interval methods and taylor model methods for odes
TRANSCRIPT
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0 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
KARLSRUHE INSTITUTE OF TECHNOLOGY (KIT)
Interval Methods and Taylor Model Methods for ODEs
Markus Neher, Dept. of Mathematics
KIT – University of the State of Baden-Wuerttemberg andNational Research Center of the Helmholtz Association www.kit.edu
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Validated Methods
1 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Also called: Verified Methods, Rigorous Methods, GuaranteedMethods, Enclosure Methods, . . .
Aim: Compute guaranteed bounds for the solution of a problem,including
Discretization errors (ODEs, PDEs, optimization)
Truncation errors (Newton’s method, summation)
Roundoff errors
Used forModelling of uncertain data
Bounding of roundoff errors
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Outline
2 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
1 Interval arithmetic
2 Interval methods for ODEs
3 Taylor model methods for ODEs
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3 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval Arithmetic
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Interval Arithmetic
4 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Compact real intervals:
IR = {x = [x, x ] | x ≤ x} (x, x ∈ R).
Basic arithmetic operations:
x ◦ y := {x ◦ y | x ∈ x , y ∈ y}, ◦ ∈ {+,−, ∗, /} (0 6∈ y for /)
x + y = [x + y , x + y ]
x − y = [x − y , x − y ]
x ∗ y = [min{xy , xy , xy , xy , }, max{xy , xy , xy , xy , }],
x / y = x ∗ [1 / y , 1 / y ]
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Interval Arithmetic
4 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Compact real intervals:
IR = {x = [x, x ] | x ≤ x} (x, x ∈ R).
Basic arithmetic operations:
x ◦ y := {x ◦ y | x ∈ x , y ∈ y}, ◦ ∈ {+,−, ∗, /} (0 6∈ y for /)
x + y = [x + y , x + y ]
x − y = [x − y , x − y ] ⇒ [0, 1]− [0, 1] = [−1, 1]
x ∗ y = [min{xy , xy , xy , xy , }, max{xy , xy , xy , xy , }],
x / y = x ∗ [1 / y , 1 / y ]
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Interval Arithmetic
4 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Compact real intervals:
IR = {x = [x, x ] | x ≤ x} (x, x ∈ R).
Basic arithmetic operations:
x ◦ y := {x ◦ y | x ∈ x , y ∈ y}, ◦ ∈ {+,−, ∗, /} (0 6∈ y for /)
x + y = [x + y , x + y ] FPIA : [ x5+ y , x 4+ y ]
x − y = [x − y , x − y ] ⇒ [0, 1]− [0, 1] = [−1, 1]
x ∗ y = [min{xy , xy , xy , xy , }, max{xy , xy , xy , xy , }],
x / y = x ∗ [1 / y , 1 / y ]
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Ranges and Inclusion Functions
5 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Range of f : D → E: Rg (f , D) := {f (x) | x ∈ D}
Inclusion function F : IR → IR of f : D ⊆ R → R:
F (x) ⊇ Rg (f , x) for all x ⊆ D
Examples:
x1 + x
, 1− 11 + x
, are inclusion functions for
f (x) =x
1 + x= 1− 1
1 + x
ex := [ex , ex ] is an inclusion function for ex
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IA: Dependency
6 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
f (x) =x
1 + x= 1− 1
1 + x, x = [1, 2]:
x1 + x
=[1, 2][2, 3]
= [13
, 1]
1− 11 + x
= 1− 1[2, 3]
= 1− [13
,12] = [
12
,23] = Rg (f , x)
Reduced overestimation: centered forms, etc.
Mean value form: Rg (f , x) ⊆ f (c) + F ′(x)(x − c), c = m(x).
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IA: Wrapping Effect
7 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Overestimation: Enclose non-interval shaped sets by intervals
Example: f : (x, y) →√
22
(x + y , y − x) (Rotation)
Interval evaluation of f on x = ([−1, 1], [−1, 1]):
−2 −1 1 2
−2
−1
1
2
x
y
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IA: Wrapping Effect
7 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Overestimation: Enclose non-interval shaped sets by intervals
Example: f : (x, y) →√
22
(x + y , y − x) (Rotation)
Interval evaluation of f on x = ([−1, 1], [−1, 1]):
−2 −1 1 2
−2
−1
1
2
x
y
−2 −1 1 2
−2
−1
1
2
x
y
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8 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval Methods for ODEs
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Validated Integration of ODEs
9 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval IVP:
u′ = f (t , u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend]
f : R×Rm → Rm sufficiently smooth, u0 ∈ IRm, tend > t0.
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Autonomous Interval IVP
10 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
u′ = f (u), u(t0) = u0 ∈ u0, t ∈ t = [t0, tend],
where D ⊂ Rm, f ∈ Cn(D), f : D → Rm, u0 ∈ IRm.
Moore’s enclosure method:
Automatic computation of Taylor coefficients
Interval iteration: For j = 1, 2, . . . :
A priori enclosure: v j ⊇ u(t) for all t ∈ [tj−1, tj ] ("Alg. I").
Truncation error: z j :=hn+1
j
(n + 1)!f (n)(v j ).
u(tj ) ∈ uj := uj−1 +n
∑k=1
hkj
k !f (k−1)(uj−1) + z j ("Algorithm II").
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Piecewise Constant A Priori Enclosure
11 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
t0 t1 t2
u0u1
u2v1
v2
t
u
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A Priori Enclosures
12 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Picard iteration: find hj , v j such that
uj−1 + [0, hj ]f (v j ) ⊆ v j
Step size restrictions: Explicit Euler steps
Improvements: Lohner 1988, Corliss & Rihm 1996, Makino 1998,Nedialkov & Jackson 2001
Alternatives: Neumaier 1994, N. 1999, N. 2007, Delanoue & Jaulin2010, Kin, Kim & Nakao 2011
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Modifications of Algorithm II
13 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Reduction of wrapping effect: Moore, Eijgenraam, Lohner, Rihm,Kuehn, Nedialkov & Jackson, . . .
Nedialkov & Jackson: Hermite-Obreshkov-Method
Rihm: Implicit methods
Petras & Hartmann, Bouissou: Runge-Kutta-Methods
Berz & Makino: Taylor models
Taylor expansion of solution w.r.t. time and initial values
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Direct Interval Method
14 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Direct method (Moore 1965): Apply mean value form to f (k):
f [0](u) = u, f [k ](u) =1k
(∂f [k−1]
∂uf
)(u) for k ≥ 1.
Let
S j−1 = I +n
∑k=1
hk0J(f [k ](uj−1)
), z j = hn+1
0 f [n](v j ),
(I: identity matrix, J(f [k ]): Jacobian of f [i ]), then for some uj−1 ∈ uj−1
u(tj ; u0) ∈ uj = uj−1 +n−1
∑k=1
hkj f [k ](uj−1) + z j + S j−1(uj−1 − uj−1).
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Global Error Propagation
15 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Wrapping effect: S j−1(uj−1 − uj−1) may overestimate
S = {Sj−1(uj−1 − uj−1) | Sj−1 ∈ S j−1, uj−1 ∈ uj−1}
→ propagate S as a parallelepiped.
u0 := m(u0), r0 = u0 − u0, B0 = I; for some nonsingular Bj−1:
uj = uj−1 +n−1
∑k=1
hkj−1f [k ](uj−1) + m(z j ),
uj = uj−1 +n−1
∑k=1
hkj−1f [k ](uj−1) + z j + (S j−1Bj−1)r j−1,
uj : approximate point solution for the central IVP
z j : local error; r j : global error
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Global Error Propagation
16 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Global error propagation:
r j =(
B−1j (S j−1Bj−1)
)r j−1 + B−1
j (z j −m(z j ))
Moore’s direct method: Bj = I
Pep method (Eijgenraam, Lohner): Bj = m(S j−1Bj−1)
QR method (Lohner): m(S j−1Bj−1) = QR, Bj := Q
Blunting method (Berz, Makino): modify Bj in the pep method suchthat condition numbers remain small
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Autonomous Linear ODE
17 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Autonomous linear system (A ∈ Rm×m):
u′ = A u, u(0) ∈ u0.
Propagation of the Global Error
r j = (B−1j TBj−1)r j−1 + B−1
j
(z j −m(z j )
), T =
n−1
∑ν=0
(hA)ν
ν!.
+ = ⊂
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Interval Methods for Linear ODEs
18 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Direct method: r j = T r j−1 + z j −m(z j ).
Optimal for local error, bad for global error (rotation)
Parallelepiped method: r j = r j−1 + (T−j )(z j − (m(z j )).
Optimal for global error; suitable for local error, if cond(T j ) is small
Bad for local error in presence of shear
QR method: r j = Rjr j−1 + QTj
(z j −m(z j )
), j = 1, 2, . . . .
Handles rotation, contraction, shear
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Wrapping Effect: Direct Interval Method
19 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
(Plots by Ned Nedialkov)
Huge overestimations in general
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Wrapping Effect: Parallelepiped Method
20 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Bj = TBj−1
−4 −3 −2 −1 0 1 2 3 4
x 10−12
−3
−2
−1
0
1
2
3
x 10−12
{ Cjr +e r ∈ r
j−1, e ∈ e
j}
{ Cjr r ∈ r
j−1 }
{ Cjr r ∈ r
j }
−6 −4 −2 0 2 4 6
x 10−12
−6
−4
−2
0
2
4
6x 10
−12
{ Cjr +e r ∈ r
j−1, e ∈ e
j}
{ Cjr r ∈ r
j−1 }
{ Cjr r ∈ r
j }
Bj often ill-conditioned, large overestimations
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Wrapping Effect: QR Method
21 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Bj = Qj , QjRj = TBj−1
−5 0 5
x 10−12
−4
−3
−2
−1
0
1
2
3
4
x 10−12
{ Cjr +e r ∈ r
j−1, e ∈ e
j }
{ Cjr r ∈ r
j−1 }
QR, Version 1QR, Version 2
−3 −2 −1 0 1 2 3
x 10−12
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
x 10−12
{ Cjr +e r ∈ r
j−1, e ∈ e
j }
{ Cjr r ∈ r
j−1 }
QR, Version 1QR, Version 2
Overestimation depends on column permutations of Bj−1
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22 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Taylor Model Methods for ODEs
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Taylor Models (I)
23 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
x ⊂ Rm, f : x → R, f ∈ Cn+1, x0 ∈ x ;
f (x) = pn,f (x − x0) + Rn,f (x − x0), x ∈ x
(pn,f Taylor polynomial, Rn,f remainder term)
Interval remainder bound of order n of f on x :
∀x ∈ x : Rn,f (x − x0) ∈ in,f
Taylor model Tn,f = (pn,f , in,f ) of order n of f :
∀x ∈ x : f (x) ∈ pn,f (x − x0) + in,f
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Taylor Models (II)
24 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Taylor model: U := pn(x) + i, x ∈ x , x ∈ IRm, i ∈ IRm
(pn: vector of m-variate polynomials of order n)
Function set: U = {f ∈ C0(x) : f (x) ∈ pn(x) + i for all x ∈ x }
Range of a TM: Rg (U ) = {z = p(x) + ξ | x ∈ x , ξ ∈ i} ⊂ Rm
Ex.: U :=(
x12 + x2
1 + x2
), x1, x2 ∈ [−1, 1]
Rg (U ):
−1 1
2
x1
x2
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TM Arithmetic
25 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Paradigm for TMA:
pn,f is processed symbolically to order n
Higher order terms are enclosed into the remainder interval of theresult
Taylor Model Methods for ODEs
Taylor expansion of solution w.r.t. time and initial values
Computation of Taylor coefficients by Picard iteration:Parameters describing initial set treated symbolically
Interval remainder bounds by fixed point iteration (Makino, 1998)
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Example: Quadratic Problem
26 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
u′ = v , u(0) ∈ [0.95, 1.05],
v ′ = u2, v(0) ∈ [−1.05,−0.95].
Taylor model method: initial set described by parameters a and b:
u0(a, b) := 1 + a, a ∈ a := [−0.05, 0.05],
v0(a, b) := −1 + b, b ∈ b := [−0.05, 0.05].
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3rd order TM Method: Enclosure of the Flow
27 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
h = 0.1, flow for τ ∈ [0, 0.1]:
U1(τ, a, b) := 1 + a− τ + bτ + 12 τ2 + aτ2 − 1
3 τ3 + i0,
V1(τ, a, b) := −1 + b + τ + 2aτ − τ2 + a2τ − aτ2 + bτ2 + 23 τ3 + j0.
Flow at t1 = 0.1:
U1(a, b) := U1(0.1, a, b) = 0.905 + 1.01a + 0.1b + i0,
V1(a, b) := V1(0.1, a, b) = −0.909 + 0.19a + 1.01b + 0.1a2 + j0
(nonlinear boundary).
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Naive TM Method
28 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval remainder terms accumulate
Linear ODEs:
Naive TM method performs similarly to the direct interval method
→ Shrink wrapping, preconditioned TM methods
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Shrink Wrapping
29 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Absorb interval term into polynomial part (Makino and Berz 2002):
(UV
)(white) vs.
(UswVsw
).
Linear ODEs: Shrink wrapping performs similarly to the pep method.
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Integration with Preconditioned Taylor Models
30 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Preconditioned integration: flow at tj :
Uj = Ul,j ◦ Ur ,j = (pl,j + i l,j ) ◦ (pr ,j + i r ,j ).
Purpose: stabilize integration as in the QR interval method
Theorem (Makino and Berz 2004)If the initial set of an IVP is given by a preconditioned Taylor model, thenintegrating the flow of the ODE only acts on the left Taylor model.
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Preconditioned TMM for linear ODE
31 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Global error:
i r ,j+1 := C−1l,j+1TCl,j i r ,j + C−1
l,j+1i l,j+1, j = 0, 1, . . . .
Cl,j+1 = TCl,j : parallelepiped preconditioning
Cl,j+1 = Qj : QR preconditioning
Other choices: curvilinear coordinates, blunting(Makino and Berz 2004)
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Integration of Quadratic Problem
32 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
u′ = v , u(0) ∈ [0.95, 1.05],
v ′ = u2, v(0) ∈ [−1.05,−0.95].
-1.5
-1
-0.5
0
0.5
1
1.5
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 1.2-2
-1.5
-1
-0.5
0
0.5
1
-2 -1.5 -1 -0.5 0 0.5 1 1.5
COSY Infinity AWA
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Summary
33 TM VII, Key West Interval Methods and Taylor Model Methods for ODEs Markus Neher, KIT
KIT
Interval arithmetic
Interval methods for ODEs
Taylor model methods for ODEs
Open problems
Treatment of high-dimensional problems
Validated implicit methods