Interval extensions are used to compute with partially known values defined by intervals

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Part III, F. Fages

Interval Methods for Reachability Analysis Examination - Course C2-19, MPRI

March 3 2015

In this examination, we study interval constraint propagation methods to verify the reachability of a set of states (defined by intervals of values for each variable) from a given set of initial states (defined similarly by intervals) in a system of Ordinary Differential Equations (ODE) defined for instance by a set of biochemical reactions with partially known initial concentrations.

Interval extensions are used to compute with partially known values defined by intervals. For instance, the function [a,b] #+ [a’,b’] = [a+a’,b+b’] is an interval extension of the addition.

Question 1. Give interval extensions for substraction and multiplication over the reals.

Answer.

[a,b] #- [a’,b’] = [a-b’, b-a’]

[a,b] #* [a’,b’] = [min(a*a’, a*b’, b*a’, b*b’), max(a*a’, a*b’, b*a’, b*b’)]

Question 2. Apply the interval extensions above to compute an interval for z = 1-x*y when x∈[-2,2] and y∈[-2,2]

Answer.

y ∈ [1,1] #- ( [-2,2] #* [-2,2] ) = [1,1] #- [-4,4] = [-3, 5]

For an ODE system

and we consider an interval extension of the solution, i.e. a function #y_i

Given interval assignments B ⃗0 and I_t (noted X₀ and T in the Figure) on ⃗0 and t respectively, one can use # ⃗⃗ to compute a refinement of the interval assignment on ⃗t as follows:

That refined domain can be computed by iterating with some time step the following

For reasoning backward, one can consider the reverse ODE with function ⃗- defined as - ⃗:

Question 3. Define similarly the operator Prune_bwd(B ⃗0, ⃗) for refining B ⃗0

Answer.

.

Question 4. Define the operator Prune_time(I_t, ⃗) for refining It

Answer.

Question 5. Give an algorithm to compute Prunetime(It, ⃗) with some time step Answer.