smart-uq: uncertainty quantification toolbox for generalized - PowerPoint PPT Presentation

ICATT Conference Darmstadt, Germany, March 14-17 2016 Carlos Ortega Absil Annalisa Riccardi Massimiliano Vasile Chiara Tardioli March 16, 2016 Strathclyde University Department of Mechanical & Aerospace Engineering smart-uq: uncertainty quantification toolbox for generalized intrusive and non intrusive polynomial algebra

SMART Project Polynomial approximation 4 Propagation of Uncertainty in Space Dynamics 3 Non-intrusive methods Intrusive methods 2 Polynomial approximation SMART-UQ: Background and Motivation SMART Project 1 Propagation of Uncertainty in Space Dynamics 1 Discussion & Conclusions Outline Discussion & Conclusions

smart project

SMART Project Polynomial approximation Propagation of Uncertainty in Space Dynamics ∙ GitHub https://github.com/space-art , C++, Doxygen quantification Optimal Control 3 Discussion & Conclusions SMART ∙ 2015 : Strathclyde Mechanical and Aerospace Research Toolboxes ∙ 2016 : open source release of SMART-UQ under the MPL license ∙ SMART-UQ for Uncertainty ∙ SMART-O2C for Optimisation and ∙ SMART-ASTRO for Astrodynamics

SMART Project Polynomial approximation Propagation of Uncertainty in Space Dynamics Hypercube sampling (LHS), low discrepancy sequence (Sobol). basis (Euler, Runge-Kutta methods) Two-body problem 4 Discussion & Conclusions SMART-UQ ∙ Sampling : random sampling, Latin ∙ Polynomial : Tchebycheff and Taylor ∙ Integrators : fixed stepsize integrators ∙ Dynamics : Lotka-Volterra, Van der Pol,

SMART Project the abstract classes problems new numerical strategies and base_sampling to integrate base_polynomial , base_integrators , base_dynamics , system Polynomial approximation integration scheme, dynamical sampling techniques, available polynomial basis, Propagation of Uncertainty in Space Dynamics 5 Discussion & Conclusions Interface ∙ User : can instantiate one of the ∙ Developer : can extend one of

SMART Project Polynomial approximation Propagation of Uncertainty in Space Dynamics SMART-UQ: Background and Motivation algebraic operators ∙ Advantages : scalability ∙ Disadvantages : requires more effort to implement and construction of the response surface ∙ Advantages : easy implementation ∙ Disadvantages : curse of dimensionality 6 Discussion & Conclusions SMART-UQ: Background and Motivation ∙ Intrusive methods : they apply inside the model, modifying ∙ Non-intrusive methods : evaluation of the model in sample points

SMART Project Polynomial approximation Integrate the work already done in the team on non-intrusive Develop a generic computer environment for multivariate polynomial Newton Models (univariate) Inclusion Methods 7 SMART-UQ: Background and Motivation Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Generalised Intrusive Polynomial expansion (GIPE) ∙ 1982 (Epstein) Ultra Arithmetic ∙ 1986 (Berz) Taylor Differential Algebra ∙ 1997 (Berz) Taylor Models ∙ 2003 (Berz) Taylor Models and Other Validated Functional ∙ 2010 (Joldes) Formal comparison between Taylor, Tchebycheff, IDEA algebra Generalized Intrusive Polynomial Expansion (GIPE) . techniques and apply them to problems in astrodynamics .

polynomial approximation

SMART Project In d variables up to degree n is the polynomial basis of choice. Polynomial approximation (1) 9 Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Multivariate Polynomials Multivariate polynomial approximation P ( x ) = p i α i ( x ) ∈ P n , d ( α i ) , ∑ i , | i |≤ n where x ∈ Ω = [ − 1 , 1 ] d ⊂ R d , i ∈ [ 0 , n ] d ⊂ N d , | i | = ∑ d r = 1 i r and α i ( x ) ∙ Ω = [ a , b ] ⊂ R d and τ : Ω → Ω → α i ( x ) = α i ( τ ( x )) , ∙ Taylor T i ( x ) = ∏ d r . r = 1 x i r ∙ Tchebycheff C i ( x ) = ∏ d r = 1 C i r ( x r ) , where C 0 ( x r ) := 1, C i r ( x r ) := cos ( i r arccos ( x r )) . They form an orthogonal basis in P n , d ( α i )

SMART Project d required to be n -th times differentiable) differentiable) is required to be more than continuous but less than i r Polynomial approximation 10 Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Polynomial Approximation f ( x ) multivariate function in d variables f ( x ) ∼ p i α i ( x ) ∈ P n , d ( α i ) , | i | = ∑ ∑ i , | i |≤ n r = 1 p i can be determined by means of hyperinterpolation techniques or algebraic manipulations of polynomials. Approximation theory ∙ Tchebycheff : uniform convergence over the interval of definition ( f ∙ Taylor : convergence in a neighborhood of the expansion point ( f is

SMART Project respectively, in the chosen basis, multivariate function d Polynomial approximation algebra. 11 Intrusive methods Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Intrusive methods Algebra Definition ( P n , d , ⊗ ) is the Algebra on the space of polynomials such that being P f ( x ) and P g ( x ) the polynomial approximation of f ( x ) and g ( x ) P f ( x ) ⊕ g ( x ) = P f ( x ) ⊗ P g ( x ) , where ⊕ ∈ { + , − , ∗ , / } and ⊗ is the corresponding operation in the = ( n + d )! ( n + d ) ∙ N = dim ( P n , d , ⊗ ) = n ! d ! ∙ Composition: h ( x ) ∈ { 1 / x , sin ( x ) , cos ( x ) , exp ( x ) , log ( x ) , ... } , f ( x ) a h ( f ( x )) ∼ H ( x ) ◦ F ( x ) , H ( x ) ∈ P n , 1 ( α i ) , F ( x ) ∈ P n , d ( α i )

SMART Project Tchebycheff basis ∙ New polynomial basis inherit a virtual method from the base class is the transformation. Polynomial approximation 12 Propagation of Uncertainty in Space Dynamics Intrusive methods Discussion & Conclusions Manipulation in Monomial basis ∙ Motivation : computationally expensive multiplication in ∙ Solution : transform the expansion of elementary functions into monomial base ϕ i . Given h ( x ) ∈ { 1 / x , sin ( x ) , cos ( x ) , exp ( x ) , log ( x ) , ... } and f ( x ) a multivariate function h ( f ( x )) ∼ τ ( H ( x )) ◦ F ϕ ( x ) , where F ϕ ( x ) is the approximation in the monomial basis of f and τ for transformation to and from monomial basis

SMART Project Expansion of the flow of an autonomous ODE At the k -th iteration in the polynomial algebra environment Forward Euler scheme: Polynomial approximation 13 Intrusive methods Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Integration of Dynamical Systems { x = f ( x ) ˙ x ( t 0 ) = x 0 Initialize x 0 as an element of the algebra X 0 ( x ) = ( α 1 1 ( x ) , . . . , α 1 d ( x )) ∈ ( P n , d ( α i ) , ⊗ ) d ∙ Real Algebra: x 1 = x 0 + dt f ( x 0 ) ∙ Polynomial Algebra: X 1 ( x ) = X 0 ( x ) + dt f ( X 0 ( x )) ∈ ( P n , d , ⊗ ) d Polynomial Expansion of the Flow X k ( x ) = X k − 1 ( x ) + dt f ( X k − 1 ( x )) ∈ ( P n , d , ⊗ ) d

SMART Project function have been overloaded evaluate(x) ) for polynomials and so on integrators, evaluate(t,state,dstate) for dynamics, integrate(ti, tend, nsteps, x0, xfinal) for of virtual functions need to be implemented (example sampling techniques or dynamics are added to the toolbox a set 14 Polynomial approximation implemented for real or polynomial evaluations Intrusive methods Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Implementation ∙ Template library : integration schemes and dynamical systems are ∙ Operator overloading : algebraic operators and elementary ∙ Abstract base classes : if new polynomial basis, integrators,

SMART Project . . ... . . . Polynomial approximation p 0 . . . Y 1 . . . Y s are the true values. . 15 Non-intrusive methods Propagation of Uncertainty in Space Dynamics (2) Discussion & Conclusions Non-intrusive methods Polynomial Interpolation The interpolation polynomial on the grid nodes is computed as F ( x ) = ∑ p i α i ( x ) , i ∈ I (Γ n , d ) Where Γ n , d is the chosen sampling scheme. The unknown coefficients p i are computed by inverting the linear system HP = Y  α 0 ( x 1 ) . . . α N ( x 1 )      H =  , P =  , Y =  ,          α 0 ( x s ) . . . α N ( x s ) p N where s = | Γ n , d | is the number of nodes, x 1 , . . . , x s are the nodes Y

SMART Project Polynomial approximation Propagation of Uncertainty in Space Dynamics Non-intrusive methods performed in the space of real numbers given a set of values (obtained through sampling and evaluation of the analysis or supplied as text file) is inherited by any polynomial 16 Discussion & Conclusions Implementation ∙ Template library : integration and polynomial evaluation are ∙ Abstract class : in the superclass the method for interpolating

propagation of uncertainty in space dynamics

SMART Project In an inertial reference frame the dynamical equations are velocity and the mass of the spacecraft varies as Polynomial approximation 18 Propagation of Uncertainty in Space Dynamics Discussion & Conclusions Two Body Problem Problem Definition r 3 x + T x = − µ m ∥ v rel ∥ v rel + ϵ ¨ m + 1 2 ρ C D A where r is the distance from the Earth, v rel is the Earth relative m = − α ∥ T ∥ ˙