Chapter 3: Discretizing Continuous Inverse Problems
Math 4/896: Seminar in Mathematics
Topic: Inverse Theory
Instructor: Thomas Shores
Department of Mathematics
AvH 10
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Some references:
1
2
3
4
5
C. Groetsch, Inverse Problems in the Mathematical Sciences,
Vieweg-Verlag, Braunschweig, Wiesbaden, 1993. (A charmer!)
M. Hanke and O. Scherzer, Inverse Problems Light: Numerical
Dierentiation, Amer. Math. Monthly, Vol 108 (2001),
512-521. (Entertaining and gets to the heart of the matter
quickly)
A. Kirsch, An Introduction to Mathematical Theory of Inverse
Problems, Springer-Verlag, New York, 1996. (Harder!
Denitely a graduate level text)
A. Tarantola, Inverse Problem Theory and Methods for Model
Parameter Estimation, SIAM, Philadelphia, 2004. (Very
substantial introduction to inverse theory at the graduate level
that emphasises statistical concepts.)
R. Aster, B. Borchers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005. (And the winner is...)
Some references:
1
2
3
4
5
C. Groetsch, Inverse Problems in the Mathematical Sciences,
Vieweg-Verlag, Braunschweig, Wiesbaden, 1993. (A charmer!)
M. Hanke and O. Scherzer, Inverse Problems Light: Numerical
Dierentiation, Amer. Math. Monthly, Vol 108 (2001),
512-521. (Entertaining and gets to the heart of the matter
quickly)
A. Kirsch, An Introduction to Mathematical Theory of Inverse
Problems, Springer-Verlag, New York, 1996. (Harder!
Denitely a graduate level text)
A. Tarantola, Inverse Problem Theory and Methods for Model
Parameter Estimation, SIAM, Philadelphia, 2004. (Very
substantial introduction to inverse theory at the graduate level
that emphasises statistical concepts.)
R. Aster, B. Borchers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005. (And the winner is...)
Some references:
1
2
3
4
5
C. Groetsch, Inverse Problems in the Mathematical Sciences,
Vieweg-Verlag, Braunschweig, Wiesbaden, 1993. (A charmer!)
M. Hanke and O. Scherzer, Inverse Problems Light: Numerical
Dierentiation, Amer. Math. Monthly, Vol 108 (2001),
512-521. (Entertaining and gets to the heart of the matter
quickly)
A. Kirsch, An Introduction to Mathematical Theory of Inverse
Problems, Springer-Verlag, New York, 1996. (Harder!
Denitely a graduate level text)
A. Tarantola, Inverse Problem Theory and Methods for Model
Parameter Estimation, SIAM, Philadelphia, 2004. (Very
substantial introduction to inverse theory at the graduate level
that emphasises statistical concepts.)
R. Aster, B. Borchers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005. (And the winner is...)
Some references:
1
2
3
4
5
C. Groetsch, Inverse Problems in the Mathematical Sciences,
Vieweg-Verlag, Braunschweig, Wiesbaden, 1993. (A charmer!)
M. Hanke and O. Scherzer, Inverse Problems Light: Numerical
Dierentiation, Amer. Math. Monthly, Vol 108 (2001),
512-521. (Entertaining and gets to the heart of the matter
quickly)
A. Kirsch, An Introduction to Mathematical Theory of Inverse
Problems, Springer-Verlag, New York, 1996. (Harder!
Denitely a graduate level text)
A. Tarantola, Inverse Problem Theory and Methods for Model
Parameter Estimation, SIAM, Philadelphia, 2004. (Very
substantial introduction to inverse theory at the graduate level
that emphasises statistical concepts.)
R. Aster, B. Borchers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005. (And the winner is...)
Some references:
1
2
3
4
5
C. Groetsch, Inverse Problems in the Mathematical Sciences,
Vieweg-Verlag, Braunschweig, Wiesbaden, 1993. (A charmer!)
M. Hanke and O. Scherzer, Inverse Problems Light: Numerical
Dierentiation, Amer. Math. Monthly, Vol 108 (2001),
512-521. (Entertaining and gets to the heart of the matter
quickly)
A. Kirsch, An Introduction to Mathematical Theory of Inverse
Problems, Springer-Verlag, New York, 1996. (Harder!
Denitely a graduate level text)
A. Tarantola, Inverse Problem Theory and Methods for Model
Parameter Estimation, SIAM, Philadelphia, 2004. (Very
substantial introduction to inverse theory at the graduate level
that emphasises statistical concepts.)
R. Aster, B. Borchers, C. Thurber, Estimation and Inverse
Problems, Elsivier, New York, 2005. (And the winner is...)
Chapter 3: Discretizing Continuous Inverse Problems
Outline
1
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Outline
1
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Chapter 3: Discretizing Continuous Inverse Problems
A Motivating Example: Integral Equations
Contanimant Transport
Let C (x , t ) be the concentration of a pollutant at point x in a
linear stream, time t , where 0 ≤ x < ∞ and 0 ≤ t ≤ T . The
dening model
∂C
∂t
C (0, t )
=
=
∂2C
∂C
−v
2
∂x
∂x
Cin (t )
D
C (x , t ) → 0, x → ∞
C (x , 0) = C0 (x )
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Solution
Solution:
C (x , T ) =
Z T
0
Cin (t ) f (x , T − t ) dt ,
where
2
x
e −(x −v τ ) /(4D τ )
3
2 πD τ
f (x , τ ) = √
The Inverse Problem
Problem:
Given simultaneous measurements at time T , to estimate the
contaminant inow history. That is, given data
di = C (xi , T ) , i = 1, 2, . . . , m,
to estimate
Cin (t ) , 0 ≤ t ≤ T .
Some Methods
More generally
Problem:
Given the IFK
d (s ) =
Z b
g (x , s ) m (x ) dx
a
and a nite sample of values d (si ), i = 1, 2, . . . , m, to estimate
parameter m (x ).
Methods we discuss at the board:
1
2
3
Quadrature
Representers
Other Choices of Trial Functions
Some Methods
More generally
Problem:
Given the IFK
d (s ) =
Z b
g (x , s ) m (x ) dx
a
and a nite sample of values d (si ), i = 1, 2, . . . , m, to estimate
parameter m (x ).
Methods we discuss at the board:
1
2
3
Quadrature
Representers
Other Choices of Trial Functions
Some Methods
More generally
Problem:
Given the IFK
d (s ) =
Z b
g (x , s ) m (x ) dx
a
and a nite sample of values d (si ), i = 1, 2, . . . , m, to estimate
parameter m (x ).
Methods we discuss at the board:
1
2
3
Quadrature
Representers
Other Choices of Trial Functions
Some Methods
More generally
Problem:
Given the IFK
d (s ) =
Z b
g (x , s ) m (x ) dx
a
and a nite sample of values d (si ), i = 1, 2, . . . , m, to estimate
parameter m (x ).
Methods we discuss at the board:
1
2
3
Quadrature
Representers
Other Choices of Trial Functions
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Outline
1
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Quadrature
Basic Ideas:
Approximate the integrals
di ≈ d (si ) =
Z b
a
g (si , x ) m (x ) dx ≡
Z b
a
gi (x ) m (x ) dx , i = 1, 2, . . . , m
(where the representers or data kernels gi (x ) = gi (si , x )) by
Selecting a set of collocation points xj , j = 1, 2, . . . , n. (It
might be wise to ensure n < m.)
Select an integration approximation method based on the
collocation points.
Use the integration approximations to obtain a linear system
G m = d in terms of the unknowns mj ≡ m (xj ),
j = 1, 2, . . . , n .
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Quadrature
Basic Ideas:
Approximate the integrals
di ≈ d (si ) =
Z b
a
g (si , x ) m (x ) dx ≡
Z b
a
gi (x ) m (x ) dx , i = 1, 2, . . . , m
(where the representers or data kernels gi (x ) = gi (si , x )) by
Selecting a set of collocation points xj , j = 1, 2, . . . , n. (It
might be wise to ensure n < m.)
Select an integration approximation method based on the
collocation points.
Use the integration approximations to obtain a linear system
G m = d in terms of the unknowns mj ≡ m (xj ),
j = 1, 2, . . . , n .
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Quadrature
Basic Ideas:
Approximate the integrals
di ≈ d (si ) =
Z b
a
g (si , x ) m (x ) dx ≡
Z b
a
gi (x ) m (x ) dx , i = 1, 2, . . . , m
(where the representers or data kernels gi (x ) = gi (si , x )) by
Selecting a set of collocation points xj , j = 1, 2, . . . , n. (It
might be wise to ensure n < m.)
Select an integration approximation method based on the
collocation points.
Use the integration approximations to obtain a linear system
G m = d in terms of the unknowns mj ≡ m (xj ),
j = 1, 2, . . . , n .
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Quadrature
Basic Ideas:
Approximate the integrals
di ≈ d (si ) =
Z b
a
g (si , x ) m (x ) dx ≡
Z b
a
gi (x ) m (x ) dx , i = 1, 2, . . . , m
(where the representers or data kernels gi (x ) = gi (si , x )) by
Selecting a set of collocation points xj , j = 1, 2, . . . , n. (It
might be wise to ensure n < m.)
Select an integration approximation method based on the
collocation points.
Use the integration approximations to obtain a linear system
G m = d in terms of the unknowns mj ≡ m (xj ),
j = 1, 2, . . . , n .
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Outline
1
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Representers
Rather than focusing on the value of m at individual points, take a
global view that m (x ) lives in a function space which is spanned by
the representer functions g1 (x ) , g2 (x ) , . . . , gn (x ) , . . .
Basic Ideas:
Make a selection of the basis functions
g1 (x ) , g2 (x ) , . . . , gn (x ) to approximate m (x ), say
m (x ) ≈
n
X
j =1
αj gj (x )
Derive a system Γm = d with a Gramian coecient matrix
Γi ,j = hgi , gj i =
Instructor: Thomas Shores Department of Mathematics
Z b
a
gi (x ) gj (x ) dx
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Representers
Rather than focusing on the value of m at individual points, take a
global view that m (x ) lives in a function space which is spanned by
the representer functions g1 (x ) , g2 (x ) , . . . , gn (x ) , . . .
Basic Ideas:
Make a selection of the basis functions
g1 (x ) , g2 (x ) , . . . , gn (x ) to approximate m (x ), say
m (x ) ≈
n
X
j =1
αj gj (x )
Derive a system Γm = d with a Gramian coecient matrix
Γi ,j = hgi , gj i =
Instructor: Thomas Shores Department of Mathematics
Z b
a
gi (x ) gj (x ) dx
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Representers
Rather than focusing on the value of m at individual points, take a
global view that m (x ) lives in a function space which is spanned by
the representer functions g1 (x ) , g2 (x ) , . . . , gn (x ) , . . .
Basic Ideas:
Make a selection of the basis functions
g1 (x ) , g2 (x ) , . . . , gn (x ) to approximate m (x ), say
m (x ) ≈
n
X
j =1
αj gj (x )
Derive a system Γm = d with a Gramian coecient matrix
Γi ,j = hgi , gj i =
Instructor: Thomas Shores Department of Mathematics
Z b
a
gi (x ) gj (x ) dx
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Example
The Most Famous Gramian of Them All:
Suppose the basis functions turn out to be gi (x ) = x i −1 ,
i = 1, 2, . . . , m, on the interval [0, 1].
Exhibit the infamous Hilbert matrix.
Example
The Most Famous Gramian of Them All:
Suppose the basis functions turn out to be gi (x ) = x i −1 ,
i = 1, 2, . . . , m, on the interval [0, 1].
Exhibit the infamous Hilbert matrix.
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Outline
1
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Chapter 3: Discretizing Continuous Inverse Problems
Other Choices of Trial Functions
Take a still more global view that m (x ) lives in a function space
which is spanned by a suitable spanning set, which might not be
the representers!
Basic Ideas:
Make a selection of the basis functions
h1 (x ) , h2 (x ) , . . . , hn (x ) to approximate m (x ), say
m (x ) ≈
n
X
j =1
αj hj (x )
Derive a system G α = d with a coecient matrix
Gi ,j = hgi , hj i =
Instructor: Thomas Shores Department of Mathematics
Z b
a
gi (x ) hj (x ) dx
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Chapter 3: Discretizing Continuous Inverse Problems
Other Choices of Trial Functions
Take a still more global view that m (x ) lives in a function space
which is spanned by a suitable spanning set, which might not be
the representers!
Basic Ideas:
Make a selection of the basis functions
h1 (x ) , h2 (x ) , . . . , hn (x ) to approximate m (x ), say
m (x ) ≈
n
X
j =1
αj hj (x )
Derive a system G α = d with a coecient matrix
Gi ,j = hgi , hj i =
Instructor: Thomas Shores Department of Mathematics
Z b
a
gi (x ) hj (x ) dx
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Chapter 3: Discretizing Continuous Inverse Problems
Other Choices of Trial Functions
Take a still more global view that m (x ) lives in a function space
which is spanned by a suitable spanning set, which might not be
the representers!
Basic Ideas:
Make a selection of the basis functions
h1 (x ) , h2 (x ) , . . . , hn (x ) to approximate m (x ), say
m (x ) ≈
n
X
j =1
αj hj (x )
Derive a system G α = d with a coecient matrix
Gi ,j = hgi , hj i =
Instructor: Thomas Shores Department of Mathematics
Z b
a
gi (x ) hj (x ) dx
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Outline
1
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Backus-Gilbert Method
Problem: we want to estimate m (x ) at a single point b
x using the
available data, and do it well. How to proceed?
Basic Ideas:
b =
Write m (b
x) ≈ m
m
X
j =1
cj dj
b
b = a A (x ) m (x ) dx
Reduce the integral conditions to m
Ideally A (x ) = δ (x − b
x ). What's the next best thing? This
leads to a quadratic programming problem.
R
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Backus-Gilbert Method
Problem: we want to estimate m (x ) at a single point b
x using the
available data, and do it well. How to proceed?
Basic Ideas:
b =
Write m (b
x) ≈ m
m
X
j =1
cj dj
b
b = a A (x ) m (x ) dx
Reduce the integral conditions to m
Ideally A (x ) = δ (x − b
x ). What's the next best thing? This
leads to a quadratic programming problem.
R
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Backus-Gilbert Method
Problem: we want to estimate m (x ) at a single point b
x using the
available data, and do it well. How to proceed?
Basic Ideas:
b =
Write m (b
x) ≈ m
m
X
j =1
cj dj
b
b = a A (x ) m (x ) dx
Reduce the integral conditions to m
Ideally A (x ) = δ (x − b
x ). What's the next best thing? This
leads to a quadratic programming problem.
R
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo
Chapter 3: Discretizing Continuous Inverse Problems
Motivating Example
Quadrature Methods
Representer Method
Generalizations
Method of Backus and Gilbert
Backus-Gilbert Method
Problem: we want to estimate m (x ) at a single point b
x using the
available data, and do it well. How to proceed?
Basic Ideas:
b =
Write m (b
x) ≈ m
m
X
j =1
cj dj
b
b = a A (x ) m (x ) dx
Reduce the integral conditions to m
Ideally A (x ) = δ (x − b
x ). What's the next best thing? This
leads to a quadratic programming problem.
R
Instructor: Thomas Shores Department of Mathematics
Math 4/896: Seminar in Mathematics Topic: Inverse Theo