Restarting Linear Solutions for Expensive Linear Solves¶
When using iterative linear solvers, it is often desirable to use the converged solution from a previous linear solve as the initial guess for the current one. There is some memory cost associated with this feature, because the solution for each quantity of interest will be saved separately. However, the benefit is reduced computational cost for the subsequent linear solves.
This feature should not be used when using the DirectSolver at the top level of your model. It won’t offer any computational savings in that situation.
If you have implemented the
solve_linear() method for an ImplicitComponent, then you will need to make sure to use the provided guess solution in your implementation. The cached solution will be put into the solution vector for you to use as an initial guess. Note that you will override those values with the final solution. In
fwd mode, the guess will be in the
d_outputs vector. In
rev mode, the guess will be in the
Below is a toy example problem that illustrates how the restart vectors should work. The restart is passed in via the
x0 argument to gmres.
from distutils.version import LooseVersion import numpy as np import scipy from scipy.sparse.linalg import gmres import openmdao.api as om class QuadraticComp(om.ImplicitComponent): """ A Simple Implicit Component representing a Quadratic Equation. R(a, b, c, x) = ax^2 + bx + c Solution via Quadratic Formula: x = (-b + sqrt(b^2 - 4ac)) / 2a """ def setup(self): self.add_input('a', val=1.) self.add_input('b', val=1.) self.add_input('c', val=1.) self.add_output('states', val=[0,0]) def setup_partials(self): self.declare_partials(of='*', wrt='*') def apply_nonlinear(self, inputs, outputs, residuals): a = inputs['a'] b = inputs['b'] c = inputs['c'] x = outputs['states'] y = outputs['states'] residuals['states'] = a * x ** 2 + b * x + c residuals['states'] = a * y + b def solve_nonlinear(self, inputs, outputs): a = inputs['a'] b = inputs['b'] c = inputs['c'] outputs['states'] = (-b + (b ** 2 - 4 * a * c) ** 0.5) / (2 * a) outputs['states'] = -b/a def linearize(self, inputs, outputs, partials): a = inputs['a'] b = inputs['b'] c = inputs['c'] x = outputs['states'] y = outputs['states'] partials['states', 'a'] = [[x**2],[y]] partials['states', 'b'] = [[x],] partials['states', 'c'] = [[1.0],] partials['states', 'states'] = [[2*a*x+b, 0],[0, a]] self.state_jac = np.array([[2*a*x+b, 0],[0, a]]) def solve_linear(self, d_outputs, d_residuals, mode): if mode == 'fwd': print("incoming initial guess", d_outputs['states']) d_outputs['states'] = gmres(self.state_jac, d_residuals['states'], x0=d_outputs['states']) elif mode == 'rev': d_residuals['states'] = gmres(self.state_jac, d_outputs['states'], x0=d_residuals['states']) p = om.Problem() p.driver = om.ScipyOptimizeDriver() p.driver.options['optimizer'] = 'SLSQP' p.model.set_input_defaults('b', val=4.) p.model.add_subsystem('quad', QuadraticComp(), promotes_inputs=['a', 'b', 'c'], promotes_outputs=['states']) p.model.add_subsystem('obj', om.ExecComp('y = (x-x)**2', x=np.ones(2))) p.model.connect('states', 'obj.x') p.model.add_design_var('a', upper=4., cache_linear_solution=True) p.model.add_constraint('states', upper=10) p.model.add_objective('obj.y') p.setup(mode='fwd') p.run_driver() print(p['a'], p['b'], p['c']) print(p['states']) print(p['obj.y'])
incoming initial guess [0. 0.]
incoming initial guess [-0.02072594 4. ] Optimization terminated successfully. (Exit mode 0) Current function value: 0.2500005345767702 Iterations: 2 Function evaluations: 2 Gradient evaluations: 2 Optimization Complete ----------------------------------- [4.] [4.] [1.] [-0.49999947 -1. ] [0.25000053]
In the example shown above, the model is solving for derivatives in ‘fwd’ mode, specified by
p.setup(mode=’fwd’). In that case the
cache_linear_solution arg may be passed when adding design variables as shown above with
p.model.add_design_var(‘a’, cache_linear_solution=True). However, if the model were to solve for derivatives in ‘rev’ mode instead, then the
cache_linear_solution arg would be applied to the objective and/or the constraint variables. For example,
p.model.add_constraint(‘states’, upper=10, cache_linear_solution=True).