Smoothing

 

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We have already met the process of smoothing in connection with the heat equation: starting with bounded initial data, f(x), the solution of the heat equation (para.1)
 
represents the effect of smoothing f(x), so that (in fact analytic) and .

A general process of smoothing can be accomplished by convolution with appropriate smoothing kernel Q(x)
 
such that is sufficiently smoother than f(x) is, and
 
With the heat kernel, the role of is played by time t > 0. A standard way to construct such filters is the following. We start with a -function supported on, say, (-1,1), such that it has a unit mass and zero first r moments,
 
Then we set and consider
 
Now, assume f is (r + 1) - differentiable in the neighborhood of x; then, since is supported on and satisfies (smoo.4) as well, we have by Taylor expansion
 
The first r moments of vanish and we are left with
 
i.e., converges to f(x) with order r+1 as . Moreover, is as smooth as is, since
 
has many bounded derivatives as Q has, i.e., starting with differentiable function f of order r + 1 in the neighborhood of x, we end up with regularized function in .

: For regularization - choose a unit mass kernel, see Figure 3.10,
 

Then is a regularization of f(x) with first order convergence rate

To increase the order of convergence, one requires more vanishing moments, (smoo.4),(which yield more oscillatory kernels). We note that this smoothing process is purely local -- it involves -neighboring values of function f, in order to yield a -regularized function with . The convergence rate here is r+1.

  
Figure 3.10: Unit mass mollifiers

We can also achieve local regularization with spectral convergence. To this end we set
 
where is a -function supported on . Figure 3.11 demonstrates such a mollifier. In this case the support of the mollifier is kept fixed; instead, by increasing m -- particularly, by allowing to increase together with N, we obtain a highly oscillatory kernel whose monomial moments satisfy (smoo.4) modulo a spectrally small error.

  
Figure 3.11: A spectral unit mass mollifier

Consider now
 
Then is because is; and the convergence rate is spectral, since by (2.1.34)
 
and since was chosen as we are left with a residual term which does not exceed

Thus, the convergence rate is as fast as the local smoothness of f permits;(in this case - the local neighborhood ). Of course, with we recover the global -regularization due to the spectral projection. The role of was to localize this process of spectral smoothing.

We can as easily implement such smoothing in the Fourier space: For example, with the heat kernel we have
 
so that for any t > 0 decay faster than exponential and hence u(x,t>0) belong to for any s (and by Sobolev embedding, therefore, is in and in fact analytic). In general we apply,
 
such that for to be in we require

and r+1 order of convergence follows with
 
Indeed, (smoo.15) implies
 

Note: Since we can deal with any unbounded f by splitting . To obtain spectral accuracy we may use
 
Clearly is and the familiar Fourier estimates give us

We emphasize that this kind of smoothing in the Fourier space need not be local; rather or are negligibly small away from a small interval centered around the origin depending on or . (This is due to the uncertainty principle.)

The smoothed version of the pseudospectral approximation of (meth_ps.16) reads
 
i.e., in each step we smooth the solution either in the real space (convolution) or in the Fourier space (cutting high modes). We claim that this smoothed version is stable hence convergent under very mild assumptions on the smoothing kernel . Specifically, (smoo.18) amounts in the Fourier space, compare (meth_ps.3)
 
The real part of the matrix in question is given by
 
where

is interpreted as the smoothed differentiation operator. Now, looking at (smoo.20) we note:

1. For p = 0 we are back at the spectral analysis, compare (meth_spec.52), (meth_spec.53) and the real part of the matrix in (smoo.20) - the aliasing free one - is bounded.
2. We are left with |p| = 1: in the unsmoothed version, these terms were unbounded since as or . With the smoothed version, these terms are bounded (and stability follows), provided we have

 

For example, consider the smoothing kernel where

This yields the smoothed differentiation symbols
 
which corresponds to the second order center differencing in (app_ps.36); stability is immediate by (smoo.21) for

Yet, this kind of smoothing reduces the overall spectral accuracy to a second one; a fourth order smoothing will be
 
In general, the accuracy is determined by the low modes while stability has to do with high ones. To entertain spectral accuracy we may consider smoothing kernels other than trigonometric polynomials ( finite difference), but rather, compare (smoo.17)
 
An increasing portion of the spectrum is differentiated exactly which yields spectral accuracy; the highest modes are not amplified because of the smoothing effect in this part of the spectrum.

We close this section noting that if the differential model contains some dissipation - e.g., the parabolic equation
 
then stability follows with no extra smoothing. The parabolic dissipation compensates for the loss of ``one derivative'' due to aliasing in first order terms. To see this we proceed as follows: multiply
 
by and sum to obtain
 
Suppressing excessive indices, , we have for the RHS of (smoo.27)
 
Now, the first sum on the right gives us the usual loss of one derivative and the second are compensates with gain of such quantity. Petrovski type stability (gain of derivatives) follows. We shall only sketch the details here. Starting with the first term on the right of (smoo.28) we have
 
while for the second term
 
and this last term dominates the RHS of (smoo.29).