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International Journal for Numerical Methods in Engineering 2002: Volume 53 , Issue CONTENTS. Digitized from IA1643935-03 . Previous issue: sim_international-journal-numerical-methods-engineering_2001-12-30_52_12 . Next issue: sim_international-journal-numerical-methods-engineering_2002-01-10_53_1 .

Collection of notebooks about quantitative finance, with interactive python code. Financial-Models-Numerical-Methods This is a collection of Jupyter notebooks based on different topics in the area of quantitative finance. Is this a tutorial? Almost! :) This is just a collection of topics and algorithms that in my opinion are interesting. It contains several topics that are not so popular nowadays, but that can be very powerful. Usually, topics such as PDE methods, Lévy processes, Fourier methods or Kalman filter are not very popular among practitioners, who prefers to work with more standard tools. The aim of these notebooks is to present these interesting topics, by showing their practical application through an interactive python implementation. Who are these notebooks for? Not for absolute beginners. These topics require a basic knowledge in stochastic calculus, financial mathematics and statistics. A basic knowledge of python programming is also necessary. In these notebooks I will not explain what is a call option, or what is a stochastic process, or a partial differential equation. However, every time I will introduce a concept, I will also add a link to the corresponding wiki page or to a reference manual.In this way, the reader will be able to immediately understand what I am talking about. These notes are for students in science, economics or finance who have followed at least one undergraduate course in financial mathematics and statistics. Self-taught students or practicioners should have read at least an introductiory book on financial mathematics. Why is it worth to read these notes? First of all, this is not a book! Every notebook is (almost) independent from the others. Therefore you can select only the notebook you are interested in! diff- Every notebook, contains python code ready to use! It is not easy to find on internet examples of financial models implemented in python which are ready to use and well documented. I think that beginners in quantitative finance will find these notebooks very useful! Moreover, Jupyter notebooks are interactive i.e. you can run the code inside the notebook. This is probably the best way to study! If you open a notebook with Github or NBviewer , sometimes mathematical formulas are not displayed correctly. For this reason, I suggest you to clone/download the repository. Is this series of notebooks complete? No! I will upload more notebooks from time to time. At the moment, I'm interested in the areas of stochastic processes, Kalman Filter, statistics and much more. I will add more interesting notebooks on these topics in the future. If you have any kind of questions, or if you find some errors, or you have suggestions for improvements, feel free to contact me. This is my linkedin page. Contents 1.1) Black-Scholes numerical methods (lognormal distribution, change of measure, Monte Carlo, Binomial method) . 1.2) SDE simulation and statistics (paths generation, Confidence intervals, Hypothesys testing, Geometric Brownian motion, Cox-Ingersoll-Ross process, Euler Maruyama method, parameters estimation) 1.3) Fourier inversion methods (derivation of inversion formula, numerical inversion, option pricing) 1.4) SDE, Heston model (correlated Brownian motions, Heston paths, Heston distribution, characteristic function, option pricing) 1.5) SDE, Lévy processes (Merton, Variance Gamma, NIG, path generation, parameter estimation) 2.1) The Black-Scholes PDE (PDE discretization, Implicit method, sparse matrix tutorial) 2.2) Exotic options (Binary options, Barrier options) 2.3) American options (PDE, Binomial method, Longstaff-Schwartz) 3.1) Merton Jump-Diffusion PIDE (Implicit-Explicit discretization, discrete convolution, model limitations, Monte Carlo, Fourier inversion, semi-closed formula ) 3.2) Variance Gamma PIDE (approximated jump-diffusion PIDE, Monte Carlo, Fourier inversion, Comparison with Black-Scholes) 3.3) Normal Inverse Gaussian PIDE (approximated jump-diffusion PIDE, Monte Carlo, Fourier inversion, properties of the Lévy measure) 4.1) Pricing with transaction costs (Davis-Panas-Zariphopoulou model, singular control problem, HJB variational inequality, indifference pricing, binomial tree, performances) 5.1) Linear regression and Kalman filter (market data cleaning, Linear regression methods, Kalman filter design, choice of parameters) A.1) Appendix: Linear equations (LU, Jacobi, Gauss-Seidel, SOR, Thomas) A.2) Appendix: Code optimization (cython, C code) A.3) Appendix: Review of Lévy processes theory (basic and important definitions, derivation of the pricing PIDE) How to run the notebooks You have two options: 1) Install docker following the instructions in install link At this point, you just need to run the script docker_start_notebook.py and you are done. This script will download the data-science docker image scipy-notebook , that will be used every time you run the script (the script will take about 10-15 minutes to download the image, ONLY the first time). You can also download a different image by modifying the script. For a list of images see here . 2) Clone the repository and open the notebooks using jupyter-notebook . If you are using an old version of python there can be compatibility problems. diff- Cython code needs to be compiled! If you are using the data science image, you can open the shell in the notebooks directory, and run the script bashpython docker_start_notebook.py after that, copy-paste the following code into the shell: bash docker exec -it Numeric_Finance bashcd work/functions/cythonpython setup.py build_ext --inplaceexit ( Numeric_Finance is the name of the docker container) If you are not using docker, just copy in the shell the following: bash cd functions/cythonpython setup.py build_ext --inplace Enjoy! To restore the repository download the bundle wget https://archive.org/download/github.com-cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05/cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05.bundle and run: git clone cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05.bundle Source: https://github.com/cantaro86/Financial-Models-Numerical-Methods Uploader: cantaro86 Upload date: 2019-11-24

Collection of notebooks about quantitative finance, with interactive python code. Financial-Models-Numerical-Methods This is a collection of Jupyter notebooks based on different topics in the area of quantitative finance. Is this a tutorial? Almost! :) This is just a collection of topics and algorithms that in my opinion are interesting. It contains several topics that are not so popular nowadays, but that can be very powerful. Usually, topics such as PDE methods, Lévy processes, Fourier methods or Kalman filter are not very popular among practitioners, who prefers to work with more standard tools. The aim of these notebooks is to present these interesting topics, by showing their practical application through an interactive python implementation. Who are these notebooks for? Not for absolute beginners. These topics require a basic knowledge in stochastic calculus, financial mathematics and statistics. A basic knowledge of python programming is also necessary. In these notebooks I will not explain what is a call option, or what is a stochastic process, or a partial differential equation. However, every time I will introduce a concept, I will also add a link to the corresponding wiki page or to a reference manual.In this way, the reader will be able to immediately understand what I am talking about. These notes are for students in science, economics or finance who have followed at least one undergraduate course in financial mathematics and statistics. Self-taught students or practicioners should have read at least an introductiory book on financial mathematics. Why is it worth to read these notes? First of all, this is not a book! Every notebook is (almost) independent from the others. Therefore you can select only the notebook you are interested in! diff- Every notebook, contains python code ready to use! It is not easy to find on internet examples of financial models implemented in python which are ready to use and well documented. I think that beginners in quantitative finance will find these notebooks very useful! Moreover, Jupyter notebooks are interactive i.e. you can run the code inside the notebook. This is probably the best way to study! If you open a notebook with Github or NBviewer , sometimes mathematical formulas are not displayed correctly. For this reason, I suggest you to clone/download the repository. Is this series of notebooks complete? No! I will upload more notebooks from time to time. At the moment, I'm interested in the areas of stochastic processes, Kalman Filter, statistics and much more. I will add more interesting notebooks on these topics in the future. If you have any kind of questions, or if you find some errors, or you have suggestions for improvements, feel free to contact me. This is my linkedin page. Contents 1.1) Black-Scholes numerical methods (lognormal distribution, change of measure, Monte Carlo, Binomial method) . 1.2) SDE simulation and statistics (paths generation, Confidence intervals, Hypothesys testing, Geometric Brownian motion, Cox-Ingersoll-Ross process, Euler Maruyama method, parameters estimation) 1.3) Fourier inversion methods (derivation of inversion formula, numerical inversion, option pricing) 1.4) SDE, Heston model (correlated Brownian motions, Heston paths, Heston distribution, characteristic function, option pricing) 1.5) SDE, Lévy processes (Merton, Variance Gamma, NIG, path generation, parameter estimation) 2.1) The Black-Scholes PDE (PDE discretization, Implicit method, sparse matrix tutorial) 2.2) Exotic options (Binary options, Barrier options) 2.3) American options (PDE, Binomial method, Longstaff-Schwartz) 3.1) Merton Jump-Diffusion PIDE (Implicit-Explicit discretization, discrete convolution, model limitations, Monte Carlo, Fourier inversion, semi-closed formula ) 3.2) Variance Gamma PIDE (approximated jump-diffusion PIDE, Monte Carlo, Fourier inversion, Comparison with Black-Scholes) 3.3) Normal Inverse Gaussian PIDE (approximated jump-diffusion PIDE, Monte Carlo, Fourier inversion, properties of the Lévy measure) 4.1) Pricing with transaction costs (Davis-Panas-Zariphopoulou model, singular control problem, HJB variational inequality, indifference pricing, binomial tree, performances) 5.1) Linear regression and Kalman filter (market data cleaning, Linear regression methods, Kalman filter design, choice of parameters) A.1) Appendix: Linear equations (LU, Jacobi, Gauss-Seidel, SOR, Thomas) A.2) Appendix: Code optimization (cython, C code) A.3) Appendix: Review of Lévy processes theory (basic and important definitions, derivation of the pricing PIDE) How to run the notebooks You have two options: 1) Install docker following the instructions in install link At this point, you just need to run the script docker_start_notebook.py and you are done. This script will download the data-science docker image scipy-notebook , that will be used every time you run the script (the script will take about 10-15 minutes to download the image, ONLY the first time). You can also download a different image by modifying the script. For a list of images see here . 2) Clone the repository and open the notebooks using jupyter-notebook . If you are using an old version of python there can be compatibility problems. diff- Cython code needs to be compiled! If you are using the data science image, you can open the shell in the notebooks directory, and run the script bashpython docker_start_notebook.py after that, copy-paste the following code into the shell: bash docker exec -it Numeric_Finance bashcd work/functions/cythonpython setup.py build_ext --inplaceexit ( Numeric_Finance is the name of the docker container) If you are not using docker, just copy in the shell the following: bash cd functions/cythonpython setup.py build_ext --inplace Enjoy! To restore the repository download the bundle wget https://archive.org/download/github.com-cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05/cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05.bundle and run: git clone cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05.bundle Source: https://github.com/cantaro86/Financial-Models-Numerical-Methods Uploader: cantaro86 Upload date: 2019-11-24

Collection of notebooks about quantitative finance, with interactive python code. Financial-Models-Numerical-Methods This is a collection of Jupyter notebooks based on different topics in the area of quantitative finance. Is this a tutorial? Almost! :) This is just a collection of topics and algorithms that in my opinion are interesting. It contains several topics that are not so popular nowadays, but that can be very powerful. Usually, topics such as PDE methods, Lévy processes, Fourier methods or Kalman filter are not very popular among practitioners, who prefers to work with more standard tools. The aim of these notebooks is to present these interesting topics, by showing their practical application through an interactive python implementation. Who are these notebooks for? Not for absolute beginners. These topics require a basic knowledge in stochastic calculus, financial mathematics and statistics. A basic knowledge of python programming is also necessary. In these notebooks I will not explain what is a call option, or what is a stochastic process, or a partial differential equation. However, every time I will introduce a concept, I will also add a link to the corresponding wiki page or to a reference manual.In this way, the reader will be able to immediately understand what I am talking about. These notes are for students in science, economics or finance who have followed at least one undergraduate course in financial mathematics and statistics. Self-taught students or practicioners should have read at least an introductiory book on financial mathematics. Why is it worth to read these notes? First of all, this is not a book! Every notebook is (almost) independent from the others. Therefore you can select only the notebook you are interested in! diff- Every notebook, contains python code ready to use! It is not easy to find on internet examples of financial models implemented in python which are ready to use and well documented. I think that beginners in quantitative finance will find these notebooks very useful! Moreover, Jupyter notebooks are interactive i.e. you can run the code inside the notebook. This is probably the best way to study! If you open a notebook with Github or NBviewer , sometimes mathematical formulas are not displayed correctly. For this reason, I suggest you to clone/download the repository. Is this series of notebooks complete? No! I will upload more notebooks from time to time. At the moment, I'm interested in the areas of stochastic processes, Kalman Filter, statistics and much more. I will add more interesting notebooks on these topics in the future. If you have any kind of questions, or if you find some errors, or you have suggestions for improvements, feel free to contact me. This is my linkedin page. Contents 1.1) Black-Scholes numerical methods (lognormal distribution, change of measure, Monte Carlo, Binomial method) . 1.2) SDE simulation and statistics (paths generation, Confidence intervals, Hypothesys testing, Geometric Brownian motion, Cox-Ingersoll-Ross process, Euler Maruyama method, parameters estimation) 1.3) Fourier inversion methods (derivation of inversion formula, numerical inversion, option pricing) 1.4) SDE, Heston model (correlated Brownian motions, Heston paths, Heston distribution, characteristic function, option pricing) 1.5) SDE, Lévy processes (Merton, Variance Gamma, NIG, path generation, parameter estimation) 2.1) The Black-Scholes PDE (PDE discretization, Implicit method, sparse matrix tutorial) 2.2) Exotic options (Binary options, Barrier options) 2.3) American options (PDE, Binomial method, Longstaff-Schwartz) 3.1) Merton Jump-Diffusion PIDE (Implicit-Explicit discretization, discrete convolution, model limitations, Monte Carlo, Fourier inversion, semi-closed formula ) 3.2) Variance Gamma PIDE (approximated jump-diffusion PIDE, Monte Carlo, Fourier inversion, Comparison with Black-Scholes) 3.3) Normal Inverse Gaussian PIDE (approximated jump-diffusion PIDE, Monte Carlo, Fourier inversion, properties of the Lévy measure) 4.1) Pricing with transaction costs (Davis-Panas-Zariphopoulou model, singular control problem, HJB variational inequality, indifference pricing, binomial tree, performances) 5.1) Linear regression and Kalman filter (market data cleaning, Linear regression methods, Kalman filter design, choice of parameters) A.1) Appendix: Linear equations (LU, Jacobi, Gauss-Seidel, SOR, Thomas) A.2) Appendix: Code optimization (cython, C code) A.3) Appendix: Review of Lévy processes theory (basic and important definitions, derivation of the pricing PIDE) How to run the notebooks You have two options: 1) Install docker following the instructions in install link At this point, you just need to run the script docker_start_notebook.py and you are done. This script will download the data-science docker image scipy-notebook , that will be used every time you run the script (the script will take about 10-15 minutes to download the image, ONLY the first time). You can also download a different image by modifying the script. For a list of images see here . 2) Clone the repository and open the notebooks using jupyter-notebook . If you are using an old version of python there can be compatibility problems. diff- Cython code needs to be compiled! If you are using the data science image, you can open the shell in the notebooks directory, and run the script bashpython docker_start_notebook.py after that, copy-paste the following code into the shell: bash docker exec -it Numeric_Finance bashcd work/functions/cythonpython setup.py build_ext --inplaceexit ( Numeric_Finance is the name of the docker container) If you are not using docker, just copy in the shell the following: bash cd functions/cythonpython setup.py build_ext --inplace Enjoy! To restore the repository download the bundle wget https://archive.org/download/github.com-cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05/cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05.bundle and run: git clone cantaro86-Financial-Models-Numerical-Methods_-_2019-11-24_19-38-05.bundle Source: https://github.com/cantaro86/Financial-Models-Numerical-Methods Uploader: cantaro86 Upload date: 2019-11-24

We study the influence of numerical methods and grid resolution on the termination of the magnetorotational instability (MRI) by means of parasitic instabilities in three-dimensional shearing-disc simulations reproducing typical conditions found in core-collapse supernovae. Whether or not the MRI is able to amplify weak magnetic fields in this context strongly depends, among other factors, on the amplitude at which its growth terminates. The qualitative results of our study do not depend on the numerical scheme. In all our models, MRI termination is caused by Kelvin-Helmholtz instabilities, consistent with theoretical predictions. Quantitatively, however, there are differences, but numerical convergence can be achieved even at relatively low grid resolutions if high-order reconstruction methods are used.

Future extreme-scale computer systems may expose silent data corruption (SDC) to applications, in order to save energy or increase performance. However, resilience research struggles to come up with useful abstract programming models for reasoning about SDC. Existing work randomly flips bits in running applications, but this only shows average-case behavior for a low-level, artificial hardware model. Algorithm developers need to understand worst-case behavior with the higher-level data types they actually use, in order to make their algorithms more resilient. Also, we know so little about how SDC may manifest in future hardware, that it seems premature to draw conclusions about the average case. We argue instead that numerical algorithms can benefit from a numerical unreliability fault model, where faults manifest as unbounded perturbations to floating-point data. Algorithms can use inexpensive "sanity" checks that bound or exclude error in the results of computations. Given a selective reliability programming model that requires reliability only when and where needed, such checks can make algorithms reliable despite unbounded faults. Sanity checks, and in general a healthy skepticism about the correctness of subroutines, are wise even if hardware is perfectly reliable.

We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations, in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the "quasi-degenerate density matrix" (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit, are manifested in finite systems as a set of low-energy "quasi-degenerate" states (in addition to the ground state). This method, its original form due to Furukawa et al.[Phys. Rev. Lett. 96, 047211 (2006)], is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the "correlation density matrix", which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss-Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide 'roadmaps' with our recommendation for which methods should be used in each situation.

Hamiltonian Boundary Value Methods are a new class of energy preserving one step methods for the solution of polynomial Hamiltonian dynamical systems. They can be thought of as a generalization of collocation methods in that they may be defined by imposing a suitable set of extended collocation conditions. In particular, in the way they are described in this note, they are related to Gauss collocation methods with the difference that they are able to precisely conserve the Hamiltonian function in the case where this is a polynomial of any high degree in the momenta and in the generalized coordinates. A description of these new formulas is followed by a few test problems showing how, in many relevant situations, the precise conservation of the Hamiltonian is crucial to simulate on a computer the correct behavior of the theoretical solutions.

The aim of this work is to provide fast and accurate approximation schemes for the Monte-Carlo pricing of derivatives in the L\'evy LIBOR model of Eberlein and \"Ozkan (2005). Standard methods can be applied to solve the stochastic differential equations of the successive LIBOR rates but the methods are generally slow. We propose an alternative approximation scheme based on Picard iterations. Our approach is similar in accuracy to the full numerical solution, but with the feature that each rate is, unlike the standard method, evolved independently of the other rates in the term structure. This enables simultaneous calculation of derivative prices of different maturities using parallel computing. We include numerical illustrations of the accuracy and speed of our method pricing caplets.

In this paper we derive accurate numerical methods for the quantum Boltzmann equation for a gas of interacting bosons. The schemes preserve the main physical features of the continuous problem, namely conservation of mass and energy, the entropy inequality and generalized Bose-Einstein distributions as steady states. These properties are essential in order to develop schemes that are able to capture the energy concentration behavior of bosons. In addition we develop fast algorithms for the numerical evaluation of the resulting quadrature formulas which allow the final schemes to be computed only in O(N^2 log N) operations instead of O(N^3).

A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is developed for systems of partial differential equations with polynomial nonlinearities. The framework is rather general and allows for an arbitrary number of dependent and independent variables with derivatives of any order. It is proved formally that second order convergence is obtained. The procedure is applied to a test case and numerical experiments are provided.

International Journal for Numerical and Analytical Methods in Geomechanics 1991: Volume 15 , Issue CONTENTS. Digitized from IA1643932-02 . Previous issue: sim_intl-journal-numerical-methods-in-geomechanics_1990-12_14_9 . Next issue: sim_intl-journal-numerical-methods-in-geomechanics_1991-01_15_1 .

We consider the Euler equation of quasi-geostrophic fluids which is widely used in weather forecast. Our goal is to study explicit volume-preserving numerical methods for very long simulations on an energy and enstrophy preserving discretization. To this purpose, we compute the average fields and estimate statistical parameters. It is observed that the statistical parameters depend on the integrals of the discretization and that the computed parameters are affected by the error in those, even if the volume of the phase space is correctly preserved. We conclude that monitoring the error in the integrals by the explicit volume preserving method can be used as an indication of how good the estimated parameters are.

The phenomenon of coherent energetic pulse propagation in exciton-phonon molecular chains such as $\alpha$-helix protein is studied using an ODE system model of Davydov-Scott type, both with numerical studies using a new unconditionally stable fourth order accurate energy-momentum conserving time discretization, and with analytical explanation of the main numerical observations. Physically natural impulsive initial data associated with the energy released by ATP hydrolysis are used, and the best current estimates of physical parameter values. In contrast to previous studies based on a proposed long wave approximation by the nonlinear Schr\"odinger (NLS) equation and focusing on initial data resembling the soliton solutions of that equation, the results here instead lead to approximation by the third derivative nonlinear Schr\"odinger equation, giving a far better fit to observed behavior. A good part of the behavior is indeed explained well by the linear part of that equation, the Airy PDE, while other significant features do not fit any PDE approximation, but are instead explained well by a linearized analysis of the ODE system. A convenient method is described for construction the highly stable, accurate conservative time discretizations used, with proof of its desirable properties for a large class of Hamiltonian systems, including a variety of molecular models.

International Journal for Numerical Methods in Engineering 1988: Volume 25 , Issue Contents. Digitized from IA1643935-06 . Previous issue: sim_international-journal-numerical-methods-engineering_1987-12_24_12 . Next issue: sim_international-journal-numerical-methods-engineering_1988-01_25_1 .

International Journal for Numerical Methods in Engineering 1988: Volume 25 , Issue Contents. Digitized from IA1643935-06 . Previous issue: sim_international-journal-numerical-methods-engineering_1987-12_24_12 . Next issue: sim_international-journal-numerical-methods-engineering_1988-01_25_1 .

The Kineticist's Workbench is a computer program currently under development whose purpose is to help chemists understand, analyze, and simplify complex chemical reaction mechanisms. This paper discusses one module of the program that numerically simulates mechanisms and constructs qualitative descriptions of the simulation results. These descriptions are given in terms that are meaningful to the working chemist (e.g., steady states, stable oscillations, and so on); and the descriptions (as well as the data structures used to construct them) are accessible as input to other programs. Keywords: Symbolic computation; Chemical kinetics.

Immersed boundary methods for computing confined fluid and plasma flows in complex geometries are reviewed. The mathematical principle of the volume penalization technique is described and simple examples for imposing Dirichlet and Neumann boundary conditions in one dimension are given. Applications for fluid and plasma turbulence in two and three space dimensions illustrate the applicability and the efficiency of the method in computing flows in complex geometries, for example in toroidal geometries with asymmetric poloidal cross-sections.

In this paper, we propose efficient and accurate numerical methods for computing the ground state and dynamics of the dipolar Bose-Einstein condensates utilising a newly developed dipole-dipole interaction (DDI) solver that is implemented with the non-uniform fast Fourier transform (NUFFT) algorithm. We begin with the three-dimensional (3D) Gross-Pitaevskii equation (GPE) with a DDI term and present the corresponding two-dimensional (2D) model under a strongly anisotropic confining potential. Different from existing methods, the NUFFT based DDI solver removes the singularity by adopting the spherical/polar coordinates in Fourier space in 3D/2D, respectively, thus it can achieve spectral accuracy in space and simultaneously maintain high efficiency by making full use of FFT and NUFFT whenever it is necessary and/or needed. Then, we incorporate this solver into existing successful methods for computing the ground state and dynamics of GPE with a DDI for dipolar BEC. Extensive numerical comparisons with existing methods are carried out for computing the DDI, ground states and dynamics of the dipolar BEC. Numerical results show that our new methods outperform existing methods in terms of both accuracy and efficiency.

The achievement of Bose-Einstein condensation (BEC) in ultracold vapors of alkali atoms has given enormous impulse to the theoretical and experimental study of dilute atomic gases in condensed quantum states inside magnetic traps and optical lattices. This article offers a short survey on mathematical models and theories as well as numerical methods for BEC based on the mean field theory. We start with the Gross-Pitaevskii equation (GPE) in three dimensions (3D) for modeling one-component BEC of the weakly interacting bosons, scale it to obtain a three-parameter model and show how to reduce it to two dimensions (2D) and one dimension (1D) GPEs in certain limiting regimes. Mathematical theories and numerical methods for ground states and dynamics of BEC are provided. Extensions to GPE with an angular momentum rotation term for a rotating BEC, to GPE with long-range anisotropic dipole-dipole interaction for a dipolar BEC and to coupled GPEs for spin-orbit coupled BECs are discussed. Finally, some conclusions are drawn and future research perspectives are discussed.

The long-germ goal of this research is to realize the potential for realistic-domain numerical models as tools for prediction and understanding of the dynamics and transport processes in the coastal ocean.

The computer program NERF (Numerical Evaluation of Reliability Functions) has been designed to evaluate the reliability functions that result from the application of reliability analysis to the fatigue of aircraft structures, in particular, those reliability functions derived by Payne and his co-workers at the Aeronautical Research Laboratories. The NERF program, although based on the Payne reliability models is capable of extension to more complex models as the need arises. This dcoument details the mathematical development of the reliability functions evaluated by NERF and describes the computer program in sufficient detail to allow desired modifications.

In this paper, we consider augmented Lagrangian (AL) algorithms for solving large-scale nonlinear optimization problems that execute adaptive strategies for updating the penalty parameter. Our work is motivated by the recently proposed adaptive AL trust region method by Curtis, Jiang, and Robinson [Math. Prog., DOI: 10.1007/s10107-014-0784-y, 2013]. The first focal point of this paper is a new variant of the approach that employs a line search rather than a trust region strategy, where a critical algorithmic feature for the line search strategy is the use of convexified piecewise quadratic models of the AL function for computing the search directions. We prove global convergence guarantees for our line search algorithm that are on par with those for the previously proposed trust region method. A second focal point of this paper is the practical performance of the line search and trust region algorithm variants in Matlab software, as well as that of an adaptive penalty parameter updating strategy incorporated into the Lancelot software. We test these methods on problems from the CUTEst and COPS collections, as well as on challenging test problems related to optimal power flow. Our numerical experience suggests that the adaptive algorithms outperform traditional AL methods in terms of efficiency and reliability. As with traditional AL algorithms, the adaptive methods are matrix-free and thus represent a viable option for solving extreme-scale problems.

The completion and testing of a fast algorithm for the calculation of two-dimensional dopant redistribution during oxide growth is described. A numerical comparison with analytical, exact solutions for low-dose implants is made which shows negligible error.

The convergence estimates obtained for the Korteweg-de Virus equation have been generalized, under the assumption that the solution u is sufficiently regular. For p 4, it is not known whether a global smooth solution exists corresponding to smooth initial data. It is in fact conjectured that for these cases, the solution may develop a singularity in finite time. A code that uses a spatially and temporally adaptive strategy has been implemented. We are currently investigating the stability of solitary type solutions. As conjectured, these solutions are highly unstable for initial amplitudes larger than one.

Research was concerned with designing and analyzing efficient and novel iterative algorithms for solving large sparse linear systems, typically arising from the discretizations of partial differential equations, which are highly parallelizable and converge fast. These include domain decomposition algorithms and multilevel preconditioners. Some basic dense linear algebra problems, including rank-revealing QR factorizations and stable Toeplitz solvers, which have applications to signal processing were considered.

An efficient linear solver plays an important role while solving partial differential equations (PDEs) and partial integro-differential equations (PIDEs) type mathematical models. In most cases, the efficiency depends on the stability and accuracy of the numerical scheme considered. In this article we consider a PIDE that arises in option pricing theory (financial problems) as well as in various scientific modeling and deal with two different topics. In the first part of the article, we study several iterative techniques (preconditioned) for the PIDE model. A wavelet basis and a Fourier sine basis have been used to design various preconditioners to improve the convergence criteria of iterative solvers. We implement a multigrid (MG) iterative method. In fact, we approximate the problem using a finite difference scheme, then implement a few preconditioned Krylov subspace methods as well as a MG method to speed up the computation. Then, in the second part in this study, we analyze the stability and the accuracy of two different one step schemes to approximate the model.

Finite element solution formulations have been carried out for a simple initial value problem based on two different variational statements; that of convolutional integral developed by Gurtin and that of large spring constants adapted by this writer for initial value problems. Numerical results indicate that both generate convergent solution to the given initial value problem of a spring-mass system subjected to a harmonic forcing function.

An indexed-based object recognition system using geometric invariance techniques has been designed, and used to recognize buildings in an image of a military site and for recognizing curved planar objects including gasless. New invariants and indexing techniques for polyhedral and curved objects with repetition or bilateral symmetry and objects with the imaged outline of a surface of revolution have been developed. A method to distinguish projectively equivalent but Euclidean distinct objects in an uncalibrated view has been investigated. A group-theoretic framework for relating quasi-invariants to invariants has been formulated. Computing invariants can be formulated as an algebraic manipulation problem involving variable elimination and solving nonlinear polynomial equations. Based on Dixon's formulation of resultants, new methods for eliminating variables have been developed and implemented. These methods are much faster and superior than other elimination techniques. A branch and prune approach for numerically solving polynomial equations has been developed. A simple algorithm for separating invariant relations among object and image features to compute invariants of object features has been designed. These algorithms can serve as a basis for building an invariant work-bench that would enable researchers to experiment with geometric configurations and investigate their geometric invariants.

In this project we have studied high order finite difference, finite element, and spectral methods for the numerical solution of discontinuous or high gradient solutions. Theoretical study of algorithm development, stability, accuracy and convergence analysis, as well as applications of the algorithms to computational fluid dynamics (both compressible and incompressible) and semiconductor device simulation are performed. Parallel implementation issues of these high order methods are also studied.

Different numerical approaches for the stray-field calculation in the context of micromagnetic simulations are investigated. We compare finite difference based fast Fourier transform methods, tensor grid methods and the finite-element method with shell transformation in terms of computational complexity, storage requirements and accuracy tested on several benchmark problems. These methods can be subdivided into integral methods (fast Fourier transform methods, tensor-grid method) which solve the stray field directly and in differential equation methods (finite-element method), which compute the stray field as the solution of a partial differential equation. It turns out that for cuboid structures the integral methods, which work on cuboid grids (fast Fourier transform methods and tensor grid methods) outperform the finite-element method in terms of the ratio of computational effort to accuracy. Among these three methods the tensor grid method is the fastest. However, the use of the tensor grid method in the context of full micromagnetic codes is not well investigated yet. The finite-element method performs best for computations on curved structures.

Exact numerical methods and stochastic simulation methods are developed to study the force stretching single RNA issue on the secondary structure level in equilibrium. By computing the force-extension curves on the constant force and the constant extension ensembles, we find the two independent methods agree with each other quite well. To show the precision of our methods in predicting unfolding experiments, the unfolding forces of different RNA molecules under different experimental conditions are calculated. We find that the ionic corrections on the RNA free energies alone might not account for the apparent differences between the theoretical calculations and the experimental data; an ionic correction to the persistent length of single-stranded RNA should be necessary.

We discuss two numerical methods, based on a path integral approach described in a previous paper (I), for solving the stochastic equations underlying the financial markets: the Monte Carlo approach, and the Green function deterministic numerical method. Then, we apply the latter to some specific financial problems. In particular, we consider the pricing of a European option, a zero-coupon bond, a caplet, an American option, and a Bermudan swaption.

Static analysis by abstract interpretation aims at automatically proving properties of computer programs. To do this, an over-approximation of program semantics, defined as the least fixpoint of a system of semantic equations, must be computed. To enforce the convergence of this computation, widening operator is used but it may lead to coarse results. We propose a new method to accelerate the computation of this fixpoint by using standard techniques of numerical analysis. Our goal is to automatically and dynamically adapt the widening operator in order to maintain precision.

Environmental models applicable for purposes of mission analysis were developed, documented, and recommended. The findings are described.

The use of multi-dimensional finite volume numerical techniques with finite thickness models for calculating aeroheating rates from measured global surface temperatures on hypersonic wind tunnel models was investigated. Both direct and inverse finite volume techniques were investigated and compared with the one-dimensional semi -infinite technique. Global transient surface temperatures were measured using an infrared thermographic technique on a 0.333-scale model of the Hyper-X forebody in the Langley Research Center 20-Inch Mach 6 Air tunnel. In these tests the effectiveness of vortices generated via gas injection for initiating hypersonic transition on the Hyper-X forebody were investigated. An array of streamwise orientated heating striations were generated and visualized downstream of the gas injection sites. In regions without significant spatial temperature gradients, one-dimensional techniques provided accurate aeroheating rates. In regions with sharp temperature gradients due to the striation patterns two-dimensional heat transfer techniques were necessary to obtain accurate heating rates. The use of the one-dimensional technique resulted in differences of 20% in the calculated heating rates because it did not account for lateral heat conduction in the model.

In this project, we have accomplished in development of algorithms to model transport and eletromagnetic processes in mesoscopic systems such as nano-electronics and biological membrane, and layered inhomogeneous media. Specifically, the following results have been obtained resulting in the publication of 6 peer-referred journal papers and a third part of a Cambridge University Press book. (1) fast integral solver for quantum dots in 3-D layered media. The fast solver is based on a window accelerated method for computing the layered Green's function and wide band Fast multipole methods for Hankel waves. (2) a new linear scaling discontinuous Galerkin density functional theory, which provide a brand new approach in combining physics-based orbitals and piece-wise polynomial finite element basis in finding the ground state energy of the DFT for quantum systems. (3) numerical methods for computation of electrostatics in ion-channel transport, (4) a new parallel solver for elliptic PDEs by combining random walk Feynmann-Kac formula and local boundary integral equations for extreme computing, (5) an improved device adaptive inflow boundary condition for Wigner quantum transport equations.

The use of multi-dimensional finite volume heat conduction techniques for calculating aeroheating rates from measured global surface temperatures on hypersonic wind tunnel models was investigated. Both direct and inverse finite volume techniques were investigated and compared with the standard one-dimensional semi-infinite technique. Global transient surface temperatures were measured using an infrared thermographic technique on a 0.333-scale model of the Hyper-X forebody in the NASA Langley Research Center 20-Inch Mach 6 Air tunnel. In these tests the effectiveness of vortices generated via gas injection for initiating hypersonic transition on the Hyper-X forebody was investigated. An array of streamwise-orientated heating striations was generated and visualized downstream of the gas injection sites. In regions without significant spatial temperature gradients, one-dimensional techniques provided accurate aeroheating rates. In regions with sharp temperature gradients caused by striation patterns multi-dimensional heat transfer techniques were necessary to obtain more accurate heating rates. The use of the one-dimensional technique resulted in differences of 20% in the calculated heating rates compared to 2-D analysis because it did not account for lateral heat conduction in the model.

AD-A244 444

Stability of multistep methods in numerical integration

A reexamination of some numerical methods is considered in light of the new class of computers which use vector streaming to achieve high computation rates. A study has been made of the effect on the relative efficiency of several numerical methods applied to a particular fluid flow problem when they are implemented on a vector computer. The method of Brailovskaya, the alternating direction implicit method, a fully implicit method, and a new method called partial implicitization have been applied to the problem of determining the steady state solution of the two-dimensional flow of a viscous imcompressible fluid in a square cavity driven by a sliding wall. Results are obtained for three mesh sizes and a comparison is made of the methods for serial computation.

A reexamination of some numerical methods is considered in light of the new class of computers which use vector streaming to achieve high computation rates. A study has been made of the effect on the relative efficiency of several numerical methods applied to a particular fluid flow problem when they are implemented on a vector computer. The method of Brailovskaya, the alternating direction implicit method, a fully implicit method, and a new method called partial implicitization have been applied to the problem of determining the steady state solution of the two-dimensional flow of a viscous imcompressible fluid in a square cavity driven by a sliding wall. Results are obtained for three mesh sizes and a comparison is made of the methods for serial computation.

A reexamination of some numerical methods is considered in light of the new class of computers which use vector streaming to achieve high computation rates. A study has been made of the effect on the relative efficiency of several numerical methods applied to a particular fluid flow problem when they are implemented on a vector computer. The method of Brailovskaya, the alternating direction implicit method, a fully implicit method, and a new method called partial implicitization have been applied to the problem of determining the steady state solution of the two-dimensional flow of a viscous imcompressible fluid in a square cavity driven by a sliding wall. Results are obtained for three mesh sizes and a comparison is made of the methods for serial computation.

The second AFSWC Hydrodynamic Conference was held at Kirtland Air Force Base for the purpose of discussing numerical techniques which have become an important and widely used tool for solving fluid flow problems. A large number of the papers presented dealt with the problems of the finite difference analogs of the differential equations of motion. A few papers discussed work toward analytic solutions of these equations. Topics included hydrodynamics, magnetohydrodynamics, radiation transport, and solid material motion.

This report presents some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere. The data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement 'noise'. The case discussed to greatest length is that of complete, global data sets on regular grids (i.e., lines of latitude and longitude, the latter, at least, separated by constant interval); the case where data are sparsely and irregularly distributed is also considered in some detail. The first section presents some basic properties of spherical harmonics, stressing their relationship to two-dimensional Fourier series. Algorithms for the evaluation of the harmonic coefficients by numerical quadratures are given here, and it is shown that the number of operations is the order of N cubed for equal angular grids, where N is the number of lines of latitude, or 'Nyquist frequency', of the grid. The second section introduces a quadratic measure for the error in the estimation of the coefficients by linear techniques. This is the error measure of least squares collocation, which is a method that can be used for harmonic analysis. Efficient algorithms for implementing collocation on the whole sphere are described. a formal relationship between collocation and least squares adjustment is used to obtain an alternative form of the collocation algorithm that is likely to be stable with dense data sets and, with a minor modification, can be used to implement least squares adjustment as well. The basic principle is that for regular grids the variance-convariance matrix of the data consists of Toeplitz-circulant blocks, so it can be both set up and inverted very efficiently.

This report presents some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere. The data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement 'noise'. The case discussed to greatest length is that of complete, global data sets on regular grids (i.e., lines of latitude and longitude, the latter, at least, separated by constant interval); the case where data are sparsely and irregularly distributed is also considered in some detail. The first section presents some basic properties of spherical harmonics, stressing their relationship to two-dimensional Fourier series. Algorithms for the evaluation of the harmonic coefficients by numerical quadratures are given here, and it is shown that the number of operations is the order of N cubed for equal angular grids, where N is the number of lines of latitude, or 'Nyquist frequency', of the grid. The second section introduces a quadratic measure for the error in the estimation of the coefficients by linear techniques. This is the error measure of least squares collocation, which is a method that can be used for harmonic analysis. Efficient algorithms for implementing collocation on the whole sphere are described. a formal relationship between collocation and least squares adjustment is used to obtain an alternative form of the collocation algorithm that is likely to be stable with dense data sets and, with a minor modification, can be used to implement least squares adjustment as well. The basic principle is that for regular grids the variance-convariance matrix of the data consists of Toeplitz-circulant blocks, so it can be both set up and inverted very efficiently.

This report presents some numerical methods for estimating spherical harmonic coefficients from data sampled on the sphere. The data may be given in the form of area means or of point values, and it may be free from errors or affected by measurement 'noise'. The case discussed to greatest length is that of complete, global data sets on regular grids (i.e., lines of latitude and longitude, the latter, at least, separated by constant interval); the case where data are sparsely and irregularly distributed is also considered in some detail. The first section presents some basic properties of spherical harmonics, stressing their relationship to two-dimensional Fourier series. Algorithms for the evaluation of the harmonic coefficients by numerical quadratures are given here, and it is shown that the number of operations is the order of N cubed for equal angular grids, where N is the number of lines of latitude, or 'Nyquist frequency', of the grid. The second section introduces a quadratic measure for the error in the estimation of the coefficients by linear techniques. This is the error measure of least squares collocation, which is a method that can be used for harmonic analysis. Efficient algorithms for implementing collocation on the whole sphere are described. a formal relationship between collocation and least squares adjustment is used to obtain an alternative form of the collocation algorithm that is likely to be stable with dense data sets and, with a minor modification, can be used to implement least squares adjustment as well. The basic principle is that for regular grids the variance-convariance matrix of the data consists of Toeplitz-circulant blocks, so it can be both set up and inverted very efficiently.

The crossed-triangle macroelement has been identified as and ideal element for Newtonian fluid flow calculations involving fluids with integral constitutive equations. In spite of an instability of the pressure approximation, these elements have been shown to have convergent velocities in Stokes flow, and there is strong evidence that pressure-smoothing schemes recover accurate pressures. Earlier studies by the author of Newtonian flow over transverse slots at low Reynolds numbers showed that excellent results could be obtained using the element. Studies of non-Newtonian flows in the same geometry showed good qualitative agreement with laboratory experiment but led to some puzzling predictions of the pressure difference between the top and bottom of the slot (the hole-pressure). In this paper, those puzzling predictions are re-examined, and the deviations from expectation are re-interpreted. They appear to make physical sense and have important ramifications for the calibration of devices which measure the primary normal-stress difference by continuous measurement of the hole-pressure. Keywords: Numerical analysis; Mathematical prediction. (Author)

The transient heat conduction equation, with Dirichlet and Neumann boundary conditions, is solved by the methods of finite-differences and finite-elements, and the numerical solutions are investigated with respect to accuracy and stability. A general six point finite-difference expression is used for which there exists a high order accurate modification. The finite-element method used is based on a stationary variational principle. Several methods for treating accuracy and convergence problems which result from a discontinuity in the initial condition are investigated. The Crank-Nicolson method is a special case of both the finite-difference and finite-element methods. The finite-difference version of the Crank-Nicolson method is shown to be more accurate than the finite-element version, especially when a discontinuity exists between the initial condition and the boundary conditions. The high order accurate schemes for both finite-differences and finite-elements are shown to be equivalent for the case of linear elements. Some of the results suggest the possibility of finding a finite-element scheme which is highly accurate in a mean square sensr over the entire solution domain. (Author)