Module Schauder

Introduction

In the following hat(x) is the piecewise linear function taking values values (0,0), (0.5,1), (1,0) on the interval [0,1] and 0 elsewhere.

The Schauder basis of level L > 0 in the interval [a,b] can be defined recursively from n = 2^L-1 classical finite elements \(\psi_i(x)\) on the grid

a + (1:n)/(n+1)*(b-a).

Assume that f is expressed as linear combination

\(f(x) = \sum_{i =1}^n c_i \psi_i(x)\)

with

\(\psi_{2j-1}(x) = hat(nx-j + 1)\) for \(j = 1 \dots 2^{L-1}\)

and

\(\psi_{2j}(x) = hat(nx-j + 1/2)\) for \(j = 1 \dots 2^{L-1}-1\)

Note that these coefficients are easy to find for the finite element basis, just

function fe_transf(f, a,b, L)
    n = 2^L-1
    return map(f, a + (1:n)/(n+1)*(b-a))
end

Then the coefficients of the same function with respect to the Schauder basis

\(f(x) = \sum_{i = 1}^n c_i \phi_i(x)\)

where for L = 2

\(\phi_2(x) = 2 hat(x)\)

\(\phi_1(x) = \psi_1(x) = hat(2x)\)

\(\phi_3(x) = \psi_3(x) = hat(2x-1)\)

can be computed directly, but also using the recursion

\(\phi_2(x) = \psi_1(x) + 2\psi_2(x) + \psi_2(x)\)

This can be implemented inplace (see pickup()), and is used throughout.

for l in 1:L-1
        b = sub(c, 2^l:2^l:n)
        b[:] *= 0.5
        a = sub(c, 2^(l-1):2^l:n-2^(l-1))
        a[:] -= b[:]
        a = sub(c, 2^l+2^(l-1):2^l:n)
        a[:] -= b[1:length(a)]
end

Reference

Schauder.pickup!(x)

Inplace computation of 2^L-1 Schauder-Faber coefficients from 2^L-1 overlapping finite-element coefficients x.

– inverse of Schauder.drop

– L = level(xj)

Schauder.drop!(x)

Inplace computation of 2^L-1 finite element coefficients from 2^L-1 Faber schauder coefficients x.

– inverse of Schauder.pickup

Schauder.finger_permute(x)

Reorders vector x or matrix A according to the reordering of the elements of a Faber-Schauder-basis from left to right, from bottom to top.

Schauder.finger_pm(L, K)

Returns the permuation used in finger_permute. Memoized reordering of faber schauder elements from low level to high level. The last K elements/rows are left untouched.

Schauder.level(x)

Gives the no. of levels of the biggest Schauder basis with less then length(x) elements.

level(x) = ilogb(size(x,1)+1)

Schauder.level(x, K)

Gives the no. of levels l of the biggest Schauder basis with less then length(x) elements and the number of additional elements n-2^l+1.

Schauder.vectoroflevels(L, K)

Gives a vector with the level of the hierarchical elements.

Schauder.hat(x)

Hat function. Piecewise linear functions with values (-inf,0), (0,0),(0.5,1), (1,0), (inf,0). – x vector or number