Reference

PDEOptimizationProblems.meta

A composite type that represents the main features of the PDE-constrained optimization problem. optimize ∫( f(θ, y, u) )dΩ subject to lvar ≤ (θ, y, u) ≤ uvar ∫( res(θ, y, u, v) )dΩ = 0 –- The following keys are valid: Problem meta

• domaindim: dimension of the domain 1/2/3 for 1D/2D/3D
• pbtype: in pbtypes
• nθ: size of the unknown vector
• ny: number of unknown function
• nu: number of control function

Solution meta

• optimalvalue: best known objective value (NaN if unknown, -Inf if unbounded problem)

Classification

• objtype: in objtypes
• contype: in contypes
• origin: in origins
• has_cvx_obj: true if the problem has a convex objective
• has_cvx_con: true if the problem has convex constraints
• has_bounds: true if the problem has bound constraints
• has_fixed_variables: true if it has fixed variables

Burger1d(;n :: Int = 512, kwargs...)

Let Ω=(0,1), we solve the one-dimensional ODE-constrained control problem: min{y,u} 0.5 ∫Ω​ |y(x) - yd(x)|^2dx + 0.5 * α * ∫Ω​ |u|^2 s.t. -ν y'' + yy' = u + h, for x ∈ Ω, y(0) = 0, y(1)=-1, for x ∈ ∂Ω, where the constraint is a 1D stationary Burger's equation over Ω, with h(x)=2(ν + x^3) and ν=0.08. The first objective measures deviation from the data y_d(x)=-x^2, while the second term regularizes the control with α = 0.01.

This example has been used in, Section 9.1, Estrin, R., Friedlander, M. P., Orban, D., & Saunders, M. A. (2020). Implementing a smooth exact penalty function for equality-constrained nonlinear optimization. SIAM Journal on Scientific Computing, 42(3), A1809-A1835.

The specificity of the problem:

• quadratic objective function;
• nonlinear constraints with AD jacobian;

Suggestions:

• FEOperatorFromTerms has only one term. We might consider splitting linear and

nonlinear terms.

Mairet, F., & Bayen, T. (2021). The promise of dawn: microalgae photoacclimation as an optimal control problem of resource allocation. Journal of Theoretical Biology, 515, 110597.

Using a photosynthetic rate proportional to the photosynthetic apparatus mass fraction.

Mairet, F., & Bayen, T. (2021). The promise of dawn: microalgae photoacclimation as an optimal control problem of resource allocation. Journal of Theoretical Biology, 515, 110597.

Using Michaelis-Menten's function for the photosynthetic rate.

controlelasticmembrane1(; n :: Int = 10, args...)

Let Ω = (-1,1)^2, we solve the following distributed Poisson control problem with Dirichlet boundary:

min{y ∈ H^10,u ∈ H^1} 0.5 ∫Ω​ |y(x) - yd(x)|^2dx + 0.5 * α * ∫Ω​ |u|^2 s.t. -Δy = h + u, for x ∈ Ω y = 0, for x ∈ ∂Ω umin(x) <= u(x) <= umax(x) where yd(x) = -x[1]^2 and α = 1e-2. The force term is h(x1,x2) = - sin( ω x1)sin( ω x2) with ω = π - 1/8. In this first case, the bound constraints are constants with umin(x) = 0.0 and umax(x) = 1.0.

controlelasticmembrane2(; n :: Int = 10, args...)

Let Ω = (-1,1)^2, we solve the following distributed Poisson control problem with Dirichlet boundary:

min{y ∈ H^10,u ∈ H^1} 0.5 ∫Ω​ |y(x) - yd(x)|^2dx + 0.5 * α * ∫Ω​ |u|^2 s.t. -Δy = h + u, for x ∈ Ω y = 0, for x ∈ ∂Ω umin(x) <= u(x) <= umax(x) where yd(x) = -x[1]^2 and α = 1e-2. The force term is h(x1,x2) = - sin( ω x1)sin( ω x2) with ω = π - 1/8. In this second case, the bound constraints are umin(x) = x1+x2 and umax(x) = x1^2+x2^2 applied at the midpoint of the cells.

incompressibleNavierStokes(; n :: Int64 = 3, kargs...)

This corresponds to the incompressible Navier-Stokes equation described in the Gridap Tutorials: https://gridap.github.io/Tutorials/stable/pages/t008incnavier_stokes/

It has no objective function and no control, just the PDE.

inversePoissonproblem2d(;n :: Int = 512, kwargs...)

Let Ω=(-1,1)^2, we solve the 2-dimensional PDE-constrained control problem: min{y ∈ H1^0, u ∈ L^∞} 0.5 ∫Ω​ |y(x) - yd(x)|^2dx + 0.5 * α * ∫Ω​ |u|^2 s.t. -∇⋅(z∇u) = h, for x ∈ Ω, u(x) = 0, for x ∈ ∂Ω. Let c = (0.2,0.2) and and define S1 = {x | ||x-c||2 ≤ 0.3 } and S2 = {x | ||x-c||1 ≤ 0.6 }. The target ud is generated as the solution of the PDE with z*(x) = 1 + 0.5 * I{S1}(x) + 0.5 * I{S2}(x). The force term here is h(x1,x2) = - sin( ω x1)sin( ω x2) with ω = π - 1/8. The control variable z represents the diffusion coefficients for the Poisson problem that we are trying to recover. Set α = 10^{-4} and discretize using P1 finite elements on a uniform mesh of 1089 triangles and employ an identical discretization for the optimization variables u, thus ncon = 1089 and npde = 961. Initial point is y0=1 and u_0 = 1. z ≥ 0 (implicit)

This example has been used in, Section 9.2, Estrin, R., Friedlander, M. P., Orban, D., & Saunders, M. A. (2020). Implementing a smooth exact penalty function for equality-constrained nonlinear optimization. SIAM Journal on Scientific Computing, 42(3), A1809-A1835.

https://arxiv.org/pdf/2103.14552.pdf Example 1. MEMBRANE Multilevel Active-Set Trust-Region (MASTR) Method for Bound Constrained Minimization Alena Kopaničáková and Rolf Krause

The solution and original problem is given in Domorádová, M., & Dostál, Z. (2007). Projector preconditioning for partially bound‐constrained quadratic optimization. Numerical Linear Algebra with Applications, 14(10), 791-806.

Let Ω=(0,1)^2, we solve the unconstrained optimization problem: min{u ∈ H1^0} 0.5 ∫_Ω​ |∇u|^2 - w u dx s.t. u(x) = 0, for x ∈ ∂Ω whre w(x)=1.0.

The minimizer of this problem is the solution of the Poisson equation: ∫_Ω​ (∇u ∇v - f*v)dx = 0, ∀ v ∈ Ω u = 0, x ∈ ∂Ω

This example has been used in Exercice 10.2.4 (p. 308) of G. Allaire, Analyse numérique et optimisation, Les éditions de Polytechnique

poisson3d(; n :: Int = 10)

This example represents a Poisson equation with Dirichlet boundary conditions over the 3d-box, (0,1)^3, and we minimize the squared H_1-norm to a manufactured solution. So, the minimal value is expected to be 0.

It is inspired from the 2nd tutorial in Gridap.jl: https://gridap.github.io/Tutorials/stable/pages/t002_validation/

poissonBoltzman2d(; n :: Int = 100)

Let Ω=(-1,1)^2, we solve the 2-dimensional PDE-constrained control problem: min{y ∈ H1^0, u ∈ L^∞} 0.5 ∫Ω​ |y(x) - yd(x)|^2dx + 0.5 * α * ∫Ω​ |u|^2 s.t. -Δy + sinh(y) = h + u, for x ∈ Ω y(x) = 0, for x ∈ ∂Ω

The force term here is h(x1,x2) = - sin( ω x1)sin( ω x2) with ω = π - 1/8. The targeted function is yd(x) = {10 if x ∈ [0.25,0.75]^2, 5 otherwise}. We discretize using P1 finite elements on a uniform mesh with 10201 triangles, resulting in a problem with n = 20002 variables and m = 9801 constraints. We use y0=1 and u0 = 1 as the initial point.

This example has been used in, Section 9.3, Estrin, R., Friedlander, M. P., Orban, D., & Saunders, M. A. (2020). Implementing a smooth exact penalty function for equality-constrained nonlinear optimization. SIAM Journal on Scientific Computing, 42(3), A1809-A1835.

The specificity of the problem:

• quadratic objective function;
• nonlinear constraints with AD jacobian;

smallestLaplacianeigenvalue(; n :: Int = 10, args...)

We solve the following problem:

min{u,z} ∫Ω​ |∇u|^2 s.t. ∫_Ω​ u^2 = 1, for x ∈ Ω u = 0, for x ∈ ∂Ω

The solution is an eigenvector of the smallest eigenvalue of the Laplacian operator, given by the value of the objective function. λ is an eigenvalue of the Laplacian if there exists u such that

Δu + λ u = 0, for x ∈ Ω u = 0, for x ∈ ∂Ω

This example has been used in Exercice 10.2.11 (p. 313) of G. Allaire, Analyse numérique et optimisation, Les éditions de Polytechnique and more eigenvalue problems can be found in Section 7.3.2

TODO:

• does the 1 work as it is? or should it be put in lcon, ucon?
• it is 1D for now.