Implemented QDEIM and ODEIM algorithms

[smallpondtom]

Checklist

• Appropriate tests were added
• Any code changes were done in a way that does not break public API
• All documentation related to code changes were updated
• The new code follows the
  contributor guidelines, in particular the SciML Style Guide and
  COLPRAC.
• Any new documentation only uses public API

Additional context

I implemented the QDEIM algorithm (Drmac and Gugercin, 2016) and ODEIM algorithm (Peherstorfer, Drmac, and Gugercin, 2020). I used the FHN model to verify my results which you can reproduce with the following code:

using ModelingToolkit, MethodOfLines, DifferentialEquations, ModelOrderReduction
using CairoMakie

## firstly construct a ModelingToolkit.PDESystem for the FitzHugh-Nagumo model
@variables x t v(..) w(..)
Dx = Differential(x)
Dxx = Dx^2
Dt = Differential(t)
const L = 1.0
const ε = 0.015
const b = 0.5
const γ = 2.0
const c = 0.05
f(v) = v * (v - 0.1) * (1.0 - v)
i₀(t) = 50000.0t^3 * exp(-15.0t)
eqs = [ε * Dt(v(x, t)) ~ ε^2 * Dxx(v(x, t)) + f(v(x, t)) - w(x, t) + c,
    Dt(w(x, t)) ~ b * v(x, t) - γ * w(x, t) + c]
bcs = [v(x, 0.0) ~ 0.0, w(x, 0) ~ 0.0, Dx(v(0, t)) ~ -i₀(t), Dx(v(L, t)) ~ 0.0]
domains = [x ∈ (0.0, L), t ∈ (0.0, 14.0)]
ivs = [x, t]
dvs = [v(x, t), w(x, t)]
pde_sys = PDESystem(eqs, bcs, domains, ivs, dvs; name = Symbol("FitzHugh-Nagumo"))

## transfer to a ModelingToolkit.ODESystem by automated discretization via MethodOfLines
N = 15 # equidistant discretization intervals
dx = (L - 0.0) / N
dxs = [x => dx]
discretization = MOLFiniteDifference(dxs, t)
ode_sys, tspan = symbolic_discretize(pde_sys, discretization)
simp_sys = structural_simplify(ode_sys)
ode_prob = ODEProblem(simp_sys, nothing, tspan)

## solve the full-order model to get snapshots
sol = solve(ode_prob, Tsit5())
snapshot_simpsys = Array(sol.original_sol)

## set POD and DEIM dimensions
# apply POD-DEIM to obtain the reduced-order model
pod_dim = deim_dim = 5
deim_sys = deim(simp_sys, snapshot_simpsys, pod_dim; deim_dim = deim_dim, interpolation_algo=:deim)
deim_prob = ODEProblem(deim_sys, nothing, tspan)
deim_sol = solve(deim_prob, Tsit5())

## retrieve the approximate solution of the original full-order model
sol_deim_x = deim_sol[x]
sol_deim_v = deim_sol[v(x, t)]
sol_deim_w = deim_sol[w(x, t)]

## apply POD-QDEIM to obtain the reduced-order model
deim_sys = deim(simp_sys, snapshot_simpsys, pod_dim; deim_dim = deim_dim, interpolation_algo=:qdeim)
deim_prob = ODEProblem(deim_sys, nothing, tspan)
deim_sol = solve(deim_prob, Tsit5())

## retrieve the approximate solution of the original full-order model
sol_qdeim_x = deim_sol[x]
sol_qdeim_v = deim_sol[v(x, t)]
sol_qdeim_w = deim_sol[w(x, t)]

## apply POD-ODEIM to obtain the reduced-order model
deim_sys = deim(simp_sys, snapshot_simpsys, pod_dim; deim_dim = deim_dim, interpolation_algo=:odeim, odeim_dim=8)
deim_prob = ODEProblem(deim_sys, nothing, tspan)
deim_sol = solve(deim_prob, Tsit5())

## retrieve the approximate solution of the original full-order model
sol_odeim_x = deim_sol[x]
sol_odeim_v = deim_sol[v(x, t)]
sol_odeim_w = deim_sol[w(x, t)]

## Plot
with_theme(theme_latexfonts()) do
    fig = Figure()
    ax = Axis3(fig[1, 1], xlabel="x", ylabel="v(x,t)", zlabel="w(x,t)", 
               title="FHN model comparison of full and reduced systems",
               xlabelsize=16, ylabelsize=16, zlabelsize=16)
    scatter!(ax, sol[x], sol[v(x, t)], sol[w(x, t)], label="Full32", markersize=8)
    scatter!(ax, sol_deim_x, sol_deim_v, sol_deim_w, label="POD5-DEIM5", markersize=8)
    scatter!(ax, sol_qdeim_x, sol_qdeim_v, sol_qdeim_w, label="POD5-QDEIM5", markersize=6)
    scatter!(ax, sol_odeim_x, sol_odeim_v, sol_odeim_w, label="POD5-ODEIM8", markersize=5)
    axislegend(ax, position=:rt, nbanks=2)
    display(fig)
end

The final results will look as follows:

Testing issue:
The test file test/deim.jl seems to fail due to something within the package itself that I couldn't quite identify. But, I did verify that the new algorithms are bug-free. Please let me know if you need further information before merging this PR!

[bowenszhu]

Thank you! I will review this after I complete

• Update dependencies versions and fix CI #89

[smallpondtom]

@bowenszhu Any updates on this?

[bowenszhu]

Apologies for the delay. I haven't forgotten about this PR, and I'll take a look as soon as I can.

[smallpondtom]

Gotcha. No worries. Thanks for the response

[smallpondtom]

Hi @bowenszhu! It's been a while but I happened to remember I made this pull request. Would it be possible to merge it? Thanks!