Custom nonlinearities

import numpy as np

import nifty8 as ift

In NIFTy, users can add hand-crafted point-wise nonlinearities that are then available for Field, MultiField, Linearization and Operator. This guide illustrates how this is done.

Suppose that we would like to use the point-wise function f(x) = x*exp(x) in an operator chain. This function is called “myptw” in the following. We introduce this function to NIFTy by implementing two functions.

First, one that takes a numpy.ndarray as an input, applies the point-wise mapping and returns the result as a numpy.ndarray of the same shape. Second, a function that takes a numpy.ndarray as an input and returns two numpy.ndarrays: the application of the nonlinearity (same as before) and the derivative.

def func(x):
    return x*np.exp(x)


def func_and_derv(x):
    expx = np.exp(x)
    return x*expx, (1+x)*expx

These two functions are then added to the NIFTy-internal dictionary that contains all implemented point-wise nonlinearities.

ift.pointwise.ptw_dict["myptw"] = func, func_and_derv

This allows us to apply this non-linearity on Fields, …

dom = ift.UnstructuredDomain(10)
fld = ift.from_random(dom)
fld = ift.full(dom, 2.)
a = fld.ptw("myptw")
b = ift.makeField(dom, func(fld.val))
ift.extra.assert_allclose(a, b)

MultiFields, …

mdom = ift.makeDomain({"bar": ift.UnstructuredDomain(10)})
mfld = ift.from_random(mdom)
a = mfld.ptw("myptw")
b = ift.makeField(mdom, {"bar": func(mfld["bar"].val)})
ift.extra.assert_allclose(a, b)

Linearizations (including the Jacobian), …

lin = ift.Linearization.make_var(fld)
a = lin.ptw("myptw").val
b = ift.makeField(dom, func(fld.val))
ift.extra.assert_allclose(a, b)

op_a = lin.ptw("myptw").jac
op_b = ift.makeOp(ift.makeField(dom, func_and_derv(fld.val)[1]))
testing_vector = ift.from_random(dom)
ift.extra.assert_allclose(op_a(testing_vector),
                          op_b(testing_vector))

and Operators.

op = ift.FieldAdapter(dom, "foo").ptw("myptw")

Please remember to always check that the gradient has been implemented correctly by comparing it to an approximation to the gradient by finite differences.

def check(func_name, eps=1e-7):
    pos = ift.from_random(ift.UnstructuredDomain(10))
    var0 = ift.Linearization.make_var(pos)
    var1 = ift.Linearization.make_var(pos+eps)
    df0 = (var1.ptw(func_name).val - var0.ptw(func_name).val)/eps
    df1 = var0.ptw(func_name).jac(ift.full(lin.domain, 1.))
    # rtol depends on how nonlinear the function is
    ift.extra.assert_allclose(df0, df1, rtol=100*eps)

check("myptw")