Abbas Edalat

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Domain-theoretic Solution of Differential
Equations

• Abbas Edalat
• Imperial College London
• www.doc.ic.ac.uk/ae
• joint work with Marko Krznaric and Andre Lieutier

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Aim

• Develop data types for ordinary differential
  equations.
• Solve initial value problems up to any given
  precision

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The Domain of Intervals of R

• Let IR a,b a, b ? R ? R
• (IR, ?) is a cpo with R as bottom ? and ?i ?0
  ai ?i ?0 ai
• (IR, ?) is a continuous Scott domain
• countable basis p,q p lt q p, q ? Q

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Data-type for Functions

• Lubs of finite and bounded collections of single-
  step functions
  
• ?1 ? i ? n(ai ? bi)
• are called step functions.

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Step Functions-An Example
R
b3
a3
b1
b2
a1
a2
0
1
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Refining the Step Functions
R
b3
a3
b1
a1
b2
a2
0
1
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Domain for Continuous Functions

• Partial order on functions 0,1 ? IR f ?
  g ? ?x ? R . f(x) ? g(x)
• (0,1 ? IR, ?) is a continuous Scott domain.
• Step functions, with ai, bi rational intervals,
  give a basis for 0,1 ? IR
• f ? If C00,1 ? ( 0,1 ? IR) is an embedding
  into a subset of maximal elements of 0,1 ? IR .

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Domain for Differentiable Functions

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

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Function and Derivative Consistency

• Theorem. (f,g) ? Cons iff there is a least
  function L(f,g) and a greatest function G(f,g)
  with the above properties in each connected
  component of dom(g) which intersects dom(f) .

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Consistency for basis elements

• (?1 ? i ? n ai ? bi , ?1 ? j ? m cj ? dj) ?
  Cons is decidable

• Updating. Up(f,g) (fg , g) where
  fg t ? L(f,g)(t) , G(f,g)(t)

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Function and Derivative Information
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Updating
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Updating Algorithm
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Updating Algorithm (left to right)
f
1
1
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Updating Algorithm (left to right)
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Updating Algorithm (right to left)
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Updating Algorithm (right to left)
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Dually for the upper boundary
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Output of the Updating Algorithm
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The Domain of Differentiable Functions

• Theorem.D1 0,1 (f,g) ? (0,1?IR)2 (f,g)
  ? Cons is a continuous Scott domain.

• Theorem. C10,1 embeds into the set of maximal
  elements of D1 0,1

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Solving Initial Value Problems

• The function v is bounded by M say in a rectangle
  K around the origin. Take positive alt1, say,
  such that -a,a ?-Ma,Ma ? K.
• The initial condition x(0) 0 is captured by the
  Scott continuous map
• f ?n ?0 fn where fn -a/2n,a
  /2n ? -Ma/2n , Ma/ 2n

• This is the initial function approximation.
• It also gives the initial derivative
  approximation
  ?t. v (t , f(t) ))

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Function and Derivative Upgrading

• Derivative upgrading
• Apv (-1,1 ? IR)2 ? (-1,1 ?
  IR)2 (f,g) ? ( f ,
  ?t. v (t , f(t) ))
• Function upgrading
• Up (-1,1 ? IR)2 ? (-1,1 ?
  IR)2

Up(f,g) (fg , g) where fg (t) L
(f,g) (t) , G (f,g) (t)

• Solve the ODE by iterating Up ? Apv on D1 -1,1
  starting with
  (f, ?t. v (t , f(t) ))
• Theorem. The domain-theoretic solution
  
  ?n ?0 (Up ? Apv )n (f, ?t. v (t , f(t) ))
  
• is the unique classical solution through
  (0,0).

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Computation of the solution for a given precision
? gt0

• 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- ? un1- ? un1 ?
  un with un - un- ? ?t. 0
• Compute the piecewise linear maps un- , un until
  the
  first n ?0 with un - un- ? ?

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Example
v is approximated by a sequence of step
functions, v0, v1, v ?i vi
.
t
The initial condition is approximated by
rectangles ai?bi (1/2,9/8) ?i ai?bi,
v
t
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Solution
At stage n we find un - and un
.
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Solution
At stage n we find un - and un
.
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Solution
At stage n we find un - and un
.
un - and un tend to the exact solutionf t ?
t2/2 1
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Computing with polynomial step functions
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Current and Further Work

• Implementation in Haskell
• Differential Calculus with Several Variables
  PDEs
• Construction of Smooth Curves and Surfaces
• Hybrid Systems, robotics,

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