DOCSLIB.ORG

Simulation of Ordinary Differential Equations

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Simulation of Ordinary Differential Equations Part II The Path-Wise Deterministic Setting 185 The mathematician’s patterns, like the paint- er’s or the poet’s must be beautiful; the ideas, like the colours or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. G H H (1877 - 1947) Random Differential Equations in Scientific Computing Chapter 6 Recap: Theory of Ordinary Dif- ferential Equations (ODEs) This chapter serves as a holistic introduction to the theory of ordinary differ- ential equations (without singularities). After providing some preliminaries, integral curves in vector fields are discussed, i.e., ordinary differential equa- tions x_ = F (t; x). We start with continuous right hand sides F and their "- approximate solutions as well as the Peano-Cauchy existence theorem and its implications. Our discussion continues with Lipschitz-continuous functions and the existence and uniqueness theorem of Picard-Lindelöf. In particular, we analyse maximal integral curves, define the three types of maximal inte- gral curves that can occur in autonomous systems and show the transforma- tion of a d-th order equation into a first order system. We deal with the exis- tence of solutions in the extended sense where the right hand side function may be discontinuous on a set of Lebesgue-measure zero. Caratheodory’s existence theorem and its implications are studied together with maximum and minimum solutions. Then, we study the broad class of linear ordinary dif- ferential equations by discussing the unique existence of their solutions and their explicit construction. Applications of the theory focus on first integrals and oscillations in the deterministic pendulum and the Volterra-Lotka system. Finally, we provide a first glance into the existence, uniqueness and exten- sion of solutions of ordinary differential equations on infinite-dimensional Banach spaces. 6.1 Key Concepts As we have seen in Chap. 3, random (ordinary) differential equations can be interpreted path-wise as (deterministic) ordinary differential equations. Thus, it pays-off to recall the basic properties of these deterministic equations, and, as we will see, it is interesting to compare the results and proof-techniques we utilized in Chap. 3 with those that are relevant for (deterministic) ordinary differential equations. Before diving into the details, let us summarize the key results of this chap- d m ter: Let J ⊆ R, U ⊆ R (d 2 N) and Λ ⊆ R (m 2 N0) be open sub- sets and F a smooth function on the Cartesian product of these to Rd, i.e., F 2 Ck(J × U × Λ; Rn) for some 1 ≤ k ≤ 1. An ordinary differential equation Chapter 6 187 Tobias Neckel & Florian Rupp (ODE) is an equation of the form x_ = F (t; x; λ) ; (6.1) where the dot indicates the differentiation with respect to the time variable t 2 J, x 2 U is a vector of state variables and λ 2 Λ is a vector of param- eters. If d > 1, the equation (6.1) is called a system of d first order (ordinary) differential equations and the differentiation is considered component-wise. In particular, during the study of bifurcation/ change of stability scenarios (cf. Sec. 8.3) the dependence on parameters will become essential. For fixed λ 2 Λ a solution (or trajectory) of (6.1) is a function ' : J0 ⊂ J ! U, J0 open, given by t 7! '(t) such that the following holds for all times t 2 J0: d' = F (t; '(t); λ) : dt The fundamental issues in ODE theory concern existence, uniqueness, con- tinuity (with respect to the parameters and the initial conditions) and exten- sibility of solutions. Fortunately, all these issues are resolved, provided the right-hand side function F has enough regularity: Every initial value problem x_ = F (t; x; λ) with initial conditions x(t0) = x0 2 U (6.2) has a unique solution1 which is smooth with respect to the initial conditions and parameters2. Moreover, the solution of an initial value problem can be extended with respect to time until it either reaches the boundary @U of the domain of definition U 2 Rd of the differential equation or blows up to infin- ity3. Let us be more precise: If there is some finite T such that limt!T k'(t; x0; λ)k ! 1, we say that the solution blows up in finite time. Such solutions will leave every compact subset of U ⊆ Rd (see [162], p. 143). Moreover, every d-th order differential equation, d > 1, is equivalent to a system of d first order differential equations, see [12], p. 105 or [162], p. 148. Finally, as we have seen in Chap. 4, (deterministic) ordinary differential equations on Banach spaces play a crucial role in understanding the mean- square solutions of random differential equations. Because the Euclidean Rd is the “simplest” Banach space, we will basically gain the same results as 1 See, for instance, [56], p. 3, theorem 1.2, or [12], p. 93, corollary 3. In [162], pp. 140, the uniqueness and existence statements are provided for locally Lipschitz-continuous right-hand sides. 2 See, for instance, [56], p. 3, theorem 1.3, or [12], p. 93, corollary 4. 3 See, for instance, [56], p. 3, theorem 1.4, and [162], pp. 143. Again, [162] provides his statements for locally Lipschitz-continuous right-hand sides. 188 Chapter 6 Random Differential Equations in Scientific Computing cited above, though the methods for establishing them will take the infinite- dimensional nature of arbitrary Banach spaces into account. When reading this chapter note the answers to the following key concepts 1. Vector fields, local Lipschitz-continuity, and integral curves. 2. Estimating bounds for the solution of ordinary differential equations by Gronwall’s lemma. 3. The theorems of (a) Cauchy-Peano (local existence w.r.t. continuous right-hand sides), (b) Picard-Lindelöf (local existence and uniqueness w.r.t. Lipschitz- continuous right-hand sides), (c) Caratheodry (local existence of solutions in the extended sense), and their applications. 4. "-approximate solutions. 5. Life-time and extension of maximal integral curves. 6. Classification of maximal integral curves in autonomous/ time- independent systems. 7. Maximum and minimum solutions of ordinary differential equations in the extended sense. 8. Fundamental system and matrix for homogeneous linear ordinary differ- ential equations. 9. Method of the variation of constant. 10. First integrals and oscillations (deterministic pendulum equation and Volterra-Lotka equation). 11. Existence, uniqueness and extension of solutions of ordinary differential equations on infinite-dimensional Banach spaces. as well as the following questions 1. How are vector fields and integral curves connected? 2. What is the method of successive approximation? 3. How and under which conditions can solutions be extended beyond a given open interval of existence? Section 6.1 189 Tobias Neckel & Florian Rupp 4. What does the lemma of Arzela-Ascoli state? 5. How can we use MATLAB to symbolically solve ordinary differential equa- tions and display the solutions graphically? 6. How can a d-th order equation be transformed into a first order system? 7. What are solutions in the extended sense? 8. How are the solution of a inhomogeneous linear ordinary differential equations constructed? The main sources for this chapter are [162], Chap. 4, for the numerics- friendly introduction of ordinary differential equations on Rd as integral curves of vector fields, as well as [82], Chap. 1, for the brief discussion of ordinary differential equations on arbitrary Banach spaces. This chapter is structured as follows: First, Sec. 6.2 equips us with the es- sential information about vector fields and their representation as well as the notion of local Lipschitz continuity and Gronwall’s lemma. Next inte- gral curves in vector fields are discussed, i.e., ordinary differential equations x_ = F (t; x). We start in Sec. 6.3 by analyzing ordinary differential equations with continuous right hand side F and studying "-approximate solutions as well as the Peano-Cauchy existence theorem and its implications. Then, in Sec. 6.4, we continue our discussion with Lipschitz-continuous functions F and the existence and uniqueness theorem of Picard-Lindelöf. It is here that we give the MATLAB commands for symbolically solving ordinary differen- tial equations and analyse maximal integral curves. In particular, we give the three types of maximal integral curves that can occur in autonomous systems and show the transformation of a d-th order equation into a first order system. Sec. 6.5 deals with existence of solutions in the extended sense where the right hand side function may be continuous except for a set of Lebesgue- measure zero. Caratheodory’s existence theorem and its implications are studied together with maximum and minimum solutions. Section 6.6 analy- ses the broad class of linear ordinary differential equations by discussing the unique existence of their solutions and explicitly constructing them. Appli- cations are given in Sec. 6.7, where we discuss in particular the deterministic pendulum and the Volterra-Lotka system. Sec. 6.8 rounds up our discussion by providing a first glance into the existence, uniqueness and extension of solutions of ordinary differential equations on infinite-dimensional Banach spaces. Finally, Sec. 6.9 wraps up the contents of this chapter. 190 Chapter 6 Random Differential Equations in Scientific Computing v(x) x Ω Figure 6.1. Sketch of the geometric/ physical interpretation of a vector field on a domain Ω. 6.2 Preliminaries Following [162], pp. 131, vector fields occur naturally in many physics and engineering contexts. Here, we recall their basic definitions and properties together with some illustrative examples. We lay the foundations for the dis- cussion of integral curves in vector fields, i.e.
Load more

Recommended publications
  • Numerical Solution of Ordinary Differential Equations
    NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS Kendall Atkinson, Weimin Han, David Stewart University of Iowa Iowa City, Iowa A JOHN WILEY & SONS, INC., PUBLICATION Copyright c 2009 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created ore extended by sales representatives or written sales materials. The advice and strategies contained herin may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.
    [Show full text]
  • Nonlinear Multi-Mode Robust Control for Small Telescopes
    NONLINEAR MULTI-MODE ROBUST CONTORL FOR SMALL TELESCOPES By WILLIAM LOUNSBURY Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical Engineering and Computer Science CASE WESTERN RESERVE UNIVERSITY January, 2015 1 Case Western Reserve University School of Graduate Studies We hereby approve the thesis of William Lounsbury, candidate for the degree of Master of Science*. Committee Chair Mario Garcia-Sanz Committee Members Marc Buchner Francis Merat Date of Defense: November 12, 2014 * We also certify that written approval has been obtained for any proprietary material contained therein 2 Contents List of Figures ……………….. 3 List of Tables ……………….. 5 Nomenclature ………………..5 Acknowledgments ……………….. 6 Abstract ……………….. 7 1. Background and Literature Review ……………….. 8 2. Telescope Model and Performance Goals ……………….. 9 2.1 The Telescope Model ……………….. 9 2.2 Tracking Precision ……………….. 14 2.3 Motion Specifications ……………….. 15 2.4 Switching Stability and Bumpiness ……………….. 15 2.5 Saturation ……………….. 16 2.6 Backlash ……………….. 16 2.7 Static Friction ……………….. 17 3. Linear Robust Control ……………….. 18 3.1 Unknown Coefficients……………….. 18 3.2 QFT Controller Design ……………….. 18 4. Regional Control ……………….. 23 4.1 Bumpless Switching ……………….. 23 4.2 Two Mode Simulations ……………….. 25 5. Limit Cycle Control ……………….. 38 5.1 Limit Cycles ……………….. 38 5.2 Three Mode Control ……………….. 38 5.3 Three Mode Simulations ……………….. 51 6. Future Work ……………….. 55 6.1 Collaboration with the Pontifical Catholic University of Chile ……………….. 55 6.2 Senior Project ……………….. 56 Bibliography ……………….. 58 3 List of Figures Figure 1: TRAPPIST Photo ……………….. 9 Figure 2: Control Design – Azimuth Aggressive ……………….. 20 Figure 3: Control Design – Azimuth Moderate ……………….. 21 Figure 4: Control Design – Azimuth Aggressive ……………….
    [Show full text]
  • Numerical Integration
    Chapter 12 Numerical Integration Numerical differentiation methods compute approximations to the derivative of a function from known values of the function. Numerical integration uses the same information to compute numerical approximations to the integral of the function. An important use of both types of methods is estimation of derivatives and integrals for functions that are only known at isolated points, as is the case with for example measurement data. An important difference between differen- tiation and integration is that for most functions it is not possible to determine the integral via symbolic methods, but we can still compute numerical approx- imations to virtually any definite integral. Numerical integration methods are therefore more useful than numerical differentiation methods, and are essential in many practical situations. We use the same general strategy for deriving numerical integration meth- ods as we did for numerical differentiation methods: We find the polynomial that interpolates the function at some suitable points, and use the integral of the polynomial as an approximation to the function. This means that the truncation error can be analysed in basically the same way as for numerical differentiation. However, when it comes to round-off error, integration behaves differently from differentiation: Numerical integration is very insensitive to round-off errors, so we will ignore round-off in our analysis. The mathematical definition of the integral is basically via a numerical in- tegration method, and we therefore start by reviewing this definition. We then derive the simplest numerical integration method, and see how its error can be analysed. We then derive two other methods that are more accurate, but for these we just indicate how the error analysis can be done.
    [Show full text]
  • Lyapunov Stability in Control System Pdf
    Lyapunov stability in control system pdf Continue This article is about the asymptomatic stability of nonlineous systems. For the stability of linear systems, see exponential stability. Part of the Series on Astrodynamics Orbital Mechanics Orbital Settings Apsis Argument Periapsy Azimut Eccentricity Tilt Middle Anomaly Orbital Nodes Semi-Large Axis True types of anomalies of two-door orbits, by the eccentricity of the Circular Orbit of the Elliptical Orbit Orbit Transmission of Orbit (Hohmann transmission of orbitBi-elliptical orbit of transmission of orbit) Parabolic orbit Hyperbolic orbit Radial Orbit Decay Orbital Equation Dynamic friction Escape speed Ke Kepler Equation Kepler Acts Planetary Motion Orbital Period Orbital Speed Surface Gravity Specific Orbital Energy Vis-viva Equation Celestial Gravitational Mechanics Sphere Influence of N-Body OrbitLagrangian Dots (Halo Orbit) Lissajous Orbits The Lapun Orbit Engineering and Efficiency Of the Pre-Flight Engineering Mass Ratio Payload ratio propellant the mass fraction of the Tsiolkovsky Missile Equation Gravity Measure help the Oberth effect of Vte Different types of stability can be discussed to address differential equations or differences in equations describing dynamic systems. The most important type is that of stability decisions near the equilibrium point. This may be what Alexander Lyapunov's theory says. Simply put, if solutions that start near the equilibrium point x e display style x_e remain close to x e display style x_ forever, then x e display style x_ e is the Lapun stable. More strongly, if x e displaystyle x_e) is a Lyapunov stable and all the solutions that start near x e display style x_ converge with x e display style x_, then x e displaystyle x_ is asymptotically stable.
    [Show full text]
  • Stability Analysis of Nonlinear Systems Using Lyapunov Theory – I
    Lecture – 33 Stability Analysis of Nonlinear Systems Using Lyapunov Theory – I Dr. Radhakant Padhi Asst. Professor Dept. of Aerospace Engineering Indian Institute of Science - Bangalore Outline z Motivation z Definitions z Lyapunov Stability Theorems z Analysis of LTI System Stability z Instability Theorem z Examples ADVANCED CONTROL SYSTEM DESIGN 2 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore References z H. J. Marquez: Nonlinear Control Systems Analysis and Design, Wiley, 2003. z J-J. E. Slotine and W. Li: Applied Nonlinear Control, Prentice Hall, 1991. z H. K. Khalil: Nonlinear Systems, Prentice Hall, 1996. ADVANCED CONTROL SYSTEM DESIGN 3 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Techniques of Nonlinear Control Systems Analysis and Design z Phase plane analysis z Differential geometry (Feedback linearization) z Lyapunov theory z Intelligent techniques: Neural networks, Fuzzy logic, Genetic algorithm etc. z Describing functions z Optimization theory (variational optimization, dynamic programming etc.) ADVANCED CONTROL SYSTEM DESIGN 4 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Motivation z Eigenvalue analysis concept does not hold good for nonlinear systems. z Nonlinear systems can have multiple equilibrium points and limit cycles. z Stability behaviour of nonlinear systems need not be always global (unlike linear systems). z Need of a systematic approach that can be exploited for control design as well. ADVANCED CONTROL SYSTEM DESIGN 5 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions System Dynamics XfX =→() fD:Rn (a locally Lipschitz map) D : an open and connected subset of Rn Equilibrium Point ( X e ) XfXee= ( ) = 0 ADVANCED CONTROL SYSTEM DESIGN 6 Dr. Radhakant Padhi, AE Dept., IISc-Bangalore Definitions Open Set A set A ⊂ n is open if for every pA∈ , ∃⊂ Bpr () A Connected Set z A connected set is a set which cannot be represented as the union of two or more disjoint nonempty open subsets.
    [Show full text]
  • On the Numerical Solution of Equations Involving Differential Operators with Constant Coefficients 1
    ON THE NUMERICAL SOLUTION OF EQUATIONS 219 The author acknowledges with thanks the aid of Dolores Ufford, who assisted in the calculations. Yudell L. Luke Midwest Research Institute Kansas City 2, Missouri 1 W. E. Milne, "The remainder in linear methods of approximation," NBS, Jn. of Research, v. 43, 1949, p. 501-511. 2W. E. Milne, Numerical Calculus, p. 108-116. 3 M. Bates, On the Development of Some New Formulas for Numerical Integration. Stanford University, June, 1929. 4 M. E. Youngberg, Formulas for Mechanical Quadrature of Irrational Functions. Oregon State College, June, 1937. (The author is indebted to the referee for references 3 and 4.) 6 E. L. Kaplan, "Numerical integration near a singularity," Jn. Math. Phys., v. 26, April, 1952, p. 1-28. On the Numerical Solution of Equations Involving Differential Operators with Constant Coefficients 1. The General Linear Differential Operator. Consider the differential equation of order n (1) Ly + Fiy, x) = 0, where the operator L is defined by j» dky **£**»%- and the functions Pk(x) and Fiy, x) are such that a solution y and its first m derivatives exist in 0 < x < X. In the special case when (1) is linear the solution can be completely determined by the well known method of varia- tion of parameters when n independent solutions of the associated homo- geneous equations are known. Thus for the case when Fiy, x) is independent of y, the solution of the non-homogeneous equation can be obtained by mere quadratures, rather than by laborious stepwise integrations. It does not seem to have been observed, however, that even when Fiy, x) involves the dependent variable y, the numerical integrations can be so arranged that the contributions to the integral from the upper limit at each step of the integration, at the time when y is still unknown at the upper limit, drop out.
    [Show full text]
  • Differential Equations We Will Use “Dsolve” to Get an Analytical Solution to the DE Y’(T) = 2Ty(T)
    Differential Equations - Theory and Applications - Version: Fall 2017 Andr´asDomokos , PhD California State University, Sacramento Contents Chapter 0. Introduction 3 Chapter 1. Calculus review. Differentiation and integration rules. 4 1.1. Derivatives 4 1.2. Antiderivatives and Indefinite Integrals 7 1.3. Definite Integrals 11 Chapter 2. Introduction to Differential Equations 15 2.1. Definitions 15 2.2. Initial value problems 20 2.3. Classifications of DEs 23 2.4. Examples of DEs modelling real-life phenomena 25 Chapter 3. First order differential equations solvable by analytical methods 27 3.1. Differential equations with separable variables 27 3.2. First order linear differential equations 31 3.3. Bernoulli's differential equations 36 3.4. Non-linear homogeneous differential equations 38 3.5. Differential equations of the form y0(t) = f(at + by(t) + c). 40 3.6. Second order differential equations reducible to first order differential equations 42 Chapter 4. General theory of differential equations of first order 45 4.1. Slope fields (or direction fields) 45 4.1.1. Autonomous first order differential equations. 49 4.2. Existence and uniqueness of solutions for initial value problems 53 4.3. The method of successive approximations 59 4.4. Numerical methods for Differential equations 62 4.4.1. The Euler's method 62 4.4.2. The improved Euler (or Heun) method 67 4.4.3. The fourth order Runge-Kutta method 68 4.4.4. NDSolve command in Mathematica 71 Chapter 5. Higher order linear differential equations 75 5.1. General theory 75 5.2. Linear and homogeneous DEs with constant coefficients 78 5.3.
    [Show full text]
  • The Original Euler's Calculus-Of-Variations Method: Key
    Submitted to EJP 1 Jozef Hanc, [email protected] The original Euler’s calculus-of-variations method: Key to Lagrangian mechanics for beginners Jozef Hanca) Technical University, Vysokoskolska 4, 042 00 Kosice, Slovakia Leonhard Euler's original version of the calculus of variations (1744) used elementary mathematics and was intuitive, geometric, and easily visualized. In 1755 Euler (1707-1783) abandoned his version and adopted instead the more rigorous and formal algebraic method of Lagrange. Lagrange’s elegant technique of variations not only bypassed the need for Euler’s intuitive use of a limit-taking process leading to the Euler-Lagrange equation but also eliminated Euler’s geometrical insight. More recently Euler's method has been resurrected, shown to be rigorous, and applied as one of the direct variational methods important in analysis and in computer solutions of physical processes. In our classrooms, however, the study of advanced mechanics is still dominated by Lagrange's analytic method, which students often apply uncritically using "variational recipes" because they have difficulty understanding it intuitively. The present paper describes an adaptation of Euler's method that restores intuition and geometric visualization. This adaptation can be used as an introductory variational treatment in almost all of undergraduate physics and is especially powerful in modern physics. Finally, we present Euler's method as a natural introduction to computer-executed numerical analysis of boundary value problems and the finite element method. I. INTRODUCTION In his pioneering 1744 work The method of finding plane curves that show some property of maximum and minimum,1 Leonhard Euler introduced a general mathematical procedure or method for the systematic investigation of variational problems.
    [Show full text]
  • Lyapunov Stability
    Appendix A Lyapunov Stability Lyapunov stability theory [1] plays a central role in systems theory and engineering. An equilibrium point is stable if all solutions starting at nearby points stay nearby; otherwise, it is unstable. It is asymptotically stable if all solutions starting at nearby points not only stay nearby, but also tend to the equilibrium point as time approaches infinity. These notions are made precise as follows. Definition A.1 Consider the autonomous system [1] x˙ = f(x), (A.1) where f : D → Rn is a locally Lipschitz map from a domain D into Rn. The equilibrium point x = 0is • Stable if, for each >0, there is δ = δ() > 0 such that x(0) <δ⇒ x(t) <, ∀t ≥ 0; • Unstable if it is not stable; • Asymptotically stable if it is stable and δ can be chosen such that x(0) <δ⇒ lim x(t) = 0. t→∞ Theorem A.1 Let x = 0 be an equilibrium point for system (A.1) and D ⊂ Rn be a domain containing x = 0. Let V : D → R be a continuously differentiable function such that V(0) = 0 and V(x)>0 in D −{0}, (A.2a) V(x)˙ ≤ 0 in D. (A.2b) Then, x = 0 is stable. Moreover, if V(x)<˙ 0 in D −{0}, (A.3) then x = 0 is asymptotically stable. H.Chen,B.Gao,Nonlinear Estimation and Control of Automotive Drivetrains, 197 DOI 10.1007/978-3-642-41572-2, © Science Press Beijing and Springer-Verlag Berlin Heidelberg 2014 198 A Lyapunov Stability A function V(x)satisfying (A.2a) is said to be positive definite.
    [Show full text]
  • Improving Numerical Integration and Event Generation with Normalizing Flows — HET Brown Bag Seminar, University of Michigan —
    Improving Numerical Integration and Event Generation with Normalizing Flows | HET Brown Bag Seminar, University of Michigan | Claudius Krause Fermi National Accelerator Laboratory September 25, 2019 In collaboration with: Christina Gao, Stefan H¨oche,Joshua Isaacson arXiv: 191x.abcde Claudius Krause (Fermilab) Machine Learning Phase Space September 25, 2019 1 / 27 Monte Carlo Simulations are increasingly important. https://twiki.cern.ch/twiki/bin/view/AtlasPublic/ComputingandSoftwarePublicResults MC event generation is needed for signal and background predictions. ) The required CPU time will increase in the next years. ) Claudius Krause (Fermilab) Machine Learning Phase Space September 25, 2019 2 / 27 Monte Carlo Simulations are increasingly important. 106 3 10− parton level W+0j 105 particle level W+1j 10 4 W+2j particle level − W+3j 104 WTA (> 6j) W+4j 5 10− W+5j 3 W+6j 10 Sherpa MC @ NERSC Mevt W+7j / 6 Sherpa / Pythia + DIY @ NERSC 10− W+8j 2 10 Frequency W+9j CPUh 7 10− 101 8 10− + 100 W +jets, LHC@14TeV pT,j > 20GeV, ηj < 6 9 | | 10− 1 10− 0 50000 100000 150000 200000 250000 300000 0 1 2 3 4 5 6 7 8 9 Ntrials Njet Stefan H¨oche,Stefan Prestel, Holger Schulz [1905.05120;PRD] The bottlenecks for evaluating large final state multiplicities are a slow evaluation of the matrix element a low unweighting efficiency Claudius Krause (Fermilab) Machine Learning Phase Space September 25, 2019 3 / 27 Monte Carlo Simulations are increasingly important. 106 3 10− parton level W+0j 105 particle level W+1j 10 4 W+2j particle level − W+3j 104 WTA (>
    [Show full text]
  • Boundary Control of Pdes
    Boundary Control of PDEs: ACourseonBacksteppingDesigns class slides Miroslav Krstic Introduction Fluid flows in aerodynamics and propulsion applications; • plasmas in lasers, fusion reactors, and hypersonic vehicles; liquid metals in cooling systems for tokamaks and computers, as well as in welding and metal casting processes; acoustic waves, water waves in irrigation systems... Flexible structures in civil engineering, aircraft wings and helicopter rotors,astronom- • ical telescopes, and in nanotechnology devices like the atomic force microscope... Electromagnetic waves and quantum mechanical systems... • Waves and “ripple” instabilities in thin film manufacturing and in flame dynamics... • Chemical processes in process industries and in internal combustion engines... • Unfortunately, even “toy” PDE control problems like heat andwaveequations(neitherof which is unstable) require some background in functional analysis. Courses in control of PDEs rare in engineering programs. This course: methods which are easy to understand, minimal background beyond calculus. Boundary Control Two PDE control settings: “in domain” control (actuation penetrates inside the domain of the PDE system or is • evenly distributed everywhere in the domain, likewise with sensing); “boundary” control (actuation and sensing are only through the boundary conditions). • Boundary control physically more realistic because actuation and sensing are non-intrusive (think, fluid flow where actuation is from the walls).∗ ∗“Body force” actuation of electromagnetic type is also possible but it has low control authority and its spatial distribution typically has a pattern that favors the near-wall region. Boundary control harder problem, because the “input operator” (the analog of the B matrix in the LTI finite dimensional model x˙ = Ax+Bu)andtheoutputoperator(theanalogofthe C matrix in y = Cx)areunboundedoperators.
    [Show full text]
  • System Stability
    7 System Stability During the description of phase portrait construction in the preceding chapter, we made note of a trajectory’s tendency to either approach the origin or diverge from it. As one might expect, this behavior is related to the stability properties of the system. However, phase portraits do not account for the effects of applied inputs; only for initial conditions. Furthermore, we have not clarified the definition of stability of systems such as the one in Equation (6.21) (Figure 6.12), which neither approaches nor departs from the origin. All of these distinctions in system behavior must be categorized with a study of stability theory. In this chapter, we will present definitions for different kinds of stability that are useful in different situations and for different purposes. We will then develop stability testing methods so that one can determine the characteristics of systems without having to actually generate explicit solutions for them. 7.1 Lyapunov Stability The first “type” of stability we consider is the one most directly illustrated by the phase portraits we have been discussing. It can be thought of as a stability classification that depends solely on a system’s initial conditions and not on its input. It is named for Russian scientist A. M. Lyapunov, who contributed much of the pioneering work in stability theory at the end of the nineteenth century. So- called “Lyapunov stability” actually describes the properties of a particular point in the state space, known as the equilibrium point, rather than referring to the properties of the system as a whole.
    [Show full text]