tf_quant_finance.math.pde.steppers.weighted_implicit_explicit.weighted_implicit_explicit_scheme

Last updated: 2023-03-16.

tf_quant_finance.math.pde.steppers.weighted_implicit_explicit.weighted_implicit_explicit_scheme#

View source

Constructs weighted implicit-explicit scheme.

tf_quant_finance.math.pde.steppers.weighted_implicit_explicit.weighted_implicit_explicit_scheme(
    theta
)

Approximates the space-discretized equation of du/dt = A(t) u(t) + b(t) as

(u(t2) - u(t1)) / (t2 - t1) = theta * (A u(t1) + b)
   + (1 - theta) (A u(t2) + b),

where A = A((t1 + t2)/2), b = b((t1 + t2)/2), and theta is a float between 0 and 1.

Note that typically A and b are evaluated at t1 and t2 in the explicit and implicit terms respectively (the two terms of the right-hand side). Instead, we evaluate them at the midpoint (t1 + t2)/2, which saves some computation. One can check that evaluating at midpoint doesn’t change the order of accuracy of the scheme: it is still second order accurate in t2 - t1 if theta = 0.5 and first order accurate otherwise.

The solution is the following: u(t2) = (1 - (1 - theta) dt A)^(-1) * (1 + theta dt A) u(t1) + dt b.

The main bottleneck here is inverting the matrix (1 - (1 - theta) dt A). This matrix is tridiagonal (each point is influenced by the two neighbouring points), and thus the inversion can be efficiently performed using tf.linalg.tridiagonal_solve.

References:#

[1] I.V. Puzynin, A.V. Selin, S.I. Vinitsky, A high-order accuracy method for numerical solving of the time-dependent Schrodinger equation, Comput. Phys. Commun. 123 (1999), 1. https://www.sciencedirect.com/science/article/pii/S0010465599002246

Args:#

  • theta: A float in range [0, 1]. A parameter used to mix implicit and explicit schemes together. Value of 0.0 corresponds to the fully implicit scheme, 1.0 to the fully explicit, and 0.5 to the Crank-Nicolson scheme.

Returns:#

A callable that consumes the following arguments by keyword:

  1. value_grid: Grid of values at time t1, i.e. u(t1).

  2. t1: Time before the step.

  3. t2: Time after the step.

  4. equation_params_fn: A callable that takes a scalar Tensor argument representing time, and constructs the tridiagonal matrix A (a tuple of three Tensors, main, upper, and lower diagonals) and the inhomogeneous term b. All of the Tensors are of the same dtype as value_grid and of the shape broadcastable with the shape of value_grid. The callable returns a Tensor of the same shape and dtype as value_grid and represents an approximate solution u(t2).

Contents