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Introduction Computing the probability of collision Conclusion A new method to compute the probability of collision for short-term space encounters R. Serra, D. Arzelier, M. Joldes, J-B. Lasserre, A. Rondepierre and B. Salvy Journ ees

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Introduction Computing the probability of collision Conclusion

A new method to compute the probability of collision for short-term space encounters

  • R. Serra, D. Arzelier, M. Joldes, J-B. Lasserre, A. Rondepierre and B. Salvy

Journ´ ees Nationales de Calcul Formel CIRM November, 2014

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Introduction Computing the probability of collision Conclusion

Fiction...

Credit Gravity (2013)

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

On-orbit collision

Figure: Cerise hit by a debris in 1996 (source : CNES/D. Ducros)

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

On-orbit collision

Figure: Space debris population model (source : ESA)

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

On-orbit collision

Context Two objects: primary P (operational satellite) and secondary S (space debris) Information about their geometry, position, velocity at a given time Affected by uncertainty Needs:

Risk assessment Design of a collision avoidance strategy

compute the probability of collision

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

On-orbit collision

Context Two objects: primary P (operational satellite) and secondary S (space debris) Information about their geometry, position, velocity at a given time Affected by uncertainty Needs:

Risk assessment Design of a collision avoidance strategy

compute the probability of collision Classical assumptions Spherical objects Gaussian probability density functions Independent probability distribution laws

Rp Rs Rc

P S

Figure: Combined spherical object

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

On-orbit collision

Context Two objects: primary P (operational satellite) and secondary S (space debris) Information about their geometry, position, velocity at a given time Affected by uncertainty Needs:

Risk assessment Design of a collision avoidance strategy

compute the probability of collision Classical assumptions Spherical objects Gaussian probability density functions Independent probability distribution laws

Rp Rs Rc

P S

Figure: Combined spherical object

Probability of collision Generally: 12-dimensional, Gaussian integrand, Complex integration domain Computation: Monte-Carlo trials and/or simplified models

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

Short-term encounter model and probability of collision

Framework: High relative velocity Assumptions:

Rectilinear relative motion No velocity uncertainty Infinite encounter time horizon

Probability of collision: 2-D integral over a disk.

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

Short-term encounter model and probability of collision

Framework: High relative velocity Assumptions:

Rectilinear relative motion No velocity uncertainty Infinite encounter time horizon

Probability of collision: 2-D integral over a disk. Formula P = 1 2πσxσy

  • B((0,0),R)

exp

  • − (x − xm)2

2σx2 − (y − ym)2 2σy2

  • dxdy,

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Introduction Computing the probability of collision Conclusion General Model Short-term encounter probability of collision

Short-term encounter model and probability of collision

Framework: High relative velocity Assumptions:

Rectilinear relative motion No velocity uncertainty Infinite encounter time horizon

Probability of collision: 2-D integral over a disk.

R x y ym combined object xm level sets

Figure: 2-D Gaussian integral over a disk

Formula P = 1 2πσxσy

  • B((0,0),R)

exp

  • − (x − xm)2

2σx2 − (y − ym)2 2σy2

  • dxdy,

where R: radius of combined object xm, ym: mean relative coordinates σx, σy: standard deviations of relative coordinates

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Existing methods

Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 uses some simplifying assumptions (σx = σy) Pro’s and Con’s

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Existing methods

Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 uses some simplifying assumptions (σx = σy) Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate

6 / 23

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Existing methods

Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 uses some simplifying assumptions (σx = σy) Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate truncation orders fixed by trial and error and by comparing with other existing software: Approximately 60,000 test cases were used to evaluate the numerical expression [...] The reference (”truth”) probability was computed with MATHCAD 11 [...] - Alfano’05

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Existing methods

Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 uses some simplifying assumptions (σx = σy) Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate truncation orders fixed by trial and error and by comparing with other existing software: Approximately 60,000 test cases were used to evaluate the numerical expression [...] The reference (”truth”) probability was computed with MATHCAD 11 [...] - Alfano’05 Fast and already used in practice

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Existing methods

Methods based on numerical integration schemes: Foster ’92, Patera ’01, Alfano ’05. Analytic methods: Chan ’97 uses some simplifying assumptions (σx = σy) provides truncation error bounds Pro’s and Con’s use truncated power series, but no rigorous proof about convergence rate truncation orders fixed by trial and error and by comparing with other existing software: Approximately 60,000 test cases were used to evaluate the numerical expression [...] The reference (”truth”) probability was computed with MATHCAD 11 [...] - Alfano’05 Fast and already used in practice Our purpose Give a ”simple”, ”analytic” formula, suitable for double-precision evaluation and effective error bounds.

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Our method - Underlying techniques

1

Laplace transform:

Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques,

Applicationes Mathematicae, 2001.

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Our method - Underlying techniques

1

Laplace transform:

Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques,

Applicationes Mathematicae, 2001.

2

D-finite functions solution of linear differential equation with polynomial coefficients power series coefficients satisfy a linear recurrence relation with polynomial coefficients Example: f(x) = exp(x) ↔ {f′ − f = 0, f(0) = 1} ↔ {(n + 1)fn+1 = fn, f0 = 1}

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Our method - Underlying techniques

1

Laplace transform:

Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques,

Applicationes Mathematicae, 2001.

2

D-finite functions solution of linear differential equation with polynomial coefficients power series coefficients satisfy a linear recurrence relation with polynomial coefficients Example: f(x) = exp(x) ↔ {f′ − f = 0, f(0) = 1} ↔ {(n + 1)fn+1 = fn, f0 = 1} cos, arccos, Airy functions, Bessel functions, ...

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Our method - Underlying techniques

1

Laplace transform:

Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques,

Applicationes Mathematicae, 2001.

2

D-finite functions solution of linear differential equation with polynomial coefficients power series coefficients satisfy a linear recurrence relation with polynomial coefficients Example: f(x) = exp(x) ↔ {f′ − f = 0, f(0) = 1} ↔ {(n + 1)fn+1 = fn, f0 = 1} cos, arccos, Airy functions, Bessel functions, ...

3

Finite-precision evaluation of power series prone to cancellation Gawronski, M¨

uller, Reinhard, Reduced Cancellation in the Evaluation of Entire Functions and Applications to the Error Function, SIAM Journal on Numerical Analysis, 2007. Chevillard, Mezzarobba, Multiple-Precision Evaluation of the Airy Ai Function with Reduced Cancellation, 21st IEEE SYMPOSIUM on Computer Arithmetic, 2013

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Our method - Underlying techniques

1

Laplace transform:

Lasserre and Zeron, Solving a Class of Multivariate Integration Problems via Laplace Techniques,

Applicationes Mathematicae, 2001.

2

D-finite functions solution of linear differential equation with polynomial coefficients power series coefficients satisfy a linear recurrence relation with polynomial coefficients Example: f(x) = exp(x) ↔ {f′ − f = 0, f(0) = 1} ↔ {(n + 1)fn+1 = fn, f0 = 1} cos, arccos, Airy functions, Bessel functions, ...

3

Finite-precision evaluation of power series prone to cancellation Gawronski, M¨

uller, Reinhard, Reduced Cancellation in the Evaluation of Entire Functions and Applications to the Error Function, SIAM Journal on Numerical Analysis, 2007. Chevillard, Mezzarobba, Multiple-Precision Evaluation of the Airy Ai Function with Reduced Cancellation, 21st IEEE SYMPOSIUM on Computer Arithmetic, 2013

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Laplace Transform

∀z ∈ R+ : g(z) := P(√z) = 1 2πσxσy

  • B((0,0),√z)

exp

  • − (x − xm)2

2σx2 − (y − ym)2 2σy2

  • dxdy,

(1) (3)

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Laplace Transform

∀z ∈ R+ : g(z) := P(√z) = 1 2πσxσy

  • B((0,0),√z)

exp

  • − (x − xm)2

2σx2 − (y − ym)2 2σy2

  • dxdy,

(1) L(g)(t) = +∞ g(z) exp(−tz)dz (2) (3)

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Laplace Transform

∀z ∈ R+ : g(z) := P(√z) = 1 2πσxσy

  • B((0,0),√z)

exp

  • − (x − xm)2

2σx2 − (y − ym)2 2σy2

  • dxdy,

(1) L(g)(t) = +∞ g(z) exp(−tz)dz (2) . . . = exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t
  • (2tσx2 + 1)(2tσy2 + 1)

(3)

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Laplace Transform

∀z ∈ R+ : g(z) := P(√z) = 1 2πσxσy

  • B((0,0),√z)

exp

  • − (x − xm)2

2σx2 − (y − ym)2 2σy2

  • dxdy,

(1) L(g)(t) = +∞ g(z) exp(−tz)dz (2) . . . = exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t
  • (2tσx2 + 1)(2tσy2 + 1)

(3)

L(g) is D-finite !

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace

g(z) L(g)(t) =

exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t√

(2tσx2+1)(2tσy2+1)

g(z) =

∞

  • i=0

li (i+1)! zi+1

ˆ L(g)(t) := t2L(g) 1

t

  • =

∞

  • i=0

li 1

t

i g(z) is: D-finite entire function of exponential type type σ =

1 2σ2

y

ˆ L(g)(t) is: D-finite Finite radius of convergence 2σ2

y

Laplace Transform expansion at ∞ Borel Transform

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace

g(z) L(g)(t) =

exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t√

(2tσx2+1)(2tσy2+1)

g(z) =

∞

  • i=0

li (i+1)! zi+1

ˆ L(g)(t) := t2L(g) 1

t

  • =

∞

  • i=0

li 1

t

i g(z) is: D-finite entire function of exponential type type σ =

1 2σ2

y

ˆ L(g)(t) is: D-finite Finite radius of convergence 2σ2

y

li satisfy a linear recurrence with polynomial coefficients. Compute everything with gfun Laplace Transform expansion at ∞ Borel Transform

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace

g(z) L(g)(t) =

exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t√

(2tσx2+1)(2tσy2+1)

g(z) =

∞

  • i=0

li (i+1)! zi+1

ˆ L(g)(t) := t2L(g) 1

t

  • =

∞

  • i=0

li 1

t

i g(z) is: D-finite entire function of exponential type type σ =

1 2σ2

y

sum prone to cancellation ˆ L(g)(t) is: D-finite Finite radius of convergence 2σ2

y

li satisfy a linear recurrence with polynomial coefficients. Compute everything with gfun Laplace Transform expansion at ∞ Borel Transform

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: σx = 115, σy = 1.41, xm = 0.15, ym = 3.88, √z = 15 g(z) =

∞

  • i=0

li (i+1)! zi+1

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: σx = 115, σy = 1.41, xm = 0.15, ym = 3.88, √z = 15 g(z) =

∞

  • i=0

li (i+1)! zi+1

g(225) = 0.16 · 10−1 + 1.5 + 16.1−250 . . . +2.2 · 1019−2.6 · 1019− . . . +4.3−0.14 − 0.60 . . .

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: σx = 115, σy = 1.41, xm = 0.15, ym = 3.88, √z = 15 g(z) =

∞

  • i=0

li (i+1)! zi+1

g(225) = 0.16 · 10−1 + 1.5 + 16.1−250 . . . +2.2 · 1019−2.6 · 1019− . . . +4.3−0.14 − 0.60 . . . Values of

  • li225i+1

(i+1)!

  • , compared to g(225) ≃ 0.1004:

Lost Digits: dg(z) ≃ log max

i

  • gizi
  • |g(z)|

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: exp(−x) =

∞

  • i=0

(−1)ixi i! exp(−20) = 1−20 . . . +1.66 · 107−1.23 · 107+ . . . +1.19 · 10−8−3.45 · 10−9 . . .

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: exp(−x) =

∞

  • i=0

(−1)ixi i! exp(−20) = 1−20 . . . +1.66 · 107−1.23 · 107+ . . . +1.19 · 10−8−3.45 · 10−9 . . . Values of

  • (−1)i20i

i!

  • , compared to exp(−20) ≃ 2.06 · 10−9:

Lost Digits: dg(z) ≃ log max

i

  • gizi
  • |g(z)|

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: exp(−x) =

∞

  • i=0

(−1)ixi i! exp(−20) = 1−20 . . . +1.66 · 107−1.23 · 107+ . . . +1.19 · 10−8−3.45 · 10−9 . . . Values of

  • (−1)i20i

i!

  • , compared to exp(−20) ≃ 2.06 · 10−9:

Lost Digits: dg(z) ≃ log max

i

  • gizi
  • |g(z)|

BUT... exp(−x) = 1 exp(x) No cancellation!

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: σx = 115, σy = 1.41, xm = 0.15, ym = 3.88, √z = 15 g(z) =

∞

  • i=0

li (i+1)! zi+1

g(225) = 0.16 · 10−1 + 1.5 + 16.1−250 . . . +2.2 · 1019−2.6 · 1019− . . . +4.3−0.14 − 0.60 . . . Values of

  • li225i+1

(i+1)!

  • , compared to g(225) ≃ 0.1004:

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: σx = 115, σy = 1.41, xm = 0.15, ym = 3.88, √z = 15 g(z) =

∞

  • i=0

li (i+1)! zi+1

g(225) = 0.16 · 10−1 + 1.5 + 16.1−250 . . . +2.2 · 1019−2.6 · 1019− . . . +4.3−0.14 − 0.60 . . . Values of

  • li225i+1

(i+1)!

  • , compared to g(225) ≃ 0.1004:

g(z) = F(z) G(z) No cancellation!

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Cancellation in finite precision power series evaluation

Example: σx = 115, σy = 1.41, xm = 0.15, ym = 3.88, √z = 15 g(z) =

∞

  • i=0

li (i+1)! zi+1

g(225) = 0.16 · 10−1 + 1.5 + 16.1−250 . . . +2.2 · 1019−2.6 · 1019− . . . +4.3−0.14 − 0.60 . . . Values of

  • li225i+1

(i+1)!

  • , compared to g(225) ≃ 0.1004:

g(z) = F(z) G(z) No cancellation! Gawronski, M¨ uller, Reinhard (2007) Method: Find F(z) and G(z) using ”complex analysis tricks” indicator function.

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace + GMR Method

g(z) = F(z) G(z) L(g)(t) =

exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t√

(2tσx2+1)(2tσy2+1)

g(z) =

∞

  • i=0

li (i+1)! zi+1

ˆ L(g)(t) := t2L(g) 1

t

  • =

∞

  • i=0

li 1

t

i g(z) is: D-finite entire function of exponential type type σ =

1 2σ2

y

ˆ L(g)(t) is: D-finite Finite radius of convergence 2σ2

y

li - P-recursive Laplace expansion at ∞ Borel Transform

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace + GMR Method

g(z) = F(z) G(z) L(g)(t) =

exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t√

(2tσx2+1)(2tσy2+1)

g(z) =

∞

  • i=0

li (i+1)! zi+1

ˆ L(g)(t) := t2L(g) 1

t

  • =

∞

  • i=0

li 1

t

i g(z) is: D-finite entire function of exponential type type σ =

1 2σ2

y

ˆ L(g)(t) is: D-finite Finite radius of convergence 2σ2

y

li - P-recursive Laplace expansion at ∞ Borel Transform

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace + GMR Method

g(z) = F(z) G(z) L(g)(t) =

exp

  • − σx2ym2+σy2xm2

2σx2σy2

+

ym2 2σy2(2tσy2+1) + xm2 2σx2(2tσx2+1)

  • t√

(2tσx2+1)(2tσy2+1)

g(z) =

∞

  • i=0

li (i+1)! zi+1

ˆ L(g)(t) := t2L(g) 1

t

  • =

∞

  • i=0

li 1

t

i g(z) is: D-finite entire function of exponential type type σ =

1 2σ2

y

indicator function:

  • g(reiθ)
  • ∼ exp (h(θ)r) for large r

h(θ) = − cos θ

2σy2

if θ ∈ [ π

2 , 3π 2 ]

if θ ∈ [0, π

2 ) ∪ ( 3π 2 , 2π].

ˆ L(g)(t) is: D-finite Finite radius of convergence 2σ2

y

li - P-recursive Laplace expansion at ∞ Borel Transform

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Zoom on Indicator functions

g(z) = F(z) G(z)

  • g(reiθ)
  • ∼ exp (h(θ)r) for large r

indicator of g: h(θ) = − cos θ

2σy2

if θ ∈ [ π

2 , 3π 2 ]

  • therwise.

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Zoom on Indicator functions

g(z) = F(z) G(z)

  • g(reiθ)
  • ∼ exp (h(θ)r) for large r

indicator of g: h(θ) = − cos θ

2σy2

if θ ∈ [ π

2 , 3π 2 ]

  • therwise.

lost precision ∼ |σ − h(0)|

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Zoom on Indicator functions

g(z) = F(z) G(z)

  • g(reiθ)
  • ∼ exp (h(θ)r) for large r

indicator of g: h(θ) = − cos θ

2σy2

if θ ∈ [ π

2 , 3π 2 ]

  • therwise.

lost precision ∼ |σ − h(0)| G(z) = exp(σz) indicator of G:

14 / 23

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Zoom on Indicator functions

G(z)g(z) = F(z)

  • g(reiθ)
  • ∼ exp (h(θ)r) for large r

indicator of g: h(θ) = − cos θ

2σy2

if θ ∈ [ π

2 , 3π 2 ]

  • therwise.

lost precision ∼ |σ − h(0)| G(z) = exp(σz) indicator of G: indicator of F:

14 / 23

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace + GMR Method

G(z)g(z) = F(z) L(g)(t − σ) G(z)g(z) =

∞

  • k=0

αk (k+1)! zk+1 ∞

  • k=0

αk 1

t

k G(z) = exp(σz) F is D-finite reduced cancellation for evaluating F, G on positive real line recurrence for αk: Laplace expansion at ∞ Borel Transform

15 / 23

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Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Sketch of the proof - Borel-Laplace + GMR Method

G(z)g(z) = F(z) L(g)(t − σ) G(z)g(z) =

∞

  • k=0

αk (k+1)! zk+1 ∞

  • k=0

αk 1

t

k G(z) = exp(σz) F is D-finite reduced cancellation for evaluating F, G on positive real line recurrence for αk:

− (32kσ4 xσ10 y + 128σ4 xσ10 y )αk+4 = (σ4 xy2 m − 2σ2 xσ2 yy2 m + σ4 yy2 m)αk +((−4σ4 xσ4 y + 8σ2 xσ6 y − 4σ8 y)k − 10σ4 xσ4 y − 6σ4 xσ2 yy2 m + 20σ2 xσ6 y + 8σ2 xσ4 yy2 m − 10σ8 y − 2σ6 yy2 m)αk+1 +((24σ4 xσ6 y − 32σ2 xσ8 y + 8σ10 y )k + 72σ4 xσ6 y + 12σ4 xσ4 yy2 m − 92σ2 xσ8 y − 8σ2 xσ6 yy2 m + 20σ10 y + 4σ8 yx2 m)αk+2 +((−48σ4 xσ8 y + 32σ2 xσ10 y )k − 168σ4 xσ8 y − 8σ4 xσ6 yy2 m + 104σ2 xσ10 y − 8σ10 y x2 m)αk+3,

Laplace expansion at ∞ Borel Transform

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SLIDE 46

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

  • Prop. Coefficients αk are positive

Suppose σx > σy. ˆ L(x) =

∞

  • k=0

αkxk satisfies ˆ L′(x) = F(x)ˆ L(x), ˆ L(0) = exp

  • −

σ2

xy2 m + σ2 yx2 m

2σ2

xσ2 y

  • 2σxσy

, where F(x) = ym2 4σy4 + σy4xm2 (x (σx2 − σy2) − 2 σx2σy2)2 − 1 −2 σy2 + x + −σx2 + σy2 2x (σx2 − σy2) − 4 σx2σy2 .

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SLIDE 47

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

  • Prop. Coefficients αk are positive

Suppose σx > σy. ˆ L(x) =

∞

  • k=0

αkxk satisfies ˆ L′(x) = F(x)ˆ L(x), ˆ L(0) = exp

  • −

σ2

xy2 m + σ2 yx2 m

2σ2

xσ2 y

  • 2σxσy

, where F(x) = ym2 4σy4 + σy4xm2 (x (σx2 − σy2) − 2 σx2σy2)2 − 1 −2 σy2 + x + −σx2 + σy2 2x (σx2 − σy2) − 4 σx2σy2 . fk = 1 +

  • 1 −

σ2

y

σ2

x

k (k + 1)

  • x2

mσ2 y

σ4

x

  • + 1 −

σ2

y

σ2

x

  • 2

(2σ2

y)k+1

+      0, k > 0 ym2 4σ4

y

, k = 0,

16 / 23

slide-48
SLIDE 48

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

  • Prop. Coefficients αk are positive

Suppose σx > σy. ˆ L(x) =

∞

  • k=0

αkxk satisfies ˆ L′(x) = F(x)ˆ L(x), ˆ L(0) = exp

  • −

σ2

xy2 m + σ2 yx2 m

2σ2

xσ2 y

  • 2σxσy

, where F(x) = ym2 4σy4 + σy4xm2 (x (σx2 − σy2) − 2 σx2σy2)2 − 1 −2 σy2 + x + −σx2 + σy2 2x (σx2 − σy2) − 4 σx2σy2 . 0 ≤ fk = 1 +

  • 1 −

σ2

y

σ2

x

k (k + 1)

  • x2

mσ2 y

σ4

x

  • + 1 −

σ2

y

σ2

x

  • 2

(2σ2

y)k+1

+      0, k > 0 ym2 4σ4

y

, k = 0,

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SLIDE 49

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

  • Prop. Coefficients αk are positive

Suppose σx > σy. ˆ L(x) =

∞

  • k=0

αkxk satisfies ˆ L′(x) = F(x)ˆ L(x), ˆ L(0) = exp

  • −

σ2

xy2 m + σ2 yx2 m

2σ2

xσ2 y

  • 2σxσy

, where F(x) = ym2 4σy4 + σy4xm2 (x (σx2 − σy2) − 2 σx2σy2)2 − 1 −2 σy2 + x + −σx2 + σy2 2x (σx2 − σy2) − 4 σx2σy2 . 0 ≤ fk = 1 +

  • 1 −

σ2

y

σ2

x

k (k + 1)

  • x2

mσ2 y

σ4

x

  • + 1 −

σ2

y

σ2

x

  • 2

(2σ2

y)k+1

+      0, k > 0 ym2 4σ4

y

, k = 0, 0 ≤ (n + 1)αn+1 =

n

  • i=0

fi · αn−i, by induction.

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SLIDE 50

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Bounds using majorant series

Let γ := 1 + 1

2

  • 1 −

σ2

y

σ2

x

+ x2

mσ2 y

σ4

x

+ ym2

σy2

  • 2σy2

, αk := α0γk and αk := α0

  • 1

2σy2 k . Then αk ≤ αk ≤ αk, ∀k ∈ N.

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SLIDE 51

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Bounds using majorant series

Let γ := 1 + 1

2

  • 1 −

σ2

y

σ2

x

+ x2

mσ2 y

σ4

x

+ ym2

σy2

  • 2σy2

, αk := α0γk and αk := α0

  • 1

2σy2 k . Then αk ≤ αk ≤ αk, ∀k ∈ N. Let ˜ PN(z) :=

N−1

  • k=0

αkzk (k + 1)! . Then we have the following error bounds: εN(z) ≤ g(z) − ˜ PN(z) ≤ εN(z), where εN(z) := 2α0σ2

ye −

z 2σ2 y

  • z

2σ2

y

N+1 (N + 1)! , εN(z) := α0 γ e

−

z 2σ2 y (zγ)N+1

N + 1! .

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SLIDE 52

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Examples

Sample 1

Case Input parameters (km) # σx σy R xm ym 1 0.05 0.025 0.005 0.01 2 0.05 0.025 0.005 0.01 3 0.075 0.025 0.005 0.01 4 0.075 0.025 0.005 0.01 5 3 1 0.01 1 6 3 1 0.01 1 7 3 1 0.01 10 8 3 1 0.01 10 9 10 1 0.01 10 10 10 1 0.01 10 11 3 1 0.05 5 12 3 1 0.05 5

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SLIDE 53

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Examples: quality η = ε10(z) − ε10(z) ε10(z) + ˜ P10(z) and plot log

  • ˜

pizi

Case 1: η = 23 Case 2: η = 22 Case 4: η = 22 Case 6: η = 47 Case 8: η = 33 Case 11: η = 34

19 / 23

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SLIDE 54

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Numerical study

Sample 1

Case Probability of Collision (-) # Alfano Patera Chan New method 1 9.742 × 10−3 9.741 × 10−3 9.754 × 10−3 9.742 × 10−3 2 9.181 × 10−3 9.181 × 10−3 9.189 × 10−3 9.181 × 10−3 3 6.571 × 10−3 6.571 × 10−3 6.586 × 10−3 6.571 × 10−3 4 6.125 × 10−3 6.125 × 10−3 6.135 × 10−3 6.125 × 10−3 5 1.577 × 10−5 1.577 × 10−5 1.577 × 10−5 1.577 × 10−5 6 1.011 × 10−5 1.011 × 10−5 1.011 × 10−5 1.011 × 10−5 7 6.443 × 10−8 6.443 × 10−8 6.443 × 10−8 6.443 × 10−8 8 3.219 × 10−27 3.216 × 10−27 3.219 × 10−27 9 3.033 × 10−6 3.033 × 10−6 3.033 × 10−6 3.033 × 10−6 10 9.656 × 10−28 9.645 × 10−28 9.656 × 10−28 11 1.039 × 10−4 1.039 × 10−4 1.039 × 10−4 1.039 × 10−4 12 1.564 × 10−9 1.564 × 10−9 1.556 × 10−9 1.564 × 10−9

− equal to reference value (from [Chan 2008]) 20 / 23

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SLIDE 55

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Examples: quality ηN = εN(z) − εN(z) εN(z) + ˜ PN(z) and plot log

  • ˜

pizi

Case 3 Alfano: η800 = 30 Case 5 Alfano: η121000 = 20 Sample 2 (from [Alfano 2009]) Case Input parameters (m) # σx σy R xm ym 3 114.25 1.41 15 0.15 3.88 5 177.8 0.038 10 2.12

  • 1.22

21 / 23

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SLIDE 56

Introduction Computing the probability of collision Conclusion Existing methods Our method Examples

Examples

Sample 2 (from [Alfano 2009])

Case Input parameters (m) # σx σy R xm ym 3 114.25 1.41 15 0.15 3.88 5 177.8 0.038 10 2.12

  • 1.22

Alfano’s test Probability of collision (-) case number Alfano New method Reference (MC) 3 0.10038 0.10038 0.10085 5 0.044712 0.045509 0.044499

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SLIDE 57

Introduction Computing the probability of collision Conclusion

Conclusion

New method Analytical formula Reduced cancellation evaluation Error bounds No simplifying assumption Current and future work Saddle-point method for ”degenerate” cases Long-term 3D encounter model Extension to polygonal cross-sections Extensive testings and comparisons with existing methods

23 / 23