Abbas Edalat

Imperial College London

www.doc.ic.ac.uk/~ae

Contains joint work with Andre Lieutier (AL)

and joint work with Marko Krznaric (MK)

Data Types for Differential EquationsData Types for Differential Equations

2

AimAim

• Develop data types for ordinary differential equations.

• Solve initial value problem up to any given precision.

• In particular for:

Hybrid System= Discrete State Machine + Continuous Process

Differential Equation

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• Let IR={ [a,b] | a, b R} {R}

• (IR, ) is a bounded complete dcpo with R as bottom: ⊔iI ai = iI ai

• a ≪ b ao b

• (IR, ⊑) is -continuous: countable basis {[p,q] | p < q & p, q Q}

• (IR, ⊑) is, thus, a continuous Scott domain.• Scott topology has basis:

↟a = {b | ao b}

x {x}

R

I R

• x {x} : R IRTopological embedding

The Domain of nonempty compact Intervals of The Domain of nonempty compact Intervals of RR

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Domain for Domain for CC00 FunctionsFunctions

• f : [0,1] R, f C0[0,1], has continuous extension

If : [0,1] IR

x {f (x)}

• Scott continuous maps [0,1] IR with: f ⊑ g x R . f(x) ⊑ g(x)is another continuous Scott domain.

• : C0[0,1] ↪ ( [0,1] IR), with f Ifis a topological embedding into a proper subset of maximal elements of [0,1] IR .

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Step FunctionsStep Functions

• Single-step function: a↘b : [0,1] IR, with a I[0,1], b IR:

b x ao x otherwise

• Lubs of finite and bounded collections of single- step functions

⊔1in(ai ↘ bi)

are called step functions.

• Step functions with ai, bi rational intervals, give a basis for [0,1] IR

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Step Functions-An ExampleStep Functions-An Example

0 1

R

b1

a3

a2

a1

b3

b2

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Refining the Step FunctionsRefining the Step Functions

0 1

R

b1

a3

a2

a1

b3

b2

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Domain for Domain for CC11 Functions (AE&AL) Functions (AE&AL)

• If h C1[0,1] , then ( Ih , Ih ) ([0,1] IR) ([0,1] IR)

• What pairs ( f, g) ([0,1] IR)2 approximate a differentiable function?

• We can approximate ( Ih, Ih ) in ([0,1] IR)2

i.e. ( f, g) ⊑ ( Ih ,Ih ) with f ⊑ Ih and g ⊑ Ih

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Interval DerivativeInterval Derivative

• The interval derivative of f: [0,1] R is defined as

IR [0,1] :dx

df

] (f)(y)Dlim , (f)(y)Dlim [

dx

df uxyxy

l if both limits are finiteotherwise

yx

f(y)f(x)lim: (f)(x)D

yx

f(y)f(x)lim : (f)(x)D

xy

xyu

l

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Function and Derivative ConsistencyFunction and Derivative Consistency

• Theorem. If (f,g) Cons, there are least and greatest functions h with the above properties in each connected component of dom(g) which intersects dom(f) .

dx

dh

• Define the consistency relation:Cons ([0,1] IR) ([0,1] IR) with(f,g) Cons if there is a continuous h: dom(g) R

with f Ih ⊑ and g ⊑

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Approximating function: f = ⊔i ai↘bi

• (⊔i ai↘bi, ⊔j cj↘dj) Cons is a finitary property:

Consistency for basis elementsConsistency for basis elements

L(f,g) = least function

G(f,g)= greatest function

• We will define L(f,g), G(f,g) in general and show that:1. (f,g) Cons iff L(f,g) G(f,g).

2. Cons is decidable on the basis.

• Upgrading. Up(f,g) := (fg , g) where fg : t [ L(f,g)(t) , G(f,g)(t) ]

fg(t)

t

Approximating derivative: g = ⊔j cj↘dj

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f

1

1

Function and Derivative Information Function and Derivative Information

g

1

2

13

f

1

1

UpdatingUpdating

g

1

2

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• Let O be a connected component of dom(g) with O dom(f) . For x , y O define:

Consistency Test and Updating for Consistency Test and Updating for (f,g)(f,g)

yxduug

xyduug

yxdx

y

x

y

)(

)(

),(

• Define: L(f,g)(x) := supyOdom(f)(f –(y) + d–+(x,y)) and G(f,g)(x) := infyOdom(f)(f +(y) + d+–(x,y))

• Theorem. (f, g) Con iff x O. L(f, g) (x) G(f, g) (x).

yxduug

xyduug

yxdx

y

x

y

)(

)(

),(

• For x dom(g), let g({x}) = [g (x), g+(x)] where g , g+: dom(g) R are lower and upper semi-continuous.

Similarly we define f , f +: dom(f) R. Write f = [f –, f +].

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Updating Linear step FunctionsUpdating Linear step Functions

• A linear single-step function: a [↘ b–, b +] : [0,1] IR, with b–, b +: ao R linear

[b–(x) , b +(x)] x ao x otherwise

We write this simply as a b ↘ with b=[b–, b +] . x

b

ba

• Hence L(f,g) is the max of k+2 linear maps. Similarly, G(f,g).

• We get a linear time algorithm for computing L(f,g), G(f,g).

myy

• Proposition. For x O, we have: L(f,g)(x) = max {f –(x) , limsup f –(y) + d–+(x , y) | ym O dom(f) }

• For (f, g) = (⊔1in ai↘bi , ⊔1jm cj↘dj) with f linear g standard,

the rational end–points of ai and cj induce a partition y0 < y1 < y2 < … < yk of the connected component O of dom(g).

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f

1

1

Updating Algorithm(AE&MK) Updating Algorithm(AE&MK)

g

1

2

17

f

1

1

Updating Algorithm (left to right)Updating Algorithm (left to right)

g

1

2

18

f

1

1

Updating Algorithm (left to right)Updating Algorithm (left to right)

g

1

2

19

f

1

1

Updating Algorithm (right to left)Updating Algorithm (right to left)

g

1

2

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f

1

1

Updating Algorithm (right to left)Updating Algorithm (right to left)

g

1

2

21

f

1

1

Updating Algorithm (similarly for Updating Algorithm (similarly for upper one)upper one)

g

1

2

22

f

1

1

Output of the Updating Algorithm Output of the Updating Algorithm

g

1

2

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• Lemma. Cons ([0,1] IR)2 is Scott closed.

• Theorem.D1 [0,1]:= { (f,g) ([0,1]IR)2 | (f,g) Cons} is a continuous Scott domain, which can be given an effective structure.

The Domain of The Domain of CC11 Functions (AE&AL)Functions (AE&AL)

• Define D1c := {(f0,f1) C1C0 | f0 = f1 }

• Theorem. : C1[0,1] C0[0,1] ([0,1] IR)2 restricts to give a topological embedding

D1c ↪ D1

(with C1 norm) (with Scott topology)

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• Theorem. In a neighbourhood of t0, there is a unique solution, which is the unique fixed point of:

P: C0 [t0-k , t0+k] C0 [t0-k , t0+k]

f t . (x0 + v(t , f(t) ) dt)

for some k>0 .

t0

t

Picard TheoremPicard Theorem

• = v(t,x) with v: R2 R continuous

x(t0) = x0 with (t0,x0) R2

and v is Lipschitz in x uniformly in t for some neighbourhood of (t0,x0).

dt

dx

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• Up⃘�Apv: (f,g) (t . (x0 + g dt , t . v(t,f(t)))

has a fixed point (f,g) with f = g = t . v(t,f(t))

t

t0

Picard Solution ReformulatedPicard Solution Reformulated

• Up: (f,g) ( t . (x0 + g(t) dt) , g )t

t0

• P: f t . (x0 + v(t , f(t)) dt)

can be considered as upgrading the information about the function f and the information about its derivative g.

t

t0

• Apv: (f,g) (f , t. v(t,f(t)))

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• To obtain Picard’s theorem with domain theory, we have to make sure that derivative updating preserves consistency.

• (f , g) is strongly consistent, (f , g) S-Cons, if h ⊒ g we have: (f , h) Cons

• Q(f,g)(x) := supyODom(f) (f –(y) + d+–(x,y))

R(f,g)x) := infyODom(f) (f +(y) + d–+(x,y))

• Theorem. If f –, f +, g–, g+: [0,1] R are bounded and g–, g+ are continuous a.e. (e.g. for polynomial step functions f and g), then (f,g) is strongly consistent iff for any connected component O of dom(g) with O dom(f) , we have: x O. Q(f,g)(x), R(f,g)(x) [f –(x) , f +(x) ]

• Thus, on basis elements strong consistency is decidable.

A domain-theoretic Picard theoremA domain-theoretic Picard theorem

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Domain-theoretic Picard theorem (AE&AL)Domain-theoretic Picard theorem (AE&AL)

• Let v : [0,1] IR IR be Scott continuous and

Apv : ([0,1] IR)2 ([0,1] IR)2

(f,g) ( f , t. v (t , f(t) ))

Up : ([0,1] IR)2 ([0,1] IR)2 Up(f,g) = (fg , g) where fg (t) = [ L (f,g) (t) , G (f,g) (t) ]

• Consider any initial value f [0,1] IR with

(f, t. v (t , f(t) ) ) S-Cons

• Then the continuous map Up � Apv has a least fixed point above (f, t.v (t , f(t))) given by

(fs, gs) = ⊔n 0 (Up � Apv )n (f, t.v (t , f(t) ) )

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• Then (f , [-a,a ] ↘ [-M ,M ] ) S-Cons, hence (f, t. v(t , f(t) ) ) S-Cons since ([-a,a ] ↘ [-M ,M ]) ⊑ t. v (t , f(t) )

• Theorem. The domain-theoretic solution

(fs, gs) = ⊔n 0 (Up � Apv )n (f, t. v (t , f(t) ))

gives the unique classical solution through (0,0).

The Classical Initial Value ProblemThe Classical Initial Value Problem

• Suppose v = Ih for a continuous h : [-1,1] R R which satisfies the Lipschitz property around (t0,x0) =(0,0).

• Then h is bounded by M say in a compact rectangle K around the origin. We can choose positive a 1 such that [-a,a] [-Ma,Ma] K.

• Put f = ⊔n 0 fn where fn = [-a/2n,a /2n] ↘ [-Ma/2n , Ma/ 2n ]

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Computation of the solution for a given precision Computation of the solution for a given precision >0

• Let (un , wn) := (Pv )n (fn , t. vn (t , fn(t) ) ) with un = [un

- ,un+]n

• We express f and v as lubs of step functions:

f = ⊔n 0 fn v = ⊔n 0 vn

• Putting Pv := Up � Apv the solution is obtained as:

• For all n 0 we have: un- un+1

- un+1+ un

+ with un+ - un

- 0

• Compute the piecewise linear maps un- , un

+ until

the first n 0 with un+ - un

-

• (fs, gs) = ⊔n 0 (Pv )n (f , t. v (t , f(t) ) ) = ⊔n 0 (Pv )

n (fn , t. vn (t , fn(t) ) ) n

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ExampleExample

1

f

g

1

1

1

v

v is approximated by a sequence of step functions, v0, v1, …

v = ⊔i vi

We solve: = v(t,x), x(t0) =x0

for t [0,1] with

v(t,x) = t and t0=1/2, x0=9/8.

dt

dx

a3

b3

a2

b2

a1

b1

v3

v2

v1

The initial condition is approximated by rectangles aibi:

{(1/2,9/8)} = ⊔i aibi,

t

t

.

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SolutionSolution

1

f

g

1

1

1

At stage n we find un

- and un +

.

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SolutionSolution

1

f

g

1

1

1 .

At stage n we find un

- and un +

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SolutionSolution

1

f

g

1

1

1 un - and un

+ tend to

the exact solution:f: t t2/2 + 1

.

At stage n we find un

- and un +

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Computing with polynomial step functionsComputing with polynomial step functions

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Current and Further WorkCurrent and Further Work

• Solving Differential Equations with Domains

• Differential Calculus with Several Variables

• Implicit and Inverse Function Theorems

• Reconstruct Geometry and Smooth Mathematics with Domain Theory

• Continuous processes, robotics,…

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THE ENDTHE END

http://www.doc.ic.ac.uk/~aehttp://www.doc.ic.ac.uk/~ae