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University of California, San Diego
UNIVERSITY OF CALIFORNIA, SAN DIEGO Computational Methods for Parameter Estimation in Nonlinear Models A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Physics with a Specialization in Computational Physics by Bryan Andrew Toth Committee in charge: Professor Henry D. I. Abarbanel, Chair Professor Philip Gill Professor Julius Kuti Professor Gabriel Silva Professor Frank Wuerthwein 2011 Copyright Bryan Andrew Toth, 2011 All rights reserved. The dissertation of Bryan Andrew Toth is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2011 iii DEDICATION To my grandparents, August and Virginia Toth and Willem and Jane Keur, who helped put me on a lifelong path of learning. iv EPIGRAPH An Expert: One who knows more and more about less and less, until eventually he knows everything about nothing. |Source Unknown v TABLE OF CONTENTS Signature Page . iii Dedication . iv Epigraph . v Table of Contents . vi List of Figures . ix List of Tables . x Acknowledgements . xi Vita and Publications . xii Abstract of the Dissertation . xiii Chapter 1 Introduction . 1 1.1 Dynamical Systems . 1 1.1.1 Linear and Nonlinear Dynamics . 2 1.1.2 Chaos . 4 1.1.3 Synchronization . 6 1.2 Parameter Estimation . 8 1.2.1 Kalman Filters . 8 1.2.2 Variational Methods . 9 1.2.3 Parameter Estimation in Nonlinear Systems . 9 1.3 Dissertation Preview . 10 Chapter 2 Dynamical State and Parameter Estimation . 11 2.1 Introduction . 11 2.2 DSPE Overview . 11 2.3 Formulation . 12 2.3.1 Least Squares Minimization . -
An Algorithm Based on Semidefinite Programming for Finding Minimax
Takustr. 7 Zuse Institute Berlin 14195 Berlin Germany BELMIRO P.M. DUARTE,GUILLAUME SAGNOL,WENG KEE WONG An algorithm based on Semidefinite Programming for finding minimax optimal designs ZIB Report 18-01 (December 2017) Zuse Institute Berlin Takustr. 7 14195 Berlin Germany Telephone: +49 30-84185-0 Telefax: +49 30-84185-125 E-mail: [email protected] URL: http://www.zib.de ZIB-Report (Print) ISSN 1438-0064 ZIB-Report (Internet) ISSN 2192-7782 An algorithm based on Semidefinite Programming for finding minimax optimal designs Belmiro P.M. Duarte a,b, Guillaume Sagnolc, Weng Kee Wongd aPolytechnic Institute of Coimbra, ISEC, Department of Chemical and Biological Engineering, Portugal. bCIEPQPF, Department of Chemical Engineering, University of Coimbra, Portugal. cTechnische Universität Berlin, Institut für Mathematik, Germany. dDepartment of Biostatistics, Fielding School of Public Health, UCLA, U.S.A. Abstract An algorithm based on a delayed constraint generation method for solving semi- infinite programs for constructing minimax optimal designs for nonlinear models is proposed. The outer optimization level of the minimax optimization problem is solved using a semidefinite programming based approach that requires the de- sign space be discretized. A nonlinear programming solver is then used to solve the inner program to determine the combination of the parameters that yields the worst-case value of the design criterion. The proposed algorithm is applied to find minimax optimal designs for the logistic model, the flexible 4-parameter Hill homoscedastic model and the general nth order consecutive reaction model, and shows that it (i) produces designs that compare well with minimax D−optimal de- signs obtained from semi-infinite programming method in the literature; (ii) can be applied to semidefinite representable optimality criteria, that include the com- mon A−; E−; G−; I− and D-optimality criteria; (iii) can tackle design problems with arbitrary linear constraints on the weights; and (iv) is fast and relatively easy to use. -
Using Mathematica As a Teaching Tool in the Undergraduate Economics Curriculum
USING MATHEMATICA AS A TEACHING TOOL IN THE UNDERGRADUATE ECONOMICS CURRICULUM Gary Hodgin* Abstract In recent years, the quantitative skills required of economics students have increased significantly. Students often experience difficulty in making the transition from their mathematics courses to the use of mathematics in their economics courses. Symbolic computation programs are mathematical tools that can potentially smooth this transition. This paper illustrates a few ways in which one symbolic computation program, Mathematica, has been used in the undergraduate economics curriculum. I. Introduction The undergraduate curriculum in economics has become increasingly quantitative. An economics major typically includes college algebra, statistics, and some form of calculus in its degree requirements.1 In addition, mathematical economics and econometrics are now standard courses in many undergraduate programs. Moreover, this quantitative emphasis generally extends to other courses across the economics curriculum, particularly the intermediate theory and managerial economics courses. For example, approximately 90 percent of the intermediate microeconomics instructors included in von Allmen and Brower's survey (1998, 279) use calculus in their courses. Although one might question whether too much emphasis is placed on quantitative skills, there appears to be consensus among economics professors that mathematics is an important ingredient in undergraduate economic education. Over the past three years, I have used a symbolic computation program entitled Mathematica in two of my intermediate microeconomics and three of my managerial economics courses. My observations in these courses suggests that some students experience difficulty in making the transition from their courses in mathematics to their use of mathematics in economics. By smoothing this transition, symbolic computation programs can assist students in learning economics. -
Julia, My New Friend for Computing and Optimization? Pierre Haessig, Lilian Besson
Julia, my new friend for computing and optimization? Pierre Haessig, Lilian Besson To cite this version: Pierre Haessig, Lilian Besson. Julia, my new friend for computing and optimization?. Master. France. 2018. cel-01830248 HAL Id: cel-01830248 https://hal.archives-ouvertes.fr/cel-01830248 Submitted on 4 Jul 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. « Julia, my new computing friend? » | 14 June 2018, IETR@Vannes | By: L. Besson & P. Haessig 1 « Julia, my New frieNd for computiNg aNd optimizatioN? » Intro to the Julia programming language, for MATLAB users Date: 14th of June 2018 Who: Lilian Besson & Pierre Haessig (SCEE & AUT team @ IETR / CentraleSupélec campus Rennes) « Julia, my new computing friend? » | 14 June 2018, IETR@Vannes | By: L. Besson & P. Haessig 2 AgeNda for today [30 miN] 1. What is Julia? [5 miN] 2. ComparisoN with MATLAB [5 miN] 3. Two examples of problems solved Julia [5 miN] 4. LoNger ex. oN optimizatioN with JuMP [13miN] 5. LiNks for more iNformatioN ? [2 miN] « Julia, my new computing friend? » | 14 June 2018, IETR@Vannes | By: L. Besson & P. Haessig 3 1. What is Julia ? Open-source and free programming language (MIT license) Developed since 2012 (creators: MIT researchers) Growing popularity worldwide, in research, data science, finance etc… Multi-platform: Windows, Mac OS X, GNU/Linux.. -
[20Pt]Algorithms for Constrained Optimization: [ 5Pt]
SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s Algorithms for Constrained Optimization: The Benefits of General-purpose Software Michael Saunders MS&E and ICME, Stanford University California, USA 3rd AI+IoT Business Conference Shenzhen, China, April 25, 2019 Optimization Software 3rd AI+IoT Business Conference, Shenzhen, April 25, 2019 1/39 SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s SOL Systems Optimization Laboratory George Dantzig, Stanford University, 1974 Inventor of the Simplex Method Father of linear programming Large-scale optimization: Algorithms, software, applications Optimization Software 3rd AI+IoT Business Conference, Shenzhen, April 25, 2019 2/39 SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s SOL history 1974 Dantzig and Cottle start SOL 1974{78 John Tomlin, LP/MIP expert 1974{2005 Alan Manne, nonlinear economic models 1975{76 MS, MINOS first version 1979{87 Philip Gill, Walter Murray, MS, Margaret Wright (Gang of 4!) 1989{ Gerd Infanger, stochastic optimization 1979{ Walter Murray, MS, many students 2002{ Yinyu Ye, optimization algorithms, especially interior methods This week! UC Berkeley opened George B. Dantzig Auditorium Optimization Software 3rd AI+IoT Business Conference, Shenzhen, April 25, 2019 3/39 SOL Optimization 1970s 1980s 1990s 2000s 2010s Summary 2020s Optimization problems Minimize an objective function subject to constraints: 0 x 1 min '(x) st ` ≤ @ Ax A ≤ u c(x) x variables 0 1 A matrix c1(x) B . C c(x) nonlinear functions @ . A c (x) `; u bounds m Optimization -
Alternatives to Python: Julia
Crossing Language Barriers with , SciPy, and thon Steven G. Johnson MIT Applied Mathemacs Where I’m coming from… [ google “Steven Johnson MIT” ] Computaonal soPware you may know… … mainly C/C++ libraries & soPware … Nanophotonics … oPen with Python interfaces … (& Matlab & Scheme & …) jdj.mit.edu/nlopt www.w.org jdj.mit.edu/meep erf(z) (and erfc, erfi, …) in SciPy 0.12+ & other EM simulators… jdj.mit.edu/book Confession: I’ve used Python’s internal C API more than I’ve coded in Python… A new programming language? Viral Shah Jeff Bezanson Alan Edelman julialang.org Stefan Karpinski [begun 2009, “0.1” in 2013, ~20k commits] [ 17+ developers with 100+ commits ] [ usual fate of all First reacBon: You’re doomed. new languages ] … subsequently: … probably doomed … sll might be doomed but, in the meanBme, I’m having fun with it… … and it solves a real problem with technical compuBng in high-level languages. The “Two-Language” Problem Want a high-level language that you can work with interacBvely = easy development, prototyping, exploraon ⇒ dynamically typed language Plenty to choose from: Python, Matlab / Octave, R, Scilab, … (& some of us even like Scheme / Guile) Historically, can’t write performance-criBcal code (“inner loops”) in these languages… have to switch to C/Fortran/… (stac). [ e.g. SciPy git master is ~70% C/C++/Fortran] Workable, but Python → Python+C = a huge jump in complexity. Just vectorize your code? = rely on mature external libraries, operang on large blocks of data, for performance-criBcal code Good advice! But… • Someone has to write those libraries. • Eventually that person may be you. -
Data Visualization in Python
Data visualization in python Day 2 A variety of packages and philosophies • (today) matplotlib: http://matplotlib.org/ – Gallery: http://matplotlib.org/gallery.html – Frequently used commands: http://matplotlib.org/api/pyplot_summary.html • Seaborn: http://stanford.edu/~mwaskom/software/seaborn/ • ggplot: – R version: http://docs.ggplot2.org/current/ – Python port: http://ggplot.yhathq.com/ • Bokeh (live plots in your browser) – http://bokeh.pydata.org/en/latest/ Biocomputing Bootcamp 2017 Matplotlib • Gallery: http://matplotlib.org/gallery.html • Top commands: http://matplotlib.org/api/pyplot_summary.html • Provides "pylab" API, a mimic of matlab • Many different graph types and options, some obscure Biocomputing Bootcamp 2017 Matplotlib • Resulting plots represented by python objects, from entire figure down to individual points/lines. • Large API allows any aspect to be tweaked • Lengthy coding sometimes required to make a plot "just so" Biocomputing Bootcamp 2017 Seaborn • https://stanford.edu/~mwaskom/software/seaborn/ • Implements more complex plot types – Joint points, clustergrams, fitted linear models • Uses matplotlib "under the hood" Biocomputing Bootcamp 2017 Others • ggplot: – (Original) R version: http://docs.ggplot2.org/current/ – A recent python port: http://ggplot.yhathq.com/ – Elegant syntax for compactly specifying plots – but, they can be hard to tweak – We'll discuss this on the R side tomorrow, both the basics of both work similarly. • Bokeh – Live, clickable plots in your browser! – http://bokeh.pydata.org/en/latest/ -
Ipython: a System for Interactive Scientific
P YTHON: B ATTERIES I NCLUDED IPython: A System for Interactive Scientific Computing Python offers basic facilities for interactive work and a comprehensive library on top of which more sophisticated systems can be built. The IPython project provides an enhanced interactive environment that includes, among other features, support for data visualization and facilities for distributed and parallel computation. he backbone of scientific computing is All these systems offer an interactive command mostly a collection of high-perfor- line in which code can be run immediately, without mance code written in Fortran, C, and having to go through the traditional edit/com- C++ that typically runs in batch mode pile/execute cycle. This flexible style matches well onT large systems, clusters, and supercomputers. the spirit of computing in a scientific context, in However, over the past decade, high-level environ- which determining what computations must be ments that integrate easy-to-use interpreted lan- performed next often requires significant work. An guages, comprehensive numerical libraries, and interactive environment lets scientists look at data, visualization facilities have become extremely popu- test new ideas, combine algorithmic approaches, lar in this field. As hardware becomes faster, the crit- and evaluate their outcome directly. This process ical bottleneck in scientific computing isn’t always the might lead to a final result, or it might clarify how computer’s processing time; the scientist’s time is also they need to build a more static, large-scale pro- a consideration. For this reason, systems that allow duction code. rapid algorithmic exploration, data analysis, and vi- As this article shows, Python (www.python.org) sualization have become a staple of daily scientific is an excellent tool for such a workflow.1 The work. -
Numericaloptimization
Numerical Optimization Alberto Bemporad http://cse.lab.imtlucca.it/~bemporad/teaching/numopt Academic year 2020-2021 Course objectives Solve complex decision problems by using numerical optimization Application domains: • Finance, management science, economics (portfolio optimization, business analytics, investment plans, resource allocation, logistics, ...) • Engineering (engineering design, process optimization, embedded control, ...) • Artificial intelligence (machine learning, data science, autonomous driving, ...) • Myriads of other applications (transportation, smart grids, water networks, sports scheduling, health-care, oil & gas, space, ...) ©2021 A. Bemporad - Numerical Optimization 2/102 Course objectives What this course is about: • How to formulate a decision problem as a numerical optimization problem? (modeling) • Which numerical algorithm is most appropriate to solve the problem? (algorithms) • What’s the theory behind the algorithm? (theory) ©2021 A. Bemporad - Numerical Optimization 3/102 Course contents • Optimization modeling – Linear models – Convex models • Optimization theory – Optimality conditions, sensitivity analysis – Duality • Optimization algorithms – Basics of numerical linear algebra – Convex programming – Nonlinear programming ©2021 A. Bemporad - Numerical Optimization 4/102 References i ©2021 A. Bemporad - Numerical Optimization 5/102 Other references • Stephen Boyd’s “Convex Optimization” courses at Stanford: http://ee364a.stanford.edu http://ee364b.stanford.edu • Lieven Vandenberghe’s courses at UCLA: http://www.seas.ucla.edu/~vandenbe/ • For more tutorials/books see http://plato.asu.edu/sub/tutorials.html ©2021 A. Bemporad - Numerical Optimization 6/102 Optimization modeling What is optimization? • Optimization = assign values to a set of decision variables so to optimize a certain objective function • Example: Which is the best velocity to minimize fuel consumption ? fuel [ℓ/km] velocity [km/h] 0 30 60 90 120 160 ©2021 A. -
Writing Mathematical Expressions with Latex
APPENDIX A Writing Mathematical Expressions with LaTeX LaTeX is extensively used in Python. In this appendix there are many examples that can be useful to represent LaTeX expressions inside Python implementations. This same information can be found at the link http://matplotlib.org/users/mathtext.html. With matplotlib You can enter the LaTeX expression directly as an argument of various functions that can accept it. For example, the title() function that draws a chart title. import matplotlib.pyplot as plt %matplotlib inline plt.title(r'$\alpha > \beta$') With IPython Notebook in a Markdown Cell You can enter the LaTeX expression between two '$$'. $$c = \sqrt{a^2 + b^2}$$ c= a+22b 537 © Fabio Nelli 2018 F. Nelli, Python Data Analytics, https://doi.org/10.1007/978-1-4842-3913-1 APPENDIX A WRITING MaTHEmaTICaL EXPRESSIONS wITH LaTEX With IPython Notebook in a Python 2 Cell You can enter the LaTeX expression within the Math() function. from IPython.display import display, Math, Latex display(Math(r'F(k) = \int_{-\infty}^{\infty} f(x) e^{2\pi i k} dx')) Subscripts and Superscripts To make subscripts and superscripts, use the ‘_’ and ‘^’ symbols: r'$\alpha_i > \beta_i$' abii> This could be very useful when you have to write summations: r'$\sum_{i=0}^\infty x_i$' ¥ åxi i=0 Fractions, Binomials, and Stacked Numbers Fractions, binomials, and stacked numbers can be created with the \frac{}{}, \binom{}{}, and \stackrel{}{} commands, respectively: r'$\frac{3}{4} \binom{3}{4} \stackrel{3}{4}$' 3 3 æ3 ö4 ç ÷ 4 è 4ø Fractions can be arbitrarily nested: 1 5 - x 4 538 APPENDIX A WRITING MaTHEmaTICaL EXPRESSIONS wITH LaTEX Note that special care needs to be taken to place parentheses and brackets around fractions. -
Ipython Documentation Release 0.10.2
IPython Documentation Release 0.10.2 The IPython Development Team April 09, 2011 CONTENTS 1 Introduction 1 1.1 Overview............................................1 1.2 Enhanced interactive Python shell...............................1 1.3 Interactive parallel computing.................................3 2 Installation 5 2.1 Overview............................................5 2.2 Quickstart...........................................5 2.3 Installing IPython itself....................................6 2.4 Basic optional dependencies..................................7 2.5 Dependencies for IPython.kernel (parallel computing)....................8 2.6 Dependencies for IPython.frontend (the IPython GUI).................... 10 3 Using IPython for interactive work 11 3.1 Quick IPython tutorial..................................... 11 3.2 IPython reference........................................ 17 3.3 IPython as a system shell.................................... 42 3.4 IPython extension API..................................... 47 4 Using IPython for parallel computing 53 4.1 Overview and getting started.................................. 53 4.2 Starting the IPython controller and engines.......................... 57 4.3 IPython’s multiengine interface................................ 64 4.4 The IPython task interface................................... 78 4.5 Using MPI with IPython.................................... 80 4.6 Security details of IPython................................... 83 4.7 IPython/Vision Beam Pattern Demo............................. -
Easybuild Documentation Release 20210907.0
EasyBuild Documentation Release 20210907.0 Ghent University Tue, 07 Sep 2021 08:55:41 Contents 1 What is EasyBuild? 3 2 Concepts and terminology 5 2.1 EasyBuild framework..........................................5 2.2 Easyblocks................................................6 2.3 Toolchains................................................7 2.3.1 system toolchain.......................................7 2.3.2 dummy toolchain (DEPRECATED) ..............................7 2.3.3 Common toolchains.......................................7 2.4 Easyconfig files..............................................7 2.5 Extensions................................................8 3 Typical workflow example: building and installing WRF9 3.1 Searching for available easyconfigs files.................................9 3.2 Getting an overview of planned installations.............................. 10 3.3 Installing a software stack........................................ 11 4 Getting started 13 4.1 Installing EasyBuild........................................... 13 4.1.1 Requirements.......................................... 14 4.1.2 Using pip to Install EasyBuild................................. 14 4.1.3 Installing EasyBuild with EasyBuild.............................. 17 4.1.4 Dependencies.......................................... 19 4.1.5 Sources............................................. 21 4.1.6 In case of installation issues. .................................. 22 4.2 Configuring EasyBuild.......................................... 22 4.2.1 Supported configuration