NestedSolve

Basic API

NestedSolve is a C++ class object needs to be instantiated (constructed) to set up a solver environment. This solver environment can be a local or member variable of the object it is used within.

The purpose of the solve environment is to store solver options, such as tolerances, and solver state, such as convergence reason or failure state of the previous solve. A solve environment can be reused for multiple solves.

The main API of NestedSolve is the nonlinear member function, which exists in two flavors:

1. NestedSolve::nonlinear(T & guess, L compute), which takes a writable reference to a variable if type T that contains an initial guess for the solution going into the function, and the solution once the call completes. compute is a lambda expression that takes references to the current solution value, the residual, and the Jacobian, and needs to update the latter two.

2. NestedSolve::nonlinear(T & guess, LR computeResidual, LJ computeJacobian) behaves largely similar to the first overload, but it takes two separate lambdas for computing the residual and the Jacobian independently. This enables the efficient use of line search and trust region algorithms internally.

Supported data types

The recommended data types for the solution/residual (T) and Jacobian are Eigen's dynamic size Matrix objects. the NestedSolve::Value<> and NestedSolve::Jacobian<> typedefs are available as shortcuts for those types.

Specializations and overloads exist for the nonlinear solve API to deal with the special cases of 1x1 systems (with the solution,residual, and Jacobians being Real scalars), and for 3x3 systems (with the solution and residual bein RealVectorValues and Jacobians being RankTwoTensor values). The correct overload is picked based on the type T of the initial guess parameter.

Checking convergence state

The state of a solve can be checked using the getSolve() method. This returns an enumeration describing how the solve converged or if it did not converge. The enumeration can take the following values:

• NestedSolve::State::NONE: No solve has begun. This is the initial value.

• NestedSolve::State::CONVERGED_ABS: The solve converged due to meeting an absolute tolerance.

• NestedSolve::State::CONVERGED_REL: The solve converged due to meeting a relative tolerance.

• NestedSolve::State::EXACT_GUESS: The solve converged due to the initial guess giving a zero residual.

• NestedSolve::State::NOT_CONVERGED: The solve did not converge.

Note that in the case that both the absolute and relative tolerances being met simultaneously, the value NestedSolve::State::CONVERGED_REL will be returned.

Basic usage

We first construct a NestedSolve object. This object can be reused for multiple solves, and could be a member of the class you are using it in.

NestedSolve solver;

Next we set the solution (vector) type. NestedSolve::Value<> is a dynamicly sized vector class from the Eigen library. Eigen uses the << operator to initialize such a vector (two components in this case).

NestedSolve::Value<> solution(2);
solution << 1.98, 1.02;

Next we set a custom relative tolerance. The default value here is 1e-8.

solver.setRelativeTolerance(1e-10);

Next we define the residual and Jacobian functions. Here we use a single lambda with three arguments:

1. The first argument is the current guess for the solution and is the _input_ to the functions, it is a vector of length N (or a scalar).

2. Argument two is a writable reference to the residual vector (of the same size as the current guess), the value of which is to be calculated by the lambda.

3. This is followed by the Jacobian matrix of the system, the derivative of the resdual vector with respect to the guess/solution.

auto compute = [&](const NestedSolve::Value<> & guess,
                   NestedSolve::Value<> & residual,
                   NestedSolve::Jacobian<> & jacobian) {
  residual(0) = guess(0) + guess(0) * guess(1) - 4;
  residual(1) = guess(0) + guess(1) - 3;

  jacobian(0, 0) = 1 + guess(1);
  jacobian(0, 1) = guess(0);
  jacobian(1, 0) = 1;
  jacobian(1, 1) = 1;
};

Note that jacobian(i,j) is the derivative of residual(i) with respect to guess(j).

Lastly we pass the initial guess along with the residual/Jacobian compute lambda to the NestedSolve::nonlinear method.

solver.nonlinear(solution, compute);

solution will now be updated from the initial guess to the actual solution of the system.

Then to check that the solve was successful, one can do the following:

if (solver.getState() == NestedSolve::State::NOT_CONVERGED)
{
  // Take some action for the case of no convergence.
}

Powell's Dogleg method solver

While the previous example used a single lambda to compute residual and Jacobian at the same time, we can instead change the code to have sparate lambdas to allow independent calculation of residual and Jacobian. This facilitates the use of solver methods that evaluate the residual more frequantly than the Jacobian, such as line search and trust region strategies.

auto computeResidual = [&](const NestedSolve::Value<> & guess,
                           NestedSolve::Value<> & residual) {
  residual(0) = guess(0) + guess(0) * guess(1) - 4;
  residual(1) = guess(0) + guess(1) - 3;
};

auto computeJacobian = [&](const NestedSolve::Value<> & guess,
                           NestedSolve::Jacobian<> & jacobian) {
  jacobian(0, 0) = 1 + guess(1);
  jacobian(0, 1) = guess(0);
  jacobian(1, 0) = 1;
  jacobian(1, 1) = 1;
};

We then use the three argument version of the nonlinear method

solver.nonlinear(solution, computeResidual, computeJacobian);