From Particle Accelerators to Celestial Dynamics: An Introduction to Differential Algebra

AstroNet II Summer School 2014 Friday, 27 June 2014 Zielona Góra (Poland) From Particle Accelerators to Celestial Dynamics: An Introduction to Differential Algebra 1, Pierluigi di Lizia 1, Roberto Armellin 2 1 Dipartimento Scienze e Tecnologie Aerospaziali, Politecnico di Milano 2 School of Engineering Sciences, University of Southampton 27 June 2014, Zielona Góra (Poland) Introduction to DA 0/56

Outline 1 What is Differential Algebra? 1 Introduction 2 History 2 Six Views of Differential Algebra 1 Multivariate Polynomials 2 Automatic Differentiation 3 Functional Analysis 4 Non-Archimedean Analysis 5 Set Theory 6 Symbolic Computation 3 Differential Algebra Algorithms 4 Applications in Aerospace Engineering 27 June 2014, Zielona Góra (Poland) Introduction to DA 1/56

What is Differential Algebra Multivariate Polynomials Non-Archimedean Analysis Automatic Differentiation Differential Algebra Set Theory Functional Analysis Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 2/56

Overview Differential Algebra A general numerical technique based on the efficient implementation of polynomials with applications in many different fields. Aspects of what we call Differential Algebra is known under various other names: Truncated Polynomial Series Algebra (TPSA) Automatic forward differentiation Jet Transport (for flow expansion) 27 June 2014, Zielona Góra (Poland) Introduction to DA 3/56

History of Differential Algebra Differential Algebra Techniques Introduced in Beam Physics (Berz, 1987) Computation of transfer maps in particle optics Extended to Rigorous Numerics (Makino & Berz, 1996) Rigorous numerical treatment including truncation and round-off errors for computer assisted proofs Wide range of applications to Celestial Mechanics (Lizia et al., 2007) Uncertainty propagation, Two-Point Boundary Value Problem, Optimal Control,... 27 June 2014, Zielona Góra (Poland) Introduction to DA 4/56

Beam & Accelerator Physics 27 June 2014, Zielona Góra (Poland) Introduction to DA 5/56

Beam & Accelerator Physics Simulation of repetitive particle optical systems: Complicated dynamical system: Goals: Thousands of magnetic and/or electrostatic elements Complicated field profiles Symplectic motion Small time scales (50, 000 revolutions per second) Design parameter optimization Long term stability properties Avoidance of chaotic motion 27 June 2014, Zielona Góra (Poland) Introduction to DA 6/56

Beam & Accelerator Physics Mathematical Tools Transfer Maps (Poincaré section) Symplectic Tracking (Iteration & Visualization) Periodic Points, Invariant Manifolds (Phase Space Structure Determination) Normal Form Transformation, Normal Form Defect Function Numerical Tool: Differential Algebra: powerful numerical method to compute and analyze high order polynomial transfer maps.(berz, 1999) 27 June 2014, Zielona Góra (Poland) Introduction to DA 7/56

Beam & Accelerator Physics I COEFFICIENT ORDER EXPONENTS 1 -.7468565055530253 1 1 0 0 0 2 -.8531012175898605 1 0 1 0 0 3 0.7955304445587349E-15 1 0 0 1 0 4 -.7955304445587349E-15 1 0 0 0 1 5 -.1525590265147758E-01 2 2 0 0 0 6 -.3240533655560984E-02 2 1 1 0 0 7 0.7698932551426080E-01 2 0 2 0 0 8 0.3731806390668192E-02 2 0 0 2 0 9 -.2654818769323237E-01 2 0 0 1 1 10 -.2557829414011944E-01 2 0 0 0 2 11 3.680711786871933 3 3 0 0 0 12-26.52260523674328 3 2 1 0 0 13 88.17307141934121 3 1 2 0 0 14-196.1802610972074 3 0 3 0 0 15 0.3755022311773308E-15 3 2 0 1 0 16 0.5539685858627665E-14 3 1 1 1 0 17 -.5012630863961773E-14 3 0 2 1 0 Partial Tevatron 18 transfer 0.2469714493559591E-15 map ( 2000 coefficients 3 2 / 0 coordinate 0 1 in 4 coordinates) 19 -.1022089130257320E-14 3 1 1 0 1 20 -.1647331523279767E-13 3 0 2 0 1 27 June 2014, Zielona Góra (Poland) Introduction to DA 8/56

Beam & Accelerator Physics x-a tracking picture of Tevatron transfer map 27 June 2014, Zielona Góra (Poland) Introduction to DA 9/56

Beam & Accelerator Physics Typical main task: Compute Poincare section of beam line pointwise: slow, hard to analyze resulting (image) data analytical: difficult and time consuming, previously done to orders 4 or 5 only DA: arbitrary order expansion of Poincaré map as polynomial (easy to analyze, fast to evaluate, fast to code) 27 June 2014, Zielona Góra (Poland) Introduction to DA 10/56

DA in Astrodynamics In 2006, Pierluigi di Lizia and Roberto Armellin from Politecnico di Milano visited MSU to study DA. They introduced the technique in the field and have applied it to various problems: Uncertainty propagation TPBVP in astrodynamics Orbit Determination MOID and collision probability computation Robust optimal control of space trajectories Global optimization of interplanetary transfers 27 June 2014, Zielona Góra (Poland) Introduction to DA 11/56

Differential Algebra A numerical technique to automatically compute high order Taylor expansions of functions f ( x 0 + δx) f ( x 0 ) + f ( x 0 ) δx + + 1 n! f (n) ( x 0 ) δx n Can be conceptualized in various ways from different view points: Multivariate Polynomials Functional Analysis Set Theory Symbolic Computation Automatic Differentiation Non-Archimedean Analysis 27 June 2014, Zielona Góra (Poland) Introduction to DA 12/56

as an Algebra of Truncated Multivariate Polynomials Multivariate Polynomials Automatic Differentiation Functional Analysis Non-Archimedean Analysis Differential Algebra Set Theory Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 13/56

as an Algebra of Truncated Multivariate Polynomials Motivation: ( ) sin(x) cos(y) + 1 f (x, y, z) = exp 1 + x 2 + y 2 + z 2 mathematical expression: instructions of operations to be performed x, y, z can be various things: real number (R), complex number (C), sometimes matrices (R n n ),... all operations must be defined in a useful way Goal Replace all algebraic operations between numbers by ones that act on (a suitably chosen subset of) polynomials instead. 27 June 2014, Zielona Góra (Poland) Introduction to DA 14/56

as an Algebra of Truncated Multivariate Polynomials 1 Ring of polynomials Natural Addition, Subtraction, Multiplication of polynomials Problems: order of polynomials not limited, grows under multiplication infinite dimensional not suited for computations 27 June 2014, Zielona Góra (Poland) Introduction to DA 15/56

as an Algebra of Truncated Multivariate Polynomials 1 Ring of polynomials Natural Addition, Subtraction, Multiplication of polynomials Problems: order of polynomials not limited, grows under multiplication infinite dimensional not suited for computations 2 Algebra of truncated polynomials Truncate all results to a fixed order n Finite dimensional space, hence computable Space n D v of polynomials of order up to n in v variables (n+v)! n!v! dimensions Not a ring or field: e.g. many nil-potent elements x, x x 2,... 27 June 2014, Zielona Góra (Poland) Introduction to DA 15/56

as an Algebra of Truncated Multivariate Polynomials Now we have +,,. Let s add division (i.e. multiplicative inverse): 1 That is: P nd v such that P 1 P = 1 Does not always exist: P(x) = x and Q(x) = (q 0 + q 1 x + q 2 x 2 + + q n x n ) P(x) Q(x) = x (q 0 + q 1 x + q 2 x 2 + + q n x n ) = q 0 x + q 1 x 2 + + q n 1 x n +q n x n+1 No constant part for any coefficients q i Exists iff P(x) = C + N(x) satisfies C 0 and N(0) = 0 in n D v n D v not a field Does not exist in ring of polynomials! 27 June 2014, Zielona Góra (Poland) Introduction to DA 16/56

as an Algebra of Truncated Multivariate Polynomials Multiplicative inverse of P(x) = C + N(x). 1 1 Calculating multiplicative inverse P(x) using naive power series expansion 1 1 + δx = 1 δx + δx 2 δx 3 +... with δx = P(x) 1. Bad: how many terms needed for convergence? 27 June 2014, Zielona Góra (Poland) Introduction to DA 17/56

as an Algebra of Truncated Multivariate Polynomials Multiplicative inverse of P(x) = C + N(x). 1 1 Calculating multiplicative inverse P(x) using naive power series expansion 1 1 + δx = 1 δx + δx 2 δx 3 +... with δx = P(x) 1. Bad: how many terms needed for convergence? 2 Instead, use smarter power series expansion with δx = N(x). 1 C + δx = 1 C δx C 2 + δx 2 C 3 δx 3 C 4 +... Non-constant part N(x) is nilpotent, i.e. N n+1 = 0 Only finitely many (n) terms of series needed 27 June 2014, Zielona Góra (Poland) Introduction to DA 17/56

as an Algebra of Truncated Multivariate Polynomials Now we have +,,, /. Let s add intrinsic functions (e.g., sin, cos, exp,...) 1 Taylor expansion around constant part Square root: 1 δx C + δx = C + 1 δx 2 2 C 8 3 +... C computed using similar arguments as for division 2 Mathematical identities to reduce problem exp(c + N(x)) = exp(c) exp(n(x)) exp(c) only for constant part (a number) exp(n(x)) only for non-linear part, hence converges after n terms 27 June 2014, Zielona Góra (Poland) Introduction to DA 18/56

as an Algebra of Truncated Multivariate Polynomials We now have +,,, /, intrinsic functions (e.g., sin, cos, exp,...). Almost there: can evaluate any function f given as a finite combination of these arithmetic operations and intrinsic functions in DA: ( ) sin(x) cos(y) + 1 f (x, y, z) = exp 1 + x 2 + y 2 + z 2 Result: (Taylor) polynomial P(x, y, z) of f (x, y, z) to arbitrary order. 27 June 2014, Zielona Góra (Poland) Introduction to DA 19/56

as an Algebra of Truncated Multivariate Polynomials Last thing: Add derivation and inverse derivation 1 operators to obtain differential algebra. x : Simple polynomial derivation w.r.t. independent variable x a 0 + a 1 x + a 2 x 2 + a 1 + a 2 x + x 1 : Simple polynomial integration w.r.t. independent variable x a 0 + a 1 x + a 2 x 2 + 0+a 0 x + a 1 x 2 + a 2 x 3 Now we can evaluate even complicated operators directly in DA: ˆ ˆ ( ) d g(x, y, z) = dy exp sin(x) cos(y) + 1 dx dz 1 + x 2 + y 2 + z 2 Result: (Taylor) polynomial Q(x, y, z) of g(x, y, z) to arbitrary order. 27 June 2014, Zielona Góra (Poland) Introduction to DA 20/56

Implementations Arithmetic can be implemented on computer. Critical points: operations must be done efficiently (for large scale computations) fixed order n and variables v are easy, arbitrary numbers of orders and variables more difficult convenient coding for user, requires overloading of operators 27 June 2014, Zielona Góra (Poland) Introduction to DA 21/56

Implementations Selected computer implementations of Differential Algebra: COSY INFINITY 9.0 (Berz, Makino 1998) rigorous (TM) and non-rigorous (DA) datatypes computing environment using COSY Script language large library of tools and utilities for beam physics and verified computation long history of active development (20+ years) DACE (Dinamica SRL, 2014) non-rigorous DA datatype comfortable native C++ and experimental Matlab interface large library of celestial mechanics routines currently developed under ESA contract (ITT 7570: Nonlinear Propagation of Uncertainties in Space Dynamics based on Taylor Differential Algebra) JACK (Thales), Jet Transport (Barcelona),... (?) 27 June 2014, Zielona Góra (Poland) Introduction to DA 22/56

Mathematical expressions DACE C++ Interface Key Concepts! All operators are overloaded for DA and double operations Independent variables (generators of the algebra) created by simple DA constructor DA(1), DA(2), Overloaded intrinsic routines available for functional programming style sin(x) instead of x.sin() Very clear and easy to understand code DA x = -1 + DA(1) + DA(2); DA y = sin(x) + 1.9;! DA z = x.sin() + 1.9;!// Creation of DA variables!!// Functional notation!!// Object oriented notation! 3!

Mathematical expressions DACE C++ Interface Key Concepts! Notation is virtually identical to built in numerical C++ types (e.g. double, int) and C/C++ math library New code can be written easily Existing code can be adapted to DA with minimal changes Does not mean old code will just run! Non-trivial code always requires some changes due to conceptual differences, not the interface! double f(double x)! {!!double res = (sin(x)+2*x)/(1+cos(x));!!for(int i = 0; i<3; i++) res = sqrt(res);!!return res;! }! Existing double code 4!

Mathematical expressions DACE C++ Interface Key Concepts! Notation is virtually identical to built in numerical C++ types (e.g. double, int) and C/C++ math library New code can be written easily Existing code can be adapted to DA with minimal changes Does not mean old code will just run! Non-trivial code always requires some changes due to conceptual differences, not the interface! DA f(da x)! {!!DA res = (sin(x)+2*x)/(1+cos(x));!!for(int i = 0; i<3; i++) res = sqrt(res);!!return res;! }! New DA code 5!

Mathematical expressions DACE C++ Interface Key Concepts! Using C++ templates allows maximum code flexibility DA class is template safe C++ automatically writes code for any data type used in function call Most versatile, and recommended, way to write functions template<class T> T f(t x)! {!!T res = (sin(x)+2*x)/(1+cos(x));!!for(int i = 0; i<3; i++) res = sqrt(res);!!return res;! }! cout << f(5.0);!// call f with double! cout << f(da(1));!// call f with DA! Hybrid template code 6!

as Automatic Differentiation Multivariate Polynomials Automatic Differentiation Functional Analysis Non-Archimedean Analysis Differential Algebra Set Theory Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 27/56

as Automatic Differentiation Accurate computation of arbitrary derivatives of a function f at given point x 0 Much faster and more accurate than e.g. divided differences Forward differentiation method 27 June 2014, Zielona Góra (Poland) Introduction to DA 28/56

as Automatic Differentiation 27 June 2014, Zielona Góra (Poland) Introduction to DA 28/56 Accurate computation of arbitrary derivatives of a function f at given point x 0 Much faster and more accurate than e.g. divided differences Forward differentiation method Let x 0 R m, f : R m R and compute P( x) = f ( x 0 + x) to some order in DA arithmetic. Then P( x) is the Taylor Expansion of f around x 0, hence contains exact derivatives dk f x0 in coefficients (up to floating point error). d x k

as Automatic Differentiation Works because binary DA operations commute with Taylor expansion T a, b F P f, g T F, G T a b f g T F G i.e. T f (x) T g (x) = T f g (x) where T f (x) is Taylor expansion of f up to DA computation order n. 27 June 2014, Zielona Góra (Poland) Introduction to DA 29/56

as Automatic Differentiation Same duality between DA and mathematical operation holds for: Each intrinsic DA function G(x) (e.g. sin, cos) G(T f (x)) = T G(f (x)). Evaluation of polynomials in DA arithmetic (function concatenation): T f g (x) = f (T g (x)) = T f (T g (x)) 27 June 2014, Zielona Góra (Poland) Introduction to DA 30/56

as an Implementation of Functional Analysis Important consequence of function concatenation: Sensitivity Let f (x), g(x), h(x) be (sufficiently smooth) functions. If the Taylor expansions T g and T h agree up to order k then the Taylor expansions T f (g) and T f (h) agree up to order k a well. Proof: Insert into Taylor expansion or by repeated application of chain rule. 27 June 2014, Zielona Góra (Poland) Introduction to DA 31/56

as an Implementation of Functional Analysis Multivariate Polynomials Automatic Differentiation Functional Analysis Non-Archimedean Analysis Differential Algebra Set Theory Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 32/56

as an Implementation of Functional Analysis Building on the commutativity of DA operations, use Taylor Polynomials as computer representations of elements of C r function space. Equivalence relation on C r : f n g f k x k = f k 0 x k f and g have same Taylor expansion to order n Factorize C r into equivalence classes by n 0 k = 0,..., n Pick Taylor Polynomial of order n as representative of each class Example: n = 3 1 + x + x2 2 + x3 6 3 exp(x) 3 exp(x) + x 3 sin(x) Use 1 + x + x2 2 + x3 6 as representative of all 3 functions 27 June 2014, Zielona Góra (Poland) Introduction to DA 33/56

as an Implementation of Functional Analysis Now use DA representation of Taylor Polynomial as computer representation of C r functions Like floating point numbers are computer representations of R. T a, b R a, b F P f, g T F, G a b T a b f g T F G Field of Math: Objects: Representation: Analysis R Floating Point Numbers Functional Analysis f C r : R n R n Differential Algebra 27 June 2014, Zielona Góra (Poland) Introduction to DA 34/56

as an Implementation of Functional Analysis Intuitive manipulation of functions in computer environment Direct computer implementation of e.g. Functional Analysis operators Example: Picard Operator for Flow Expansion Fixed point operator to solve ODE x = f (x), x(0) = x 0. P : C r C r ˆ t P [x(τ)] = x 0 + f (x(τ)) dτ 0 Can be implemented straight forward using DA Integration operator is just algebraic operation Operator P converges to fixed point in n D v in at most n steps (!) 27 June 2014, Zielona Góra (Poland) Introduction to DA 35/56

as an Implementation of Functional Analysis DA order function The DA order function O : n D v N { } is the minimum order of all non-zero terms in a monomial, or if the polynomial is zero. Examples: O(2 x 3 y + 4x 2 3) = 0 O(2 x 3 y + 4x 2 ) = 2 O(2 x 3 y) = 4 O(0) = 27 June 2014, Zielona Góra (Poland) Introduction to DA 36/56

as an Implementation of Functional Analysis DA fixed point theorem Let A : n D v n D v be an operator on n D v such that x, y n D v Then O(Ax Ay) > O(x y). 1 A has a unique fixed point x 0 n D v, i.e. Ax 0 = x 0 2 For any x n D v the sequence x, Ax, A 2 x, A 3 x,... converges to x 0 order by order, i.e. O(A n x x 0 ) n 27 June 2014, Zielona Góra (Poland) Introduction to DA 37/56

as Non-Archimedean Analysis Multivariate Polynomials Automatic Differentiation Functional Analysis Non-Archimedean Analysis Differential Algebra Set Theory Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 38/56

as Non-Archimedean Analysis To understand convergence properties on n D v, a view from Non-Archimedean Analysis is useful. Order on n D v can be introduced by comparing coefficients one after another. E.g. 1 + 2x 3x 2 < 1 + 2x + 2x 2 Thus n D v has infinitesimal elements: 1 + x > K (0 + x) = Kx K R i.e. any multiple of x is still smaller than 1 + x. 27 June 2014, Zielona Góra (Poland) Introduction to DA 39/56

as Non-Archimedean Analysis Axiom of Archimedes ε > 0 n N s.t. 1/n < ε every positive ε can be multiplied by some n such that ε n is larger than 1 Non-Archimedean Analysis: Drop axiom to allow infinitesimals The Levi-Civita Field Rigorous mathematical treatment of algebra with infinitesimals Provides rigorous theoretical underpinning for DA Useful to develop fast implementations of the computation of basic DA algorithms contracting operators: number of correct orders increases by 1 super-convergent operators: number of correct orders doubles 27 June 2014, Zielona Góra (Poland) Introduction to DA 40/56

as a Representation of Sets and Manifolds Multivariate Polynomials Automatic Differentiation Functional Analysis Non-Archimedean Analysis Differential Algebra Set Theory Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 41/56

as a Representation of Sets and Manifolds DA objects can be considered as representation of very general sets: Consider DA as a (structured) set by looking at image of domain [ 1, 1] n under a polynomial map Can approximate very complicated sets very well Much better approximation of set valued functions than Interval Arithmetic Domain Initial Set Final Set 1,1 δ x,δ y S f -1,-1 -δ x,-δ y P=f(S) 27 June 2014, Zielona Góra (Poland) Introduction to DA 42/56

as a Representation of Sets and Manifolds DA objects can be considered as representation of very general sets: Consider DA as a (structured) set by looking at image of domain [ 1, 1] n under a polynomial map Can approximate very complicated sets very well Much better approximation of set valued functions than Interval Arithmetic Domain Initial Set Final Set 1,1 δ x,δ y S f -1,-1 -δ x,-δ y P=f(S) 27 June 2014, Zielona Góra (Poland) Introduction to DA 42/56

as a Representation of Sets and Manifolds Set theoretical view of DA allows computation only on selected (sub)sets fast propagation of sets of initial conditions bounding of resulting sets DA representation of sets has a structure Manifolds Instead of one single map, consider many maps each covering part of the initial set Natural representation of the mathematical concept of a manifold by representing the charts of the atlas as DA objects Calculations on manifold very easy 27 June 2014, Zielona Góra (Poland) Introduction to DA 43/56

Taylor Models Extension of DA techniques to automatically compute rigorous bounds of truncation errors DA: f ( x 0 + δx) f ( x 0 ) + f ( x 0 ) δx + + 1 n! f (n) ( x 0 ) δx n TM: f ( x 0 + δx) f ( x 0 ) + f ( x 0 ) δx + + 1 n! f (n) ( x 0 ) δx n +[ ε, ε] Combined with polynomial bounders provides highly accurate, rigorous bounds for range of f over given domains. Applications in verified numerics: Global Optimization Global Fixed Point Finder Verified Integration Manifold Enclosures = Computer Assisted Proofs 27 June 2014, Zielona Góra (Poland) Introduction to DA 44/56

ODE flow expansion Several methods to compute flow expansion ϕ( x 0, t): Picard Operator Iteration (as shown before) DA evaluation of classical numerical schemes Runge Kutta (e.g. RK45, DP78) Adams-Bashforth Never: variational equations! Result of each method: P(δ x, δt) = ϕ( x 0 + δ x, t + δt) First order of P corresponds to state transition matrix Extremely useful to propagate entire sets of initial conditions 27 June 2014, Zielona Góra (Poland) Introduction to DA 45/56

ODE flow expansion Uncertainty propagation of reference value r = (r 1, r 2,... ) and uncertainty vector σ = (σ 1, σ 2,... ): 1 Set up the initial condition as the set P initial (δ x) = r + 3 σ T δ x r 1 σ 1 δx 1 = r 2 + 3 σ 2 δx 2.. such that on [ 1, 1] n it covers the entire 3 σ set. 2 Propagate polynomial P initial (δ x) using an integration scheme implemented in DA. 3 Result: P final (δ x) describing the 3 σ set after propagation. 27 June 2014, Zielona Góra (Poland) Introduction to DA 46/56

ODE flow expansion Example: simple Euler scheme with fixed step size h x i+1 = x i + f (x i ) h 1 Perform steps in DA arithmetic starting with x 0 = P initial (δ x) x 1 = x 0 + f (x 0 ) h 2 Now x 1 = x 1 (δ x) is the polynomial expansion of propagated set at time h. 3 Repeat until final time In reality, use higher order scheme (e.g. DP78). 27 June 2014, Zielona Góra (Poland) Introduction to DA 47/56

DA-based ODE flow expansion order: 6 ODE flow expansion Uncertainty box on initial position: 0.00 Propagation of set of initial conditions in Kepler dynamics (set view): y [AU] 2 1.5 1 0.5 0 0.5 (a) t i = 929.8 day t 0 = 0 Com Intel Any s can b order 1 1.5 2 3 2 1 0 1 x [AU] 27 June 2014, Zielona Góra (Poland) Introduction to DA 48/56 Fa

ODE flow expansion Advanced uncertainty propagation techniques: Domain Splitting: nonlinear dynamics cause sets to grow. Automatically decompose polynomial into smaller polynomials covering subsets to ensure convergence. Taylor integrator: arbitrary order integrator using the Taylor flow expansion. Instead of numerical scheme, use Taylor flow expansion and compute and evaluate at each time step. Verified integration: Taylor integrator extended with verified Taylor Models. Compute verified enclosure of flow including truncation and round-off errors in each step. Yields verified enclosure of set (computer assisted proof). 27 June 2014, Zielona Góra (Poland) Introduction to DA 49/56

as Symbolic Computation Multivariate Polynomials Automatic Differentiation Functional Analysis Non-Archimedean Analysis Differential Algebra Set Theory Symbolic Computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 50/56

as Symbolic Computation DA can be viewed as a hybrid of symbolic and numeric computation: Polynomials are treated symbolically up to order n Results are again polynomials, not a more complicated symbolic expression Combines speed of numerical with many advantages of symbolic computation 27 June 2014, Zielona Góra (Poland) Introduction to DA 51/56

Algorithms Some basic DA algorithms: Function inversion to arbitrary order Constraint satisfaction Taylor expansion of ODE flow in time in initial conditions (no need for variational equations) Poincare map computation (combining ODE flow & constraint satisfaction) Two Point Boundary Value Problem (TPBVP) Normal Form transformation Manifold expansion 27 June 2014, Zielona Góra (Poland) Introduction to DA 52/56

Aerospace Applications Over past 8 years, DA techniques have been applied to many problems: Uncertainty propagation Fast Monte Carlo methods Propagation of statistics MOID & Collision probability Robustness analysis and optimization Orbit determination Dynamics structure determination (LCS) Map propagation method Manifold representation local manifold generation manifold globalization Lambert problem (extension methods) Robust optimal control Global optimization of interplanetary transfers 27 June 2014, Zielona Góra (Poland) Introduction to DA 53/56

References (Aerospace) Uncertainty propagation in astrodynamics: Pierluigi Di Lizia and Roberto Armellin s PhD theses (2007-2008) Armellin R., Di Lizia P., Bernelli Zazzera F., Berz M. Asteroid Close Encounters Characterization Using Differential Algebra: the Case of Apophis, Celestial Mechanics and Dynamical Astronomy, 2010 Valli M., Armellin R., Di Lizia P., Lavagna M., Nonlinear Mapping of Uncertainties in Celestial Mechanics, Journal of Guidance Control and Dynamics, 2013 TPBVP in astrodynamics: Pierluigi Di Lizia and Roberto Armellin s PhD theses (2007-2008) Di Lizia P., Armellin R., Lavagna M., Application of High Order Expansions of Two-Point Boundary Value Problems to Astrodynamics, Celestial Mechanics and Dynamical Astronomy, 2008 Orbit Determination: Armellin R., Di Lizia P., Lavagna M., High-Order Expansion of the Solution of Preliminary Orbit Determination Problem, Celestial Mechanics and Dynamical Astronomy, 2012 MOID and collision probability computation: Armellin R., Di Lizia P., Berz M., Makino K., Computing the Critical Points of the Distance Function Between Two Keplerian Orbits via Rigorous Global Optimization, Celestial Mechanics and Dynamical Astronomy, 2010 Armellin R., Morselli A., Di Lizia P., Lavagna M., Rigorous Computation of Orbital Conjunctions, Advances in Space Research, 2012 Morselli A., Armellin R., Di Lizia P., Bernelli Zazzera F., A High Order Method for Orbital Conjunctions Analysis: Sensitivity to Initial Uncertainties, Advances in Space Research, 2014 27 June 2014, Zielona Góra (Poland) Introduction to DA 54/56

References (Aerospace) Robust optimal control of space trajectories: Pierluigi Di Lizia and Roberto Armellin s PhD theses (2007-2008) Di Lizia P., Armellin R., Ercoli Finzi A., Berz M., High-Order Robust Guidance of Interplanetary Trajectories Based on Differential Algebra, Journal of Aerospace Engineering, Sciences and Applications, 2008 Di Lizia P., Armellin R., Bernelli Zazzera F., Berz M., High Order Optimal Control of Space Trajectories with Uncertain Boundary Conditions, Acta Astronautica, 2014 Di Lizia P., Armellin R., Morselli A., Bernelli Zazzera F., High Order Optimal Feedback Control of Space Trajectories with Bounded Control, Acta Astronautica, 2014 Global optimization of interplanetary transfers: Armellin R., Di Lizia P., Topputo F., Lavagna M., Bernelli Zazzera F., Berz M., Gravity Assist Space Pruning Based on Differential Algebra, Celestial Mechanics and Dynamical Astronomy, 2010 Armellin R., Di Lizia P., Makino K., Berz M., Rigorous Global Optimization of Impulsive Planet-to-Planet Transfers in the Patched-Conics Approximation, Engineering Optimization, 2012 Awarded grants on DA application to space-related problems: Global Trajectory Optimization: Can We Prune the Solution Space when considering Deep Space Manouvres? ESA Contract, 2007 Predicting Asteroid Trajectories using Validated Integrators and Determining Impact Leading Conditions. ESA Contract, 2008 Trajectory optimisation under uncertainties: a differential algebra approach. ESA Contract, 2011 Nonlinear Propagation of Uncertainties in Space Dynamics based on Taylor Differential Algebra, ESA contract 2014 27 June 2014, Zielona Góra (Poland) Introduction to DA 55/56

References (Differential Algebra) Berz, M. 1987. The method of power series tracking for the mathematical description of beam dynamics. Nuclear Instruments and Methods, A258, 431 437. Berz, M. 1999. Modern Map Methods in Particle Beam Physics. San Diego: Academic Press. Also available at http://bt.pa.msu.edu/pub. Lizia, P. Di, Zazzera, F. Bernelli, & Berz, M. 2007. High Order Integration and Sensitivity Analysis of Multibody Systems using Differential Algebra. AIDAA, 19. Makino, K., & Berz, M. 1996. Remainder Differential Algebras and their Applications. Pages 63 74 of: Berz, M., Bischof, C., Corliss, G., & Griewank, A. (eds), Computational Differentiation: Techniques, Applications, and Tools. Philadelphia: SIAM. 27 June 2014, Zielona Góra (Poland) Introduction to DA 56/56