Nonlinear Problems
The Four Types of Nonlinear Problems
NonlinearSolve.jl tackles four related types of nonlinear systems:
- Interval rootfinding problems. I.e., find the $t \in [t_0, t_f]$ such that $f(t) = 0$.
- Systems of nonlinear equations, i.e., find the $u$ such that $f(u) = 0$.
- Steady state problems, i.e., find the $u$ such that $u' = f(u,t)$ has reached steady state, i.e., $0 = f(u, ∞)$.
- The nonlinear least squares problem, which is an under/over-constrained nonlinear system which might not be satisfiable, i.e. there may be no $u$ such that $f(u) = 0$, and thus we find the $u$ which minimizes $\|f(u)\|_2^2$ in the least squares sense.
The first is for solving scalar rootfinding problems, i.e., finding a single number, and requires that a bracketing interval is known. For a bracketing interval, one must have that the sign of $f(t_0)$ is opposite the sign of $f(t_f)$, thus guaranteeing a root in the interval.
Interval rootfinding problems allow for $f$ to return an array, in which case the interval rootfinding problem is interpreted as finding the first $t$ such that any of the components of the array hit zero.
The second type of nonlinear system can be multidimensional, and thus no ordering nor boundaries are assumed to be known. For a system of nonlinear equations, $f$ can return an array, and the solver seeks the value of $u$ for which all outputs of $f$ are simultaneously zero.
The third type is equivalent to a nonlinear system, but with the extra interpretation of having a potentially preferred unique root. That is, when there are multiple u such that $f(u) = 0$, the NonlinearProblem does not have a preferred solution, while for the SteadyStateProblem the preferred solution is the $u(∞)$ that would arise from solving the ODE $u' = f(u,t)$.
The fourth type is an overdetermined nonlinear system, which has more constraints than free variables, and thus is usually not possible to solve exactly. In these contexts, it is usually convenient to minimize the Euclidean norm, as it is continuously differentiable if the original function is.
Most solvers for SteadyStateProblem do not guarantee the preferred solution and instead will solve for some u in the set of solutions. The documentation of the nonlinear solvers will note if they return the preferred solution.
Problem Construction Details
The problem types are defined and documented by SciMLBase:
IntervalNonlinearProblemNonlinearProblemSteadyStateProblemNonlinearLeastSquaresProblemHomotopyProblem