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from nose import tools |
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import numpy as np |
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from scipy import stats |
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from .. import basis_functions |
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from .. import problems |
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from .. import solvers |
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def analytic_solution(y, nL, alpha): |
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""" |
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Analytic solution to the differential equation describing the signaling |
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equilbrium of the Spence (1974) model. |
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""" |
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D = ((1 + alpha) / 2) * (nL / yL(nL, alpha)**-alpha)**2 - yL(nL, alpha)**(1 + alpha) |
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return y**(-alpha) * (2 * (y**(1 + alpha) + D) / (1 + alpha))**0.5 |
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def spence_model(y, n, alpha, **params): |
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return [(n**-1 - alpha * n * y**(alpha - 1)) / y**alpha] |
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def initial_condition(y, n, nL, alpha, **params): |
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return [n - nL] |
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def yL(nL, alpha): |
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return (nL**2 * alpha)**(1 / (1 - alpha)) |
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def initial_mesh(yL, yH, num, problem): |
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ys = np.linspace(yL, yH, num=num) |
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ns = problem.params['nL'] + np.sqrt(ys) |
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return ys, ns |
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random_seed = np.random.randint(2147483647) |
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params = {'nL': 1.0, 'alpha': 0.15} |
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test_problem = problems.IVP(initial_condition, 1, 1, params, spence_model) |
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def test_bspline_collocation(): |
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"""Tests B-spline collocation.""" |
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bspline_basis = basis_functions.BSplineBasis() |
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solver = solvers.Solver(bspline_basis) |
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boundary_points = (yL(**params), 10) |
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ys, ns = initial_mesh(*boundary_points, num=250, problem=test_problem) |
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tck, u = bspline_basis.fit([ns], u=ys, k=5, s=0) |
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knots, coefs, k = tck |
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initial_coefs = np.hstack(coefs) |
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basis_kwargs = {'knots': knots, 'degree': k, 'ext': 2} |
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nodes = np.linspace(*boundary_points, num=249) |
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solution = solver.solve(basis_kwargs, boundary_points, initial_coefs, |
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nodes, test_problem) |
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# check that solver terminated successfully |
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msg = "Solver failed!\nSeed: {}\nModel params: {}\n" |
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tools.assert_true(solution.result.success, |
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msg=msg.format(random_seed, test_problem.params)) |
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# compute the residuals |
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normed_residuals = solution.normalize_residuals(ys) |
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# check that residuals are close to zero on average |
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tools.assert_true(np.mean(normed_residuals) < 1e-6, |
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msg=msg.format(random_seed, test_problem.params)) |
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# check that the numerical and analytic solutions are close |
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numeric_soln = solution.evaluate_solution(ys) |
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analytic_soln = analytic_solution(ys, **test_problem.params) |
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tools.assert_true(np.mean(numeric_soln - analytic_soln) < 1e-6) |
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davidrpugh /
pyCollocation
