the Higgs boson and gravitational waves, accessible to undergraduate students in ...
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This book is the first to make chi-squared model testing, one of the data analysis methods used to discover the Higgs boson and gravitational waves, accessible to undergraduate students in introductory physics laboratory courses. By including uncertainties in the curve fitting, chi-squared data analysis improves on the centuries old ordinary least squares and linear regression methods and combines best fit parameter estimation and model testing in one method. A toolkit of essential statistical and experimental concepts is developed from the ground up with novel features to interest even those familiar with the material. The presentation of one- and two-parameter chi-squared model testing, requiring only elementary probability and algebra, is followed by case studies that apply the methods to simple introductory physics lab experiments. More challenging topics, requiring calculus, are addressed in an advanced topics chapter. This self-contained and student-friendly introduction to chi-squared analysis and model testing includes a glossary, end-of-chapter problems with complete solutions, and software scripts written in several popular programming languages, that the reader can use for chi-squared model testing. In addition to introductory physics lab students, this accessible introduction to chi-squared analysis and model testing will be of interest to all who need to learn chi-squared model testing, e.g. beginning researchers in astrophysics and particle physics, beginners in data science, and lab students in other experimental sciences.Less
This book is the first to make chi-squared model testing, one of the data analysis methods used to discover the Higgs boson and gravitational waves, accessible to undergraduate students in introductory physics laboratory courses. By including uncertainties in the curve fitting, chi-squared data analysis improves on the centuries old ordinary least squares and linear regression methods and combines best fit parameter estimation and model testing in one method. A toolkit of essential statistical and experimental concepts is developed from the ground up with novel features to interest even those familiar with the material. The presentation of one- and two-parameter chi-squared model testing, requiring only elementary probability and algebra, is followed by case studies that apply the methods to simple introductory physics lab experiments. More challenging topics, requiring calculus, are addressed in an advanced topics chapter. This self-contained and student-friendly introduction to chi-squared analysis and model testing includes a glossary, end-of-chapter problems with complete solutions, and software scripts written in several popular programming languages, that the reader can use for chi-squared model testing. In addition to introductory physics lab students, this accessible introduction to chi-squared analysis and model testing will be of interest to all who need to learn chi-squared model testing, e.g. beginning researchers in astrophysics and particle physics, beginners in data science, and lab students in other experimental sciences.
Lionel Raff, Ranga Komanduri, Martin Hagan, and Satish Bukkapatnam
- Published in print:
- 2012
- Published Online:
- November 2020
- ISBN:
- 9780199765652
- eISBN:
- 9780197563113
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780199765652.003.0013
- Subject:
- Chemistry, Physical Chemistry
The use of neural networks (NNs) to predict an outcome or the output results as a function of a set of input parameters has been gaining wider acceptance with the ...
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The use of neural networks (NNs) to predict an outcome or the output results as a function of a set of input parameters has been gaining wider acceptance with the advance in computer technology as well as with an increased awareness of the potential of NNs. A neural network is first trained to learn the underlying functional relationship between the output and the input parameters by providing it with a large number of data points, where each data point corresponds to a set of output and input parameters. Sumpter and Noid demonstrated the use of NNs to map the vibrational motion derived from the vibrational spectra onto a PES with relatively high accuracy. In another application, Sumpter et al. trained an NN to learn the relation between the phase-space points along a trajectory and the mode energies for stretching, torsion, and bending vibrations of H2O2. Likewise, Nami et al. demonstrated the use of NNs to determine the TiO2 deposition rates in a chemical vapor deposition (CVD) process from the knowledge of a range of deposition conditions. In view of the success achieved in obtaining interpolated values of the PESs for multi-atomic systems using an NN trained by the ab initio energy values for a large number of configurations, it is reasonable to ask whether we can successfully compute the results of an MD trajectory for a chemical reaction using an NN trained by the data obtained by previous MD simulations. If this can be done successfully, it becomes possible to execute a small number of trajectories, M, and then utilize the results of these trajectories as a database to train an NN to predict the final results of a very large number of trajectories N, where N >> M, that can be used to increase the statistical accuracy of the MD calculations and to further explore the dependence of the trajectory results upon a wide variety of variables without actually having to perform any further numerical integrations. In effect, the NN replaces the computationally laborious numerical integrations.
Less
The use of neural networks (NNs) to predict an outcome or the output results as a function of a set of input parameters has been gaining wider acceptance with the advance in computer technology as well as with an increased awareness of the potential of NNs. A neural network is first trained to learn the underlying functional relationship between the output and the input parameters by providing it with a large number of data points, where each data point corresponds to a set of output and input parameters. Sumpter and Noid demonstrated the use of NNs to map the vibrational motion derived from the vibrational spectra onto a PES with relatively high accuracy. In another application, Sumpter et al. trained an NN to learn the relation between the phase-space points along a trajectory and the mode energies for stretching, torsion, and bending vibrations of H2O2. Likewise, Nami et al. demonstrated the use of NNs to determine the TiO2 deposition rates in a chemical vapor deposition (CVD) process from the knowledge of a range of deposition conditions. In view of the success achieved in obtaining interpolated values of the PESs for multi-atomic systems using an NN trained by the ab initio energy values for a large number of configurations, it is reasonable to ask whether we can successfully compute the results of an MD trajectory for a chemical reaction using an NN trained by the data obtained by previous MD simulations. If this can be done successfully, it becomes possible to execute a small number of trajectories, M, and then utilize the results of these trajectories as a database to train an NN to predict the final results of a very large number of trajectories N, where N >> M, that can be used to increase the statistical accuracy of the MD calculations and to further explore the dependence of the trajectory results upon a wide variety of variables without actually having to perform any further numerical integrations. In effect, the NN replaces the computationally laborious numerical integrations.
I. S. Duff, A. M. Erisman, and J. K. Reid
- Published in print:
- 2017
- Published Online:
- April 2017
- ISBN:
- 9780198508380
- eISBN:
- 9780191746420
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198508380.003.0015
- Subject:
- Mathematics, Numerical Analysis
We consider solving a problem that is closely related to one that we have previously solved or whose parts have previously been solved. We study backward error analysis for sparse problems, obtaining ...
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We consider solving a problem that is closely related to one that we have previously solved or whose parts have previously been solved. We study backward error analysis for sparse problems, obtaining entries of the inverse of a sparse matrix, sparsity in nonlinear problems, solution methods based on orthogonal transformations, and hybrid methods that combine iterative and direct methods.Less
We consider solving a problem that is closely related to one that we have previously solved or whose parts have previously been solved. We study backward error analysis for sparse problems, obtaining entries of the inverse of a sparse matrix, sparsity in nonlinear problems, solution methods based on orthogonal transformations, and hybrid methods that combine iterative and direct methods.
Arthur Lupia
- Published in print:
- 2016
- Published Online:
- November 2020
- ISBN:
- 9780190263720
- eISBN:
- 9780197559598
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780190263720.003.0022
- Subject:
- Education, Educational Policy and Politics
In 2012, a Fairleigh Dickinson University (FDU) survey made headlines. The headlines questioned Fox News viewers’ intelligence. The Nation’s headline ...
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In 2012, a Fairleigh Dickinson University (FDU) survey made headlines. The headlines questioned Fox News viewers’ intelligence. The Nation’s headline read: “It’s Official: Watching Fox Makes You Stupider.” It claimed that “[a] ccording to a new study by Farleigh Dickinson University, Fox viewers are the least knowledgeable audience of any outlet, and they know even less about politics and current events than people who watch no news at all.” It concluded that Fox News “fails the fundamental test of journalism: Are you informing your audience?” The Huffington Post (2012) claimed that “people who only watch Fox News are less informed than all other news consumers.” The New York Times’ Timothy Egan (2014) repeated the assertion. Conservative-leaning publications interpreted FDU’s findings differently. The Examiner’s headline read “Democrats Use Biased ‘Study’ to Smear Fox News.” It claimed that the pollsters “abandoned all integrity to vindictively trash Fox News and peddle the partisan smear that anyone who watches ‘right-wing propaganda’ (anything that includes multiple sides of the story) is stupid.” FDU’s report on its Public Mind Poll (2012) focused not on how respondents answered individual recall questions, but on an aggregate PK scale that FDU manufactured. Like nearly all published PK scales, FDU’s scale was formed by adding the number of correct answers respondents gave to a small set of recall questions. Such scales typically range in value from zero-to-five or zero-to-seven, with the high number representing the total number of recall questions included in the scale. If a respondent answers no questions correctly, they get a score of zero. If they answer all questions correctly, they get the highest possible score. PK scales are regularly used to represent “the range of factual information about politics that is stored in long-term memory.” FDU’s report and the subsequent media reports are based on the finding that Fox News viewers scored lower on FDU’s PK scale than did viewers of other networks. In this chapter, I examine this case and other claims that are based on PK scales.
Less
In 2012, a Fairleigh Dickinson University (FDU) survey made headlines. The headlines questioned Fox News viewers’ intelligence. The Nation’s headline read: “It’s Official: Watching Fox Makes You Stupider.” It claimed that “[a] ccording to a new study by Farleigh Dickinson University, Fox viewers are the least knowledgeable audience of any outlet, and they know even less about politics and current events than people who watch no news at all.” It concluded that Fox News “fails the fundamental test of journalism: Are you informing your audience?” The Huffington Post (2012) claimed that “people who only watch Fox News are less informed than all other news consumers.” The New York Times’ Timothy Egan (2014) repeated the assertion. Conservative-leaning publications interpreted FDU’s findings differently. The Examiner’s headline read “Democrats Use Biased ‘Study’ to Smear Fox News.” It claimed that the pollsters “abandoned all integrity to vindictively trash Fox News and peddle the partisan smear that anyone who watches ‘right-wing propaganda’ (anything that includes multiple sides of the story) is stupid.” FDU’s report on its Public Mind Poll (2012) focused not on how respondents answered individual recall questions, but on an aggregate PK scale that FDU manufactured. Like nearly all published PK scales, FDU’s scale was formed by adding the number of correct answers respondents gave to a small set of recall questions. Such scales typically range in value from zero-to-five or zero-to-seven, with the high number representing the total number of recall questions included in the scale. If a respondent answers no questions correctly, they get a score of zero. If they answer all questions correctly, they get the highest possible score. PK scales are regularly used to represent “the range of factual information about politics that is stored in long-term memory.” FDU’s report and the subsequent media reports are based on the finding that Fox News viewers scored lower on FDU’s PK scale than did viewers of other networks. In this chapter, I examine this case and other claims that are based on PK scales.