We are very happy to announce the publication in the Journal of Open Source Software of the paper PDENLPModels.jl: A NLPModel API for optimization problems with PDE-constraints. It is accessible in open access here.
PDENLPModels.jl is a Julia package that specializes the NLPModel API for modeling and discretizing optimization problems with mixed algebraic and PDE in the constraints.
We consider optimization problems of the form: find functions \(y, u\) and \(κ \in \mathbb{R}^n\) satisfying
\[ \begin{array}{lll} \underset{y, u, \theta}{\text{minimize}} \int_\Omega J(y, u, \theta)d\Omega \ \text{ subject to} & e(y, u, \theta) = 0, & \text{(governing PDE on $\Omega$)} \\ & l_{yu} \leq (y, u) \leq u_{yu}, & \text{(functional bound constraints)} \\ & l_{\theta} \leq \theta \leq u_{\theta}, & \text{(bound constraints)} \end{array} \]
The main challenges in modeling such a problem are to be able to discretize the domain and generate corresponding discretizations of the objective and constraints, and their evaluate derivatives with respect to all variables. We use Gridap.jl to define the domain, meshes, function spaces, and finite-element families to approximate unknowns, and to model functionals and sets of PDEs in a weak form. PDENLPModels extends Gridap.jl's differentiation facilities to also obtain derivatives useful for optimization, i.e., first and second derivatives of the objective and constraint functions with respect to controls and finite-dimensional variables.
After discretization of the domain \(\Omega\), the integral, and the derivatives, the resulting problem is a nonlinear optimization problem. PDENLPModels exports the GridapPDENLPModel type, an instance of an AbstractNLPModel, as defined in NLPModels.jl, which provides access to objective and constraint function values, to their first and second derivatives, and to any information that a solver might request from a model. The role of NLPModels.jl is to define an API that users and solvers can rely on. It is the role of other packages to implement facilities that create models compliant with the NLPModels API. We refer to juliasmoothoptimizers.github.io for tutorials on the NLPModel API.
As such, PDENLPModels offers an interface between generic PDE-constrained optimization problems and cutting-edge optimization solvers such as Artelys Knitro via NLPModelsKnitro.jl, Ipopt via NLPModelsIpopt.jl , DCISolver.jl, Percival.jl, and any solver accepting an AbstractNLPModel as input, see JuliaSmoothOptimizers.
The following example shows how to solve a Poisson control problem with Dirichlet boundary conditions using DCISolver.jl: find functions \(y \in H^1_0\) and \(u \in H^1\) satisfying
\[ \begin{array}{lll} \underset{y, u}{\text{minimize}} \int_{(-1,1)^2} \frac{1}{2}\|y_d - y\|^2 +\frac{\alpha}{2}\|u\|^2 d\Omega \quad \text{subject to} & \Delta y - u - h = 0, & \text{on } \Omega.\\ & y = 0, & \text{on } \partial\Omega, \end{array} \]
for some given functions \(y_d:(-1,1)^2 \rightarrow \mathbb{R}\) and \(h:(-1,1)^2 \rightarrow \mathbb{R}\), and \(\alpha > 0\).
using DCISolver, Gridap, PDENLPModels
# Cartesian discretization of Ω=(-1,1)² in 100² squares.
Ω = (-1, 1, -1, 1)
model = CartesianDiscreteModel(Ω, (100, 100))
fe_y = ReferenceFE(lagrangian, Float64, 2) # Finite-elements for the state
Xpde = TestFESpace(model, fe_y; dirichlet_tags = "boundary")
Ypde = TrialFESpace(Xpde, x -> 0.0) # y is 0 over ∂Ω
fe_u = ReferenceFE(lagrangian, Float64, 1) # Finite-elements for the control
Xcon = TestFESpace(model, fe_u)
Ycon = TrialFESpace(Xcon)
dΩ = Measure(Triangulation(model), 1) # Gridap's integration machinery
# Define the objective function f
yd(x) = -x[1]^2
f(y, u) = ∫(0.5 * (yd - y) * (yd - y) + 0.5 * 1e-2 * u * u) * dΩ
# Define the constraint operator in weak form
h(x) = -sin(7π / 8 * x[1]) * sin(7π / 8 * x[2])
c(y, u, v) = ∫(∇(v) ⊙ ∇(y) - v * u - v * h) * dΩ
# Define an initial guess for the discretized problem
x0 = zeros(num_free_dofs(Ypde) + num_free_dofs(Ycon))
# Build a GridapPDENLPModel, which implements the NLPModel API.
name = "Control elastic membrane"
nlp = GridapPDENLPModel(x0, f, dΩ, Ypde, Ycon, Xpde, Xcon, c, name = name)
dci(nlp, verbose = 1) # solve the problem with DCI
Migot, T., Orban D., & Siqueira A. S. PDENLPModels.jl: A NLPModel API for optimization problems with PDE-constraints Journal of Open Source Software 7(80), 4736 (2022). 10.21105/joss.04736