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

Introduction

The procedure is as follows. Consider the diffusion process \((x_t\colon 0 \le t \le T)\) given by

\(dx_t = b(x_t) dt + dw_t\)

where the drift b is expressed as linear combination

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

(see Module Schauder) and prior distribution on the coefficients

\(c_i \sim N(0,\xi_i)\)

Then the posterior distribution of \(b\) given observations \(x_t\) is given by

\(c_i | x_s \sim N(W^{-1}\mu, W^{-1})\) \(W = \Sigma + (\operatorname{diag}(\xi))^{-1}\),

with the nxn-matrix

\(\Sigma_{ij} = \int_0^T \phi_i(x_t)\phi_j(x_t) dt\)

and the n-vector

\(\mu_i = \int_0^T \phi_i(x_t) d x_t\).

Using the recursion detailed in Module Schauder, one rather computes

\(\Sigma^\prime_{ij} = \int_0^T \psi_i(x_t)\psi_j(x_t) dt\)

and the n-vector

\(\mu^\prime_i = \int_0^T \psi_i(x_t) d x_t\)

and uses pickup_mu!(mu) and pickup_Sigma!(Sigma) to obtain \(\mu\) and \(\Sigma\).

Optional additional basis functions

One can extend the basis by additional functions, implemented are variants. B1 includes a constant, B2 two linear functions

B1

\(\phi_1 \dots \phi_n, c\)

B2

\(\phi_1 \dots \phi_n, \max(1-x, 0), \max(x, 0)\)

To compute mu, use

mu = pickup_mu!(fe_mu(y,L, 0))
mu = fe_muB1(mu, y);

or

mu = pickup_mu!(fe_mu(y,L, 0))
mu = fe_muB2(mu, y);

Reference

Functions taking y` without parameter [a,b] expect ``y to be shifted into the intervall [0,1].

Schauder.pickup_mu!(mu)

computes mu from mu’

Schauder.drop_mu!(mu)

Computes mu’ from mu.

Schauder.pickup_Sigma!(Sigma)

Transforms Sigma’ into Sigma.

Schauder.drop_Sigma!(Sigma)

Transforms Sigma into Sigma’.

Schauder.fe_mu(y, L, K)

Computes mu’ from the observations y using 2^L-1 basis elements and returns a vector with K trailing zeros (in case one ones to customize the basis.

Schauder.fe_muB1(mu, y)

Append \(\mu_{n+1} = \int_0^T \phi_{n+1} d x_t\) with \(\phi_{n+1} = 1\).

Schauder.fe_muB2(mu, y)

Append \(\mu_{n+1} = \int_0^T \phi_{n+1} d x_t\) with \(\phi_{n+1} = \max(1-x, 0)\) and \(\mu_{n+2} = \int_0^T \phi_{n+2} d x_t\) with \(\phi_{n+2} = \max(x, 0)\)

Schauder.fe_Sigma(y, dt, L)

Computes the matrix Sigma’ from the observations y uniformly spaced at distance dt using 2^L-1 basis elements.

Schauder.bayes_drift(x, dt, a, b, L, xirem, beta, B)

Performs estimation of drift on observations x in [a,b] spaced at distance dt using the Schauder basis of level L and level wise coefficients decaying at rate beta. A Brownian motion like prior is obtained for beta= 0.5. The K remaining optional basiselements have variance xirem.

The result is returned as [designp coeff se] where coeff are coefficients of finite elements with maximum at the designpoints designp and standard error se.

Observations outside [a,b] may influence the result through phi_{n+1}, ..., phi_{n+K}