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I have a dynamics problem where I need to fetch data (e.g. using linear interpolation) within the derivative. Is such a thing possible?
using ReachabilityAnalysis, Interpolations
tspan = (0.0, 10.0)
f(x) = x^2 # real f
xs = range(tspan[1], tspan[2], length = 20);
data = f.(xs)
Lin_interp = LinearInterpolation(data, XS) # interpolated f
@taylorize function f_real(du, u, p, t)
du[1] = f(t)
end
@taylorize function f_interp(du, u, p, t)
du[1] = Lin_interp(xs)
end
X0 = Interval(0, 0.1)
prob = @ivp(x' = f_real(x), x(0) ∈ X0, dim=1)
prob_interp = @ivp(x' = f_interp(x), x(0) ∈ X0, dim=1)
sol = solve(prob, tspan = (0.0, 10.0), alg = TMJets21a(abstol = 1e-10))
sol_interp = solve(prob_interp, tspan = (0.0, 10.0), alg = TMJets21a(abstol = 1e-10))
plot(sol,vars = (0, 1))
plot(sol_interp, vars = (0, 1))
When running solve with the interpolated version, it tries to evaluate the function with a Taylor series.
My initial thoughts on doing this is that we could take the bounding box of the Taylor model, and evaluate the interpolator with it. Perhaps using interval arithmetic, but for the case of linear interpolations I believe we can get the exact bounds, as either the endpoints or the maximum and minimum data value within the range. i.e. something like this:
using IntervalArithmetic: interval
function interpolate(Lin_interp, x::Interval)
grid = Lin_interp.itp.knots[1]
data = Lin_interp.itp.coefs
data_in_x = data[grid .∈ x]
x_lo = Lin_interp(x.lo)
x_hi = Lin_interp(x.hi)
points = [data_in_x; x_lo; x_hi]
return interval(minimum(points), maximum(points))
end
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