integrate._bvp

Boundary value problem solver.

Module Contents

Classes

BVPResult()

Functions

estimate_fun_jac(fun,x,y,p,f0=None) Estimate derivatives of an ODE system rhs with forward differences.
estimate_bc_jac(bc,ya,yb,p,bc0=None) Estimate derivatives of boundary conditions with forward differences.
compute_jac_indices(n,m,k) Compute indices for the collocation system Jacobian construction.
stacked_matmul(a,b) Stacked matrix multiply: out[i,:,:] = np.dot(a[i,:,:], b[i,:,:]).
construct_global_jac(n,m,k,i_jac,j_jac,h,df_dy,df_dy_middle,df_dp,df_dp_middle,dbc_dya,dbc_dyb,dbc_dp) Construct the Jacobian of the collocation system.
collocation_fun(fun,y,p,x,h) Evaluate collocation residuals.
prepare_sys(n,m,k,fun,bc,fun_jac,bc_jac,x,h) Create the function and the Jacobian for the collocation system.
solve_newton(n,m,h,col_fun,bc,jac,y,p,B,bvp_tol) Solve the nonlinear collocation system by a Newton method.
print_iteration_header()
print_iteration_progress(iteration,residual,total_nodes,nodes_added)
estimate_rms_residuals(fun,sol,x,h,p,r_middle,f_middle) Estimate rms values of collocation residuals using Lobatto quadrature.
create_spline(y,yp,x,h) Create a cubic spline given values and derivatives.
modify_mesh(x,insert_1,insert_2) Insert nodes into a mesh.
wrap_functions(fun,bc,fun_jac,bc_jac,k,a,S,D,dtype) Wrap functions for unified usage in the solver.
solve_bvp(fun,bc,x,y,p=None,S=None,fun_jac=None,bc_jac=None,tol=0.001,max_nodes=1000,verbose=0) Solve a boundary-value problem for a system of ODEs.
estimate_fun_jac(fun, x, y, p, f0=None)

Estimate derivatives of an ODE system rhs with forward differences.

df_dy : ndarray, shape (n, n, m)
Derivatives with respect to y. An element (i, j, q) corresponds to d f_i(x_q, y_q) / d (y_q)_j.
df_dp : ndarray with shape (n, k, m) or None
Derivatives with respect to p. An element (i, j, q) corresponds to d f_i(x_q, y_q, p) / d p_j. If p is empty, None is returned.
estimate_bc_jac(bc, ya, yb, p, bc0=None)

Estimate derivatives of boundary conditions with forward differences.

dbc_dya : ndarray, shape (n + k, n)
Derivatives with respect to ya. An element (i, j) corresponds to d bc_i / d ya_j.
dbc_dyb : ndarray, shape (n + k, n)
Derivatives with respect to yb. An element (i, j) corresponds to d bc_i / d ya_j.
dbc_dp : ndarray with shape (n + k, k) or None
Derivatives with respect to p. An element (i, j) corresponds to d bc_i / d p_j. If p is empty, None is returned.
compute_jac_indices(n, m, k)

Compute indices for the collocation system Jacobian construction.

See construct_global_jac for the explanation.

stacked_matmul(a, b)

Stacked matrix multiply: out[i,:,:] = np.dot(a[i,:,:], b[i,:,:]).

In our case a[i, :, :] and b[i, :, :] are always square.

construct_global_jac(n, m, k, i_jac, j_jac, h, df_dy, df_dy_middle, df_dp, df_dp_middle, dbc_dya, dbc_dyb, dbc_dp)

Construct the Jacobian of the collocation system.

There are n * m + k functions: m - 1 collocations residuals, each containing n components, followed by n + k boundary condition residuals.

There are n * m + k variables: m vectors of y, each containing n components, followed by k values of vector p.

For example, let m = 4, n = 2 and k = 1, then the Jacobian will have the following sparsity structure:

1 1 2 2 0 0 0 0 5 1 1 2 2 0 0 0 0 5 0 0 1 1 2 2 0 0 5 0 0 1 1 2 2 0 0 5 0 0 0 0 1 1 2 2 5 0 0 0 0 1 1 2 2 5

3 3 0 0 0 0 4 4 6 3 3 0 0 0 0 4 4 6 3 3 0 0 0 0 4 4 6

Zeros denote identically zero values, other values denote different kinds of blocks in the matrix (see below). The blank row indicates the separation of collocation residuals from boundary conditions. And the blank column indicates the separation of y values from p values.

Refer to [1]_ (p. 306) for the formula of n x n blocks for derivatives of collocation residuals with respect to y.

n : int
Number of equations in the ODE system.
m : int
Number of nodes in the mesh.
k : int
Number of the unknown parameters.
i_jac, j_jac : ndarray

Row and column indices returned by compute_jac_indices. They represent different blocks in the Jacobian matrix in the following order (see the scheme above):

  • 1: m - 1 diagonal n x n blocks for the collocation residuals.
  • 2: m - 1 off-diagonal n x n blocks for the collocation residuals.
  • 3 : (n + k) x n block for the dependency of the boundary conditions on ya.
  • 4: (n + k) x n block for the dependency of the boundary conditions on yb.
  • 5: (m - 1) * n x k block for the dependency of the collocation residuals on p.
  • 6: (n + k) x k block for the dependency of the boundary conditions on p.
df_dy : ndarray, shape (n, n, m)
Jacobian of f with respect to y computed at the mesh nodes.
df_dy_middle : ndarray, shape (n, n, m - 1)
Jacobian of f with respect to y computed at the middle between the mesh nodes.
df_dp : ndarray with shape (n, k, m) or None
Jacobian of f with respect to p computed at the mesh nodes.
df_dp_middle: ndarray with shape (n, k, m - 1) or None
Jacobian of f with respect to p computed at the middle between the mesh nodes.
dbc_dya, dbc_dyb : ndarray, shape (n, n)
Jacobian of bc with respect to ya and yb.
dbc_dp: ndarray with shape (n, k) or None
Jacobian of bc with respect to p.
J : csc_matrix, shape (n * m + k, n * m + k)
Jacobian of the collocation system in a sparse form.
[1]J. Kierzenka, L. F. Shampine, “A BVP Solver Based on Residual Control and the Maltab PSE”, ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001.
collocation_fun(fun, y, p, x, h)

Evaluate collocation residuals.

This function lies in the core of the method. The solution is sought as a cubic C1 continuous spline with derivatives matching the ODE rhs at given nodes x. Collocation conditions are formed from the equality of the spline derivatives and rhs of the ODE system in the middle points between nodes.

Such method is classified to Lobbato IIIA family in ODE literature. Refer to [1]_ for the formula and some discussion.

col_res : ndarray, shape (n, m - 1)
Collocation residuals at the middle points of the mesh intervals.
y_middle : ndarray, shape (n, m - 1)
Values of the cubic spline evaluated at the middle points of the mesh intervals.
f : ndarray, shape (n, m)
RHS of the ODE system evaluated at the mesh nodes.
f_middle : ndarray, shape (n, m - 1)
RHS of the ODE system evaluated at the middle points of the mesh intervals (and using y_middle).
[1]J. Kierzenka, L. F. Shampine, “A BVP Solver Based on Residual Control and the Maltab PSE”, ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001.
prepare_sys(n, m, k, fun, bc, fun_jac, bc_jac, x, h)

Create the function and the Jacobian for the collocation system.

solve_newton(n, m, h, col_fun, bc, jac, y, p, B, bvp_tol)

Solve the nonlinear collocation system by a Newton method.

This is a simple Newton method with a backtracking line search. As advised in [1]_, an affine-invariant criterion function F = ||J^-1 r||^2 is used, where J is the Jacobian matrix at the current iteration and r is the vector or collocation residuals (values of the system lhs).

The method alters between full Newton iterations and the fixed-Jacobian iterations based

There are other tricks proposed in [1]_, but they are not used as they don’t seem to improve anything significantly, and even break the convergence on some test problems I tried.

All important parameters of the algorithm are defined inside the function.

n : int
Number of equations in the ODE system.
m : int
Number of nodes in the mesh.
h : ndarray, shape (m-1,)
Mesh intervals.
col_fun : callable
Function computing collocation residuals.
bc : callable
Function computing boundary condition residuals.
jac : callable
Function computing the Jacobian of the whole system (including collocation and boundary condition residuals). It is supposed to return csc_matrix.
y : ndarray, shape (n, m)
Initial guess for the function values at the mesh nodes.
p : ndarray, shape (k,)
Initial guess for the unknown parameters.
B : ndarray with shape (n, n) or None
Matrix to force the S y(a) = 0 condition for a problems with the singular term. If None, the singular term is assumed to be absent.
bvp_tol : float
Tolerance to which we want to solve a BVP.
y : ndarray, shape (n, m)
Final iterate for the function values at the mesh nodes.
p : ndarray, shape (k,)
Final iterate for the unknown parameters.
singular : bool
True, if the LU decomposition failed because Jacobian turned out to be singular.
[1]U. Ascher, R. Mattheij and R. Russell “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”
print_iteration_header()
print_iteration_progress(iteration, residual, total_nodes, nodes_added)
class BVPResult
estimate_rms_residuals(fun, sol, x, h, p, r_middle, f_middle)

Estimate rms values of collocation residuals using Lobatto quadrature.

The residuals are defined as the difference between the derivatives of our solution and rhs of the ODE system. We use relative residuals, i.e. normalized by 1 + np.abs(f). RMS values are computed as sqrt from the normalized integrals of the squared relative residuals over each interval. Integrals are estimated using 5-point Lobatto quadrature [1]_, we use the fact that residuals at the mesh nodes are identically zero.

In [2] they don’t normalize integrals by interval lengths, which gives a higher rate of convergence of the residuals by the factor of h**0.5. I chose to do such normalization for an ease of interpretation of return values as RMS estimates.

rms_res : ndarray, shape (m - 1,)
Estimated rms values of the relative residuals over each interval.
[1]http://mathworld.wolfram.com/LobattoQuadrature.html
[2]J. Kierzenka, L. F. Shampine, “A BVP Solver Based on Residual Control and the Maltab PSE”, ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001.
create_spline(y, yp, x, h)

Create a cubic spline given values and derivatives.

Formulas for the coefficients are taken from interpolate.CubicSpline.

sol : PPoly
Constructed spline as a PPoly instance.
modify_mesh(x, insert_1, insert_2)

Insert nodes into a mesh.

Nodes removal logic is not established, its impact on the solver is presumably negligible. So only insertion is done in this function.

x : ndarray, shape (m,)
Mesh nodes.
insert_1 : ndarray
Intervals to each insert 1 new node in the middle.
insert_2 : ndarray
Intervals to each insert 2 new nodes, such that divide an interval into 3 equal parts.
x_new : ndarray
New mesh nodes.

insert_1 and insert_2 should not have common values.

wrap_functions(fun, bc, fun_jac, bc_jac, k, a, S, D, dtype)

Wrap functions for unified usage in the solver.

solve_bvp(fun, bc, x, y, p=None, S=None, fun_jac=None, bc_jac=None, tol=0.001, max_nodes=1000, verbose=0)

Solve a boundary-value problem for a system of ODEs.

This function numerically solves a first order system of ODEs subject to two-point boundary conditions:

dy / dx = f(x, y, p) + S * y / (x - a), a <= x <= b
bc(y(a), y(b), p) = 0

Here x is a 1-dimensional independent variable, y(x) is a n-dimensional vector-valued function and p is a k-dimensional vector of unknown parameters which is to be found along with y(x). For the problem to be determined there must be n + k boundary conditions, i.e. bc must be (n + k)-dimensional function.

The last singular term in the right-hand side of the system is optional. It is defined by an n-by-n matrix S, such that the solution must satisfy S y(a) = 0. This condition will be forced during iterations, so it must not contradict boundary conditions. See [2]_ for the explanation how this term is handled when solving BVPs numerically.

Problems in a complex domain can be solved as well. In this case y and p are considered to be complex, and f and bc are assumed to be complex-valued functions, but x stays real. Note that f and bc must be complex differentiable (satisfy Cauchy-Riemann equations [4]), otherwise you should rewrite your problem for real and imaginary parts separately. To solve a problem in a complex domain, pass an initial guess for y with a complex data type (see below).

fun : callable
Right-hand side of the system. The calling signature is fun(x, y), or fun(x, y, p) if parameters are present. All arguments are ndarray: x with shape (m,), y with shape (n, m), meaning that y[:, i] corresponds to x[i], and p with shape (k,). The return value must be an array with shape (n, m) and with the same layout as y.
bc : callable
Function evaluating residuals of the boundary conditions. The calling signature is bc(ya, yb), or bc(ya, yb, p) if parameters are present. All arguments are ndarray: ya and yb with shape (n,), and p with shape (k,). The return value must be an array with shape (n + k,).
x : array_like, shape (m,)
Initial mesh. Must be a strictly increasing sequence of real numbers with x[0]=a and x[-1]=b.
y : array_like, shape (n, m)
Initial guess for the function values at the mesh nodes, i-th column corresponds to x[i]. For problems in a complex domain pass y with a complex data type (even if the initial guess is purely real).
p : array_like with shape (k,) or None, optional
Initial guess for the unknown parameters. If None (default), it is assumed that the problem doesn’t depend on any parameters.
S : array_like with shape (n, n) or None
Matrix defining the singular term. If None (default), the problem is solved without the singular term.
fun_jac : callable or None, optional

Function computing derivatives of f with respect to y and p. The calling signature is fun_jac(x, y), or fun_jac(x, y, p) if parameters are present. The return must contain 1 or 2 elements in the following order:

  • df_dy : array_like with shape (n, n, m) where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d (y_q)_j.
  • df_dp : array_like with shape (n, k, m) where an element (i, j, q) equals to d f_i(x_q, y_q, p) / d p_j.

Here q numbers nodes at which x and y are defined, whereas i and j number vector components. If the problem is solved without unknown parameters df_dp should not be returned.

If fun_jac is None (default), the derivatives will be estimated by the forward finite differences.

bc_jac : callable or None, optional

Function computing derivatives of bc with respect to ya, yb and p. The calling signature is bc_jac(ya, yb), or bc_jac(ya, yb, p) if parameters are present. The return must contain 2 or 3 elements in the following order:

  • dbc_dya : array_like with shape (n, n) where an element (i, j) equals to d bc_i(ya, yb, p) / d ya_j.
  • dbc_dyb : array_like with shape (n, n) where an element (i, j) equals to d bc_i(ya, yb, p) / d yb_j.
  • dbc_dp : array_like with shape (n, k) where an element (i, j) equals to d bc_i(ya, yb, p) / d p_j.

If the problem is solved without unknown parameters dbc_dp should not be returned.

If bc_jac is None (default), the derivatives will be estimated by the forward finite differences.

tol : float, optional
Desired tolerance of the solution. If we define r = y' - f(x, y) where y is the found solution, then the solver tries to achieve on each mesh interval norm(r / (1 + abs(f)) < tol, where norm is estimated in a root mean squared sense (using a numerical quadrature formula). Default is 1e-3.
max_nodes : int, optional
Maximum allowed number of the mesh nodes. If exceeded, the algorithm terminates. Default is 1000.
verbose : {0, 1, 2}, optional

Level of algorithm’s verbosity:

  • 0 (default) : work silently.
  • 1 : display a termination report.
  • 2 : display progress during iterations.

Bunch object with the following fields defined: sol : PPoly

Found solution for y as scipy.interpolate.PPoly instance, a C1 continuous cubic spline.
p : ndarray or None, shape (k,)
Found parameters. None, if the parameters were not present in the problem.
x : ndarray, shape (m,)
Nodes of the final mesh.
y : ndarray, shape (n, m)
Solution values at the mesh nodes.
yp : ndarray, shape (n, m)
Solution derivatives at the mesh nodes.
rms_residuals : ndarray, shape (m - 1,)
RMS values of the relative residuals over each mesh interval (see the description of tol parameter).
niter : int
Number of completed iterations.
status : int

Reason for algorithm termination:

  • 0: The algorithm converged to the desired accuracy.
  • 1: The maximum number of mesh nodes is exceeded.
  • 2: A singular Jacobian encountered when solving the collocation system.
message : string
Verbal description of the termination reason.
success : bool
True if the algorithm converged to the desired accuracy (status=0).

This function implements a 4-th order collocation algorithm with the control of residuals similar to [1]_. A collocation system is solved by a damped Newton method with an affine-invariant criterion function as described in [3].

Note that in [1]_ integral residuals are defined without normalization by interval lengths. So their definition is different by a multiplier of h**0.5 (h is an interval length) from the definition used here.

New in version 0.18.0.

[1]J. Kierzenka, L. F. Shampine, “A BVP Solver Based on Residual Control and the Maltab PSE”, ACM Trans. Math. Softw., Vol. 27, Number 3, pp. 299-316, 2001.
[2]L.F. Shampine, P. H. Muir and H. Xu, “A User-Friendly Fortran BVP Solver”.
[3]U. Ascher, R. Mattheij and R. Russell “Numerical Solution of Boundary Value Problems for Ordinary Differential Equations”.
[4]Cauchy-Riemann equations on Wikipedia.

In the first example we solve Bratu’s problem:

y'' + k * exp(y) = 0
y(0) = y(1) = 0

for k = 1.

We rewrite the equation as a first order system and implement its right-hand side evaluation:

y1' = y2
y2' = -exp(y1)
>>> def fun(x, y):
...     return np.vstack((y[1], -np.exp(y[0])))

Implement evaluation of the boundary condition residuals:

>>> def bc(ya, yb):
...     return np.array([ya[0], yb[0]])

Define the initial mesh with 5 nodes:

>>> x = np.linspace(0, 1, 5)

This problem is known to have two solutions. To obtain both of them we use two different initial guesses for y. We denote them by subscripts a and b.

>>> y_a = np.zeros((2, x.size))
>>> y_b = np.zeros((2, x.size))
>>> y_b[0] = 3

Now we are ready to run the solver.

>>> from scipy.integrate import solve_bvp
>>> res_a = solve_bvp(fun, bc, x, y_a)
>>> res_b = solve_bvp(fun, bc, x, y_b)

Let’s plot the two found solutions. We take an advantage of having the solution in a spline form to produce a smooth plot.

>>> x_plot = np.linspace(0, 1, 100)
>>> y_plot_a = res_a.sol(x_plot)[0]
>>> y_plot_b = res_b.sol(x_plot)[0]
>>> import matplotlib.pyplot as plt
>>> plt.plot(x_plot, y_plot_a, label='y_a')
>>> plt.plot(x_plot, y_plot_b, label='y_b')
>>> plt.legend()
>>> plt.xlabel("x")
>>> plt.ylabel("y")
>>> plt.show()

We see that the two solutions have similar shape, but differ in scale significantly.

In the second example we solve a simple Sturm-Liouville problem:

y'' + k**2 * y = 0
y(0) = y(1) = 0

It is known that a non-trivial solution y = A * sin(k * x) is possible for k = pi * n, where n is an integer. To establish the normalization constant A = 1 we add a boundary condition:

y'(0) = k

Again we rewrite our equation as a first order system and implement its right-hand side evaluation:

y1' = y2
y2' = -k**2 * y1
>>> def fun(x, y, p):
...     k = p[0]
...     return np.vstack((y[1], -k**2 * y[0]))

Note that parameters p are passed as a vector (with one element in our case).

Implement the boundary conditions:

>>> def bc(ya, yb, p):
...     k = p[0]
...     return np.array([ya[0], yb[0], ya[1] - k])

Setup the initial mesh and guess for y. We aim to find the solution for k = 2 * pi, to achieve that we set values of y to approximately follow sin(2 * pi * x):

>>> x = np.linspace(0, 1, 5)
>>> y = np.zeros((2, x.size))
>>> y[0, 1] = 1
>>> y[0, 3] = -1

Run the solver with 6 as an initial guess for k.

>>> sol = solve_bvp(fun, bc, x, y, p=[6])

We see that the found k is approximately correct:

>>> sol.p[0]
6.28329460046

And finally plot the solution to see the anticipated sinusoid:

>>> x_plot = np.linspace(0, 1, 100)
>>> y_plot = sol.sol(x_plot)[0]
>>> plt.plot(x_plot, y_plot)
>>> plt.xlabel("x")
>>> plt.ylabel("y")
>>> plt.show()