math: solve_pde function for PDE problems

Summary:

Use solve_pde to solve PDEs.

Description:

The design involves cmdstan, stan, and math. See also https://github.com/stan-dev/stan/issues/2567. This issue is about Math repo only.

In order to solve PDE, similar to ODE solver in Stan, the PDE is provided through a functor. Unlike ODE solver, we cannot ship a PDE library with Stan Math. Alternatively, the user is asked to provide his own PDE solver. His workflow could be the following.

• provide a functor that uses external PDE solver. The functor has the following signature

  inline std::vector<std::vector<double> >
  operator()(const std::vector<double>& theta,
             const bool need_sens,
             const std::vector<double>& x_r,
             const std::vector<int>& x_i,
             std::ostream* msgs = nullptr);

The functor returns PDE solution(quantity of interest) when need_sens=0, and returns PDE solution together with its gradient with respect to theta(parameters) when need_sens=1.

• call solve_pde, with the above functor as the first argument. Within the solve_pde function, the above functor calculates PDE solution and sensitivity, andvar variables are constructed based on these values.
• provide Makefile for the external PDE solver. See separate issue #934 .

The user/PDE developer does not need to know utilize stan::math::var to write the above functor. He just needs to decide the behavior of the PDE solver: how the solution and the sensitivity are calculated. solve_pde should be able to take these values and construct proper var items, possibly using precomputed_gradient.

Additional Information:

Provide any additional information here.

Current Version:

v2.17.0