A Problem in Helioseismology:
determining the interior of the
sun from its acoustic spectrum.
WILLIAM RUNDELL
Texas A&M University
Problem: Determine the critical coefficients characterizing the interior of the sun
from its “five minute” oscillations.
Problem: Determine the critical coefficients characterizing the interior of the sun
from its “five minute” oscillations.
= 12,
= 8 (equatorial)
= 5,
= 12,
= 0 (equatorial)
= 3 (equatorial)
= 0 (equatorial)
= 3,
= 0 (equatorial)
= 5,
= 1,
= 8 (polar)
We assume that the sun is spherical and rotationally invariant with unit radius.
We assume that the sun is spherical and rotationally invariant with unit radius.
. separating
The acoustic vibrational amplitude is
the variables in the
Laplacian gives a radial equation of the form
the natural frequencies.
!
for the normal modes of vibrations with
We assume that the sun is spherical and rotationally invariant with unit radius.
. separating
The acoustic vibrational amplitude is
the variables in the
Laplacian gives a radial equation of the form
the natural frequencies.
for the normal modes of vibrations with
!
( )
( )+
.
)=
(
and the density
Two functions of importance; the propagation speed
as well as .
( )
( )
( )
4
3
=
)
= (1
(
and
=
( ) 1
4 ( )
( + 1)
)=
( )
( )=
depends on both
,
2
depends only on
We assume that the sun is spherical and rotationally invariant with unit radius.
. separating
The acoustic vibrational amplitude is
the variables in the
Laplacian gives a radial equation of the form
the natural frequencies.
for the normal modes of vibrations with
!
( )
( )+
.
)=
(
and the density
Two functions of importance; the propagation speed
as well as .
( )
( )
( )
4
3
=
)
= (1
(
and
=
( ) 1
4 ( )
( + 1)
)=
( )
( )=
depends on both
,
2
depends only on
The most realistic boundary condition is of the form
the parameter is also to be determined.
where
There are enormous amounts of data: GONG (Global Oscillation Network Group)
data consists of
’s with from 0 to 1,000 and from 1 to about 50-100.
Accuracy is very good in an absolute scale, but poor from a usable standpoint.
No other spectral information is readily available.
There are enormous amounts of data: GONG (Global Oscillation Network Group)
data consists of
’s with from 0 to 1,000 and from 1 to about 50-100.
Accuracy is very good in an absolute scale, but poor from a usable standpoint.
No other spectral information is readily available.
If we use the Liouville transform on
we can remove the term containing , modifying the , but also modifying the
singular term. If we ignore this, then a canonical form might be
There are enormous amounts of data: GONG (Global Oscillation Network Group)
data consists of
’s with from 0 to 1,000 and from 1 to about 50-100.
Accuracy is very good in an absolute scale, but poor from a usable standpoint.
No other spectral information is readily available.
If we use the Liouville transform on
we can remove the term containing , modifying the , but also modifying the
singular term. If we ignore this, then a canonical form might be
If the terms in colour are removed, this is a standard Sturm-Liouville problem:
is uniquely determined by the eigenvalues
, together with a
second sequence: norming constants, end-point values, or a second spectral
.
sequence corresponding to another boundary condition at
!
There are enormous amounts of data: GONG (Global Oscillation Network Group)
data consists of
’s with from 0 to 1,000 and from 1 to about 50-100.
Accuracy is very good in an absolute scale, but poor from a usable standpoint.
No other spectral information is readily available.
If we use the Liouville transform on
we can remove the term containing , modifying the , but also modifying the
singular term. If we ignore this, then a canonical form might be
If the terms in colour are removed, this is a standard Sturm-Liouville problem:
is uniquely determined by the eigenvalues
, together with a
second sequence: norming constants, end-point values, or a second spectral
.
sequence corresponding to another boundary condition at
!
The helioseismology application does not allow a change in the boundary
values or the measurement of anything other than eigenvalue data.
There are enormous amounts of data: GONG (Global Oscillation Network Group)
data consists of
’s with from 0 to 1,000 and from 1 to about 50-100.
Accuracy is very good in an absolute scale, but poor from a usable standpoint.
No other spectral information is readily available.
If we use the Liouville transform on
we can remove the term containing , modifying the , but also modifying the
singular term. If we ignore this, then a canonical form might be
What we must do is use the existence of the singular term, and the
resulting eigenvalue sequence for different values to compensate.
There are enormous amounts of data: GONG (Global Oscillation Network Group)
data consists of
’s with from 0 to 1,000 and from 1 to about 50-100.
Accuracy is very good in an absolute scale, but poor from a usable standpoint.
No other spectral information is readily available.
If we use the Liouville transform on
we can remove the term containing , modifying the , but also modifying the
singular term. If we ignore this, then a canonical form might be
What we must do is use the existence of the singular term, and the
resulting eigenvalue sequence for different values to compensate.
We can say nothing about the critical question of uniqueness for the full problem.
Break into two simpler problems - each containing only one of the coloured terms.
Optimal Method for the Regular Sturm Liouville Problem
Optimal Method for the Regular Sturm Liouville Problem
where
sin
Optimal Method for the Regular Sturm Liouville Problem
where
sin
Optimal Method for the Regular Sturm Liouville Problem
Set to be the spectrum from
to obtain
and evaluate at
.
,
sin
sin
This equation then gives
where
sin
Optimal Method for the Regular Sturm Liouville Problem
Set to be the spectrum from
to obtain
and evaluate at
.
,
sin
sin
This equation then gives
to get
from
Use
where
sin
Optimal Method for the Regular Sturm Liouville Problem
sin
where
,
Set to be the spectrum from
to obtain
and evaluate at
.
to get
from
Use
Solve Cauchy Problem from
using
,
as “initial data” to
.
and hence
determine
sin
sin
This equation then gives
Optimal Method for the Regular Sturm Liouville Problem
sin
where
,
Set to be the spectrum from
to obtain
and evaluate at
.
to get
from
Use
Solve Cauchy Problem from
using
,
as “initial data” to
.
and hence
determine
to recover
Iterate
sin
sin
This equation then gives
Case 1: Including only the singular term
We consider the eigenvalue problem for
Case 1: Including only the singular term
.
,
.
For fixed , (1) has a countable sequence of eigenvalues,
We consider the eigenvalue problem for
Case 1: Including only the singular term
.
is the classical Inverse Sturm-Liouville problem:
,
.
For fixed , (1) has a countable sequence of eigenvalues,
.
We consider the eigenvalue problem for
Case 1: Including only the singular term
.
.
!
from (some subset of) the spectral data
The goal is to recover
,
The eigenvalues have the following asymptotic values
For fixed , (1) has a countable sequence of eigenvalues,
.
We consider the eigenvalue problem for
Case 1: Including only the singular term
.
.
!
from (some subset of) the spectral data
The goal is to recover
,
The eigenvalues have the following asymptotic values
It is always instructive to look at the simplest case.
For fixed , (1) has a countable sequence of eigenvalues,
We consider the eigenvalue problem for
Case 1: Including only the singular term
.
and we take Dirichlet conditions at
.
, then the eigenfunctions are
If
,
For fixed , (1) has a countable sequence of eigenvalues,
where
is the spherical Bessel function.
The eigenvalues are the positive roots of
.
For nonzero we expect the eigenvalues and eigenfunctions to have similar proper.
ties - at least for a sufficiently small
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be determined from two spectral sequences, namely
!
Problem 1: Can
for
?
!
be determined from two spectral sequences, namely
Problem 1: Can
for
?
Carlson and Shubin showed that the set of potentials sharing the same two spectral
is an odd integer.
sequences is locally of finite dimension provided that
There are positive answers (and reconstructions) for cases with small :
for example,
,
,
,
(Rundell, Sacks).
!
!
!
The forward map.
We formulate the inverse spectral problem as a nonlinear operator equation; for each
value of
define to be the solution of
The forward map.
We formulate the inverse spectral problem as a nonlinear operator equation; for each
value of
define to be the solution of
Then define
by
The forward map.
We formulate the inverse spectral problem as a nonlinear operator equation; for each
value of
define to be the solution of
by
Then define
, does the equation
!
!
for some
The question: if
uniquely determine ?
The forward map.
We formulate the inverse spectral problem as a nonlinear operator equation; for each
value of
define to be the solution of
by
Then define
, does the equation
!
!
for some
The question: if
uniquely determine ?
It is natural to attempt to solve (2) by some version of Newton’s method
.
and this requires some insight into the structure of the linearized map
be a fixed function then
Let
Lemma 1.
be a fixed function then
Let
Lemma 1.
, then
, that is if
We must show local injectivity of
for
.
be a fixed function then
Let
Lemma 1.
, then
, that is if
We must show local injectivity of
for
.
be a fixed function then
Let
Lemma 1.
, then
, that is if
We must show local injectivity of
.
for
is uniquely determined by the asymptotics of the eigen-
The mean value
values, for any fixed .
by a preliminary calculation we may always assume that
.
with
Hence need only consider those
.
be a fixed function then
Let
Lemma 1.
, then
, that is if
We must show local injectivity of
for
,
Note that if
, then this becomes
sin
or
cos
.
be a fixed function then
Let
Lemma 1.
, then
, that is if
We must show local injectivity of
for
, then this becomes
,
Note that if
sin
or
is odd.
implies
Thus
cos
.
by
Lemma 2. For each positive integer , define
!
, The function
.
Then
is bounded and one to one on
and
element in the nullspace of
is the only
by
Lemma 2. For each positive integer , define
!
We can chain these step operators together,
, The function
.
Then
is bounded and one to one on
and
element in the nullspace of
is the only
by
Lemma 2. For each positive integer , define
, The function
.
We can chain these step operators together,
define the operators
Lemma 3. For each
by
and
with
Then for any
,
cos
!
Then
is bounded and one to one on
and
element in the nullspace of
is the only
by
Lemma 2. For each positive integer , define
, The function
.
We can chain these step operators together,
define the operators
Lemma 3. For each
by
and
with
Then for any
,
cos
This leads to
for
then
,
If
Lemma 4.
implies
!
Then
is bounded and one to one on
and
element in the nullspace of
is the only
This would actually be enough to conclude that
,
but in fact
and so we are missing the frequencies below
.
This would actually be enough to conclude that
,
but in fact
and so we are missing the frequencies below
Here is what we get in the case of
.
.
).
,(
()
+ 24
12
( )
T [ ]=
()
( )
4
T [ ]= ( )
,
where
This would actually be enough to conclude that
,
but in fact
and so we are missing the frequencies below
Here is what we get in the case of
.
.
).
,(
()
!
Next step is to use the conditions
are zero.
constants
+ 24
12
( )
T [ ]=
()
( )
4
T [ ]= ( )
,
where
to show that the three
This would actually be enough to conclude that
,
but in fact
and so we are missing the frequencies below
.
Here is what we get in the case of
.
).
,(
to show that the three
and
.
Finally, we show that
odd operators on
, implies
!
Next step is to use the conditions
are zero.
constants
()
+ 24
12
( )
T [ ]=
()
( )
4
T [ ]= ( )
,
where
, where
and
are the even and
This would actually be enough to conclude that
,
but in fact
and so we are missing the frequencies below
.
Here is what we get in the case of
.
).
,(
to show that the three
and
.
Finally, we show that
odd operators on
, implies
!
Next step is to use the conditions
are zero.
constants
()
+ 24
12
( )
T [ ]=
()
( )
4
T [ ]= ( )
,
where
and
, where
are the even and
We have accomplished this for several pairs of values and can show that the restriction of
odd can be removed.
Reconstructions with 5% error in
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This case is far from complete
This case is far from complete
We still have to show that the there is no finite dimensional set of ’s with the
same two spectra for larger values of .
This case is far from complete
We still have to show that the there is no finite dimensional set of ’s with the
same two spectra for larger values of .
We must extend the uniqueness result to be global rather than local.
It would be nice to use the conversion to an overposed hyperbolic problem as in
the regular case, or a similar “clean” technique.
This case is far from complete
We still have to show that the there is no finite dimensional set of ’s with the
same two spectra for larger values of .
We must extend the uniqueness result to be global rather than local.
It would be nice to use the conversion to an overposed hyperbolic problem as in
the regular case, or a similar “clean” technique.
For the equation
Do we need 3, 4, or an infinite number of different values?
both and ?
is it possible to recover
This case is far from complete
We still have to show that the there is no finite dimensional set of ’s with the
same two spectra for larger values of .
We must extend the uniqueness result to be global rather than local.
It would be nice to use the conversion to an overposed hyperbolic problem as in
the regular case, or a similar “clean” technique.
For the equation
Do we need 3, 4, or an infinite number of different values?
both and ?
is it possible to recover
Not all spectral sequences
for different values carry the same information content about (we would prefer small ). It is certainly the case that
the error in the spectra also varies with . If we use more data than is necessary,
how do take all of this into account?
!
Case 2: Omitting the singular term, but two potentials
Case 2: Omitting the singular term, but two potentials
We consider the eigenvalue problem
, normalized with
and .
, and ask
with, say, the boundary condition
whether it is possible to recover both
Case 2: Omitting the singular term, but two potentials
We consider the eigenvalue problem
, normalized with
and .
, and ask
with, say, the boundary condition
whether it is possible to recover both
Since we no longer have different values, we generate two spectra by changing
:
corresponding to
and
to
the boundary conditions at
.
determines the pair
?
!
!
Is it possible that the pair
!
!
Case 2: Omitting the singular term, but two potentials
We consider the eigenvalue problem
, and ask
, normalized with
and .
with, say, the boundary condition
whether it is possible to recover both
Since we no longer have different values, we generate two spectra by changing
the boundary conditions at
:
corresponding to
and
to
.
!
!
?
determines the pair
Is it possible that the pair
!
!
We can write (4) in the form of a quadratic eigenvalue problem:
where
,
,
. These are all self-adjoint,
,
, are also positive operators. Under these
and provided
,
conditions the spectrum is real, positive, and consists of two sequences
with
,
and
,
.
!
!
For our particular operators these sequences have the asymptotic form
For our particular operators these sequences have the asymptotic form
!
!
.
,
cos
by
,
as finite term Fourier series
Represent
and for a given set , define the map
,
Suppose we are given the two pairs of sequences
arising from Dirichlet and Neumann conditions at
(1)
(1)
˜ (1)
˜ (1)
,
,
are the solutions of
where
and boundary conditions
corresponding to
.
!
For our particular operators these sequences have the asymptotic form
!
!
,
.
,
cos
by
,
as finite term Fourier series
Represent
and for a given set , define the map
Suppose we are given the two pairs of sequences
arising from Dirichlet and Neumann conditions at
(1)
(1)
˜ (1)
˜ (1)
,
,
are the solutions of
where
is injective.
the map
!
Theorem. For any
and boundary conditions
corresponding to
.
For our particular operators these sequences have the asymptotic form
!
!
,
.
,
cos
by
,
as finite term Fourier series
Represent
and for a given set , define the map
Suppose we are given the two pairs of sequences
arising from Dirichlet and Neumann conditions at
(1)
(1)
˜ (1)
˜ (1)
,
,
are the solutions of
where
.
!
is injective.
the map
Theorem. For any
and boundary conditions
corresponding to
Proof: Use the asymptotic expansions to show that the block matrix representation
is diagonally dominant.
of
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This case is also far from complete
This case is also far from complete
It would be nice to extend the uniqueness result to be global rather than local.
I see no means of using the prefered technology for the regular inverse Sturm
Liouville problem.
This case is also far from complete
It would be nice to extend the uniqueness result to be global rather than local.
I see no means of using the prefered technology for the regular inverse Sturm
Liouville problem.
Of course, we have to include the singular potential
values instead of different boundary conditions at
.
and use different