Minimax optimization of the residual of a differential equation.

We can’t solve differential equations $Ly = 0$ exactly usually. We need to work in some finite subspace of the full function space $ y(t) = \sum_i a_i f_i(t)$. A common criteria is to find a solution that is closest to obeying the differential equation in a least squares sense, say $ \min (Ly)^2 $. This is nice because it leads to linear system of equations. However, a minimax objective $\min \max |Ly| $ is also feasible using the sum of squares method. See here for more

We can write down the optimization problem using a finite polynomial parametrization of our solution. We relax the condition of being some of squares everywhere to instead just a region of interest by adding a term that makes the inequality stricter in the domain and looser outside the domain. The domain is described by a polynomial expression $t (1 - t) $ which is positive when $ 0 \leq t \leq 1$ and negative otherwise. Here is an example for

\[\frac{d^2 y}{dt^2}=-y\] \[y(0)=1\] \[y'(0) = 0\]

with exact solution $ \cos(t) $

using JuMP
using SumOfSquares
using DynamicPolynomials
using SCS

@polyvar t
T = monomials(t, 0:4)
model = SOSModel(with_optimizer(SCS.Optimizer))
@variable(model, y, Poly(T))
@variable(model, α)
dy = differentiate(y, t)
ddy = differentiate(dy, t)
domain = t*(π/2-t)
@variable(model, λ_1 , SOSPoly(T))
@variable(model, λ_2 , SOSPoly(T))
@constraint(model, y(t => 0) == 1)
@constraint(model, dy(t => 0) == 0)
@constraint(model, ddy + y - λ_1*domain >= -α)
@constraint(model, α >= ddy + y + λ_2*domain)

@objective(model, Min, α)


# $$ 0.027642031145745472t^{4} + 0.021799794207213376t^{3} - 0.5066442977156951t^{2} + 3.506190174561713e-8t + 1.0000000041335204 $$

using Plots
xs = 0:0.01:π/2; exact_y = cos.(xs); approx_y = map(x -> value(y)(t => x), xs)# These are the plotting data

original link:

Huh. This doesn’t embed very well. Maybe you’re better off just clicking into the thing. It’s nice not to let things rot too long though. shrug

Other ideas: Can I not come up with some scheme to use Sum of Squares for rigorous upper and lower bound regions like in ? Maybe a next post.