On improvements in Exact Real Arithmetic for Initial Value Problems - PowerPoint PPT Presentation

On improvements in Exact Real Arithmetic for Initial Value Problems Franz Brauße 1 Margarita Korovina 2 Norbert Müller 1 1 Universität Trier 2 IIS Novosibirsk CCC, Kochel, 2015-09-17

1 Background and Setting Motivation iRRAM 2 Algorithm 3 Radius of Convergance Picard-Lindelöf’s method Improved by Integrals Iterative Improvement 4 Countering wrapping effects Lipschitz bounds reducing wrapping Taylor Models 5 Future Work and References Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 2 / 20

Background and Setting Motivation Why another approach to solving IVPs? Goal • Provide reliable solutions up to arbitrary accuracy, efficiently! Setting: Computable Analysis • Theoretical foundation for computations with continuous objects: Real Numbers, Functions, Sets, . . . Result: IVP-solver in iRRAM • C++ implementation of concepts from Computable Analysis. • x ∈ R is represented as sequence ( c i + e i I ) i converging to x where c i , e i dyadic and I = [− 1, 1 ] . Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 3 / 20

Background and Setting iRRAM Provide methods / tools suitable for use by engineers w/o in-depth knowledge about Real computation, but who assume to perform those. iRRAM works on names: • Cauchy: REAL • τ TM : TM , linear multivariate poly w/ interval coeffs. T : I k → R d , k can vary over course of computation due to polishing • τ A for other wrappings A ? Algorithms dependant on actual rep- Algorithms work independent of con- resentation of Reals (low level): crete backend/representation (high level): • arithmetic • elementary funs • lim • solving PIVP systems (including • evaluation of power series bounds to R, M, L ) • Lipschitzify Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 4 / 20

Background and Setting iRRAM Real Computation computation − − − − − − − → − accuracy x ± 2 − 21 y ± 2 − 19 − ↑ − − x ± 2 − 53 y ± 2 − 50 z ± 2 − 45 · · · − ← x ± 2 − 140 y ± 2 − 136 z ± 2 − 128 · · · . . . ... . . . . . . • Computable Analysis: complete table • Numerical Computation: horizontal line • iRRAM: finite path Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 5 / 20

Background and Setting iRRAM Real Computation computation − − − − − − − → − accuracy x ± 2 − 21 y ± 2 − 19 − ↑ − − x ± 2 − 53 y ± 2 − 50 z ± 2 − 45 · · · − ← x ± 2 − 140 y ± 2 − 136 z ± 2 − 128 · · · . . . ... . . . . . . • Computable Analysis: complete table • Numerical Computation: horizontal line • iRRAM: finite path Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 5 / 20

Background and Setting iRRAM Real Computation computation − − − − − − − → − accuracy x ± 2 − 21 y ± 2 − 19 − ↑ − − x ± 2 − 53 y ± 2 − 50 z ± 2 − 45 · · · − ← x ± 2 − 140 y ± 2 − 136 z ± 2 − 128 · · · . . . ... . . . . . . • Computable Analysis: complete table • Numerical Computation: horizontal line • iRRAM: finite path Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 5 / 20

Background and Setting Polynomial ODE systems Polynomial ODE system f : R × R d → R d in d + 1 variables, then solution d -dimensional polynomial � y : R → R d described by ODE system � d y ( t ) = � d t � f ( t, � y ( t )) Usually we also have an initial value � y 0 = � y ( t 0 ) at some t 0 . • virtually all real-world ODE systems are describable by poly right hand side � f • example systems: Van-der-Pol oscillator, n -Body problem, double pendulum, Lorentz attractor, . . . Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 6 / 20

Background and Setting Polynomial ODE systems Example: Van-der-Pol equation, α = 3 y 1 = y 2 y 1 ( 0 ) = 1 ˙ y 2 = αy 2 − y 1 − αy 2 ˙ 1 y 2 y 2 ( 0 ) = 1 vdp-3 4 2 y 2 ( t ) y 2 ( t ) 0 -2 -4 020406080 t 2 1 0 -1 100 -2 y 1 ( t ) Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 7 / 20

Algorithm 1 Background and Setting Motivation iRRAM 2 Algorithm 3 Radius of Convergance Picard-Lindelöf’s method Improved by Integrals Iterative Improvement 4 Countering wrapping effects Lipschitz bounds reducing wrapping Taylor Models 5 Future Work and References Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 8 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20

Algorithm Approach to solving such IVP systems Classical power series method, e.g. derivable through Picard iteration 3m + 1 initial value � y m � → sequence of Taylor series coeffs ( � a n ) n a n � ≤ MR − n ❀ 3m + 2 bounds R on radius of convergence and M : � � bounded truncation error ❀ power series for � y ( t ) while t ∈ ( t m ± R ) y m + 1 = � a n t n 3m + 3 choose t m + 1 and evaluate � n � m + 1 solution y ( t ) initial condition evaluation recursion inside of on circle of coefficients convergence sum power function series summation time t Brauße, Korovina, Müller (UT, IIS) Using ERA for IVPs CCC, Kochel, 2015-09-17 9 / 20