DiscreteFiltering

Discrete filtering toolbox

S!(A, u, t; kwargs...)

Solve ODE for given operator and IC (mutating form, not differentiable).

S(A, u₀, t; kwargs...)

Solve ODE for given operator and IC. This form is differentiable.

S_stencil!(A, u, t; kwargs...)

Solve ODE for given operator and IC (mutating form, not differentiable).

circulant(n, inds, stencil)

Create circulant SparseMatrixCSC.

create_data_dns(A, c, ξ, t)

Create DNS solution (numerical approximation).

create_data_exact(c, ξ, t)

Create exact unfiltered solutions.

create_data_filtered(W, A, dns)

Create discrete filtered solution from DNS

create_gaussian(h)

Create Gaussian filter with filter radius h.

The kernel is given by

\[ G(x, \xi) = \sqrt{\frac{3}{2 \pi h^2(x)}} \mathrm{e}^{-\frac{3 (\xi - x)^2}{2 h^2(x)}},\]

and the local transfer function by

\[ \hat{G}_k(x) = \mathrm{e}^{-\frac{2 \pi^2}{3} k^2 h^2(x)}.\]

create_initial_state(A_ref)

Create initial state for the ADAM optimizer.

create_loss_fit(u, t; n_sample = size(u, 2), n_time = length(t) - 1, kwargs...)

Create embedded loss function on dataset, evaluating at n_sample random initial conditions and n_time random time steps at each call.

The keyword arguments kwargs are passed to the ODE-solver S.

create_loss_mixed(losses, weights)

Create a composed loss function $L(A) = \sum_i w_i L_i(A)$.

create_loss_prior(A_ref)

Create a prior loss functino comparing the operator A to a reference operator A_ref.

create_tophat(h)

Create top-hat filter with filter radius h.

The kernel is given by

\[ G(x, \xi) = \begin{cases} \frac{1}{2h(x)} \quad & |\xi - x| \leq h(x), \\ 0 \quad & \text{otherwise}, \end{cases}\]

and the local transfer function by

\[ \hat{G}_k(x) = \frac{\sin \left ( 2 \pi k h(x) \right )}{2 \pi k h(x)}.\]

filter_matrix(F, x, ξ)

Create filter matrix with filter F from fine grid ξ to coarse grid x.

fit_embedded(
    state,
    loss;
    α = 0.001,
    β₁ = 0.9,
    β₂ = 0.999,
    ϵ = 1e-8,
    n_iter = 100,
    testloss,
    ntestloss = 10,
    doplot = true,
)

Fit operator to data while embedded in ODE solver using the ADAM optimizer.

If doplot, a real time plot of the validation performance is plotted.

https://arxiv.org/abs/1412.6980

interpolation_matrix(x, y)

Create matrix for interpolating from grid $x \in \mathbb{R}^N$ to $y \in \mathbb{R}^M$.

plotmat(A; kwargs...)

Plot matrix.

relerr(A, uₜ, t; kwargs...)

Relative error, averaged over time and data samples.

relerrs(A, uₜ, t; kwargs...)

Time dependent relative errors, averaged over data samples.

spectral_relerr(A, uₜ, t; kwargs...)

Time dependent spectral relative errors, averaged over data samples.