lv_sens.sensitivity

Functions

J(theta, xdata, dt, N, x0)	Compute cost function for a specific parameter a by first solving the forward problem.
grad_adj(theta, xdata, dt, N, x0)	Compute the cost function gradient via the adjoint.
grad_fd(theta, xdata, dt, N, x0[, h])	Approximate the cost function gradient with finite differences.
optimize(theta0, x_data, dt, N, x0, grad_type)	Optimize the inverse problem

lv_sens.sensitivity.J(theta: ndarray[Any, dtype[float64]], xdata: ndarray[Any, dtype[float64]], dt: float, N: int, x0: ndarray[Any, dtype[float64]]) → float

Compute cost function for a specific parameter a by first solving the forward problem.

Parameters:

• theta – parameter

• xdata – “measurement” data

• dt – time step

• N – number of timesteps

• x0 – initial condition of forward solve

Returns:

cost function value

lv_sens.sensitivity.grad_adj(theta: ndarray[Any, dtype[float64]], xdata: ndarray[Any, dtype[float64]], dt: float, N: int, x0: ndarray[Any, dtype[float64]]) → ndarray[Any, dtype[float64]]

Compute the cost function gradient via the adjoint.

Parameters:

• theta – parameter

• xdata – ‘measurement’ data

• dt – time step size

• N – number of timesteps

• x0 – initial condition of forward solve

Returns:

cost function gradient

lv_sens.sensitivity.grad_fd(theta: ndarray[Any, dtype[float64]], xdata: ndarray[Any, dtype[float64]], dt: float, N: int, x0: ndarray[Any, dtype[float64]], h: float = 1e-07) → ndarray[Any, dtype[float64]]

Approximate the cost function gradient with finite differences.

Parameters:

• theta – parameter

• xdata – “measurement” data

• dt – time step size

• N – number of timesteps

• x0 – initial condition of forward solve

• h – finite differente step size (parameter)

Returns:

cost function gradient approximation

lv_sens.sensitivity.optimize(theta0: ndarray[Any, dtype[float64]], x_data: ndarray[Any, dtype[float64]], dt: float, N: int, x0: ndarray[Any, dtype[float64]], grad_type: str) → OptimizeResult

Optimize the inverse problem

\[\hat\theta = \textrm{argmin}\, J(\theta)\]

for cost function

\[J(\theta) = \frac{1}{2}\sum_{i=1}^N |x_i(\theta) - y_i|\]

where \(x\) is the solution of the ODE LV() at time step \(i\) and \(y_i\) is a noisy measurement of \(x_i\).

Parameters:

• theta0 – parameter initial guess

• x_data – data array

• dt – time step

• N – number of time steps

• x0 – initial guess

• grad_type – gradient type of the cost function

Returns:

scipy result object