This document provides an introduction to inverse problems and their applications. It summarizes integral equations like Volterra and Fredholm equations of the first and second kind. It also describes inverse problems for partial differential equations, including inverse convection-diffusion, Poisson, and Laplace problems. Applications mentioned include medical imaging, non-destructive testing, and geophysics. Bibliographic references are provided.

1. Σ YSTEMS
Introduction to Inverse Problems
Dimitrios Papadopoulos
Delta Pi Systems
Thessaloniki, Greece

2. Overview
Integral equations
◮ Volterra equations of the first and second type
◮ Fredholm equations of the first and second type
Inverse Problems for PDEs
◮ Inverse convection-diffusion problems
◮ Inverse Poisson problem
◮ Inverse Laplace problem
Applications
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3. Integral equations
◮ Volterra equation of the first kind
t
g(t) = K(t, s)f (s)ds (1)
a
◮ Volterra equation of the second kind
t
f (t) = K(t, s)f (s)ds + g(t) (2)
a
b−a
◮ Mesh with uniform spacing: ti = a + ih, i = 0, 1, . . . , N, h ≡ N
◮ Quadrature rule: trapezoidal
ti i−1
1 1
Z X
K(ti , s)f (s)ds = h( Ki0 f0 + Kij fj + Kii fi ) (3)
a 2 j=1
2
i−1
1 1 X
(1 − hKii )fi = h( Ki0 f0 + Kij fj ) + gi i = 1, . . . , N (4)
2 2 i=1
f0 = g0
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4. Integral equations (cntd)
◮ Fredholm equation of the first kind
b
g(t) = K(t, s)f (s)ds (5)
a
◮ Fredholm equation of the second kind
b
f (t) = λ K(t, s)f (s)ds + g(t) (6)
a
◮ Gaussian quadrature:
N
X
f (ti ) = λ wj K(ti , sj ) + g(ti ) (7)
j=1
˜
with Kij = Kij wj in matrix form:
˜
(I − λK)f = g (8)
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5. Inverse Problems
◮ One Dimensional Convection
u′ (x) = f (x) in (0, 1], u(0) = 0 (9)
with u : [0, 1] → R. Find f : [0, 1] → R which minimizes the total error
1
J(f ) = [u(1) − u(1)]2 + µ
¯ f (x)2 dx (10)
0
where u(1) is the observed boundary value and µ ≥ 0 is a regularization
¯
constant.
1
Solution: f (x) = u(1)x for x ∈ [0, 1]
¯
1+µ
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6. Inverse Problems (cntd)
◮ One Dimensional Diffusion
−u′′ (x) = f (x) in (0, 1), u′ (0) = 0, u′ (1) + u(1) = 0 (11)
with u : [0, 1] → R. Find f (x) : [0, 1] → R which minimizes the total error
1
J(f ) = (u(0) − u(0))2 + (u(1) − u(1))2 + µ
¯ ¯ f (x)2 dx (12)
0
where u(0), u(1) are the observed boundary values and µ ≥ 0 is a
¯ ¯
regularization constant.
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7. Inverse Problems (cntd)
◮ Poisson Equation
ˆ
Given u(x) = U (x) for x ∈ Γ, find f (x) for x ∈ Ω
−∆u = f in Ω
(13)
∂n u + κu = 0 on Γ
In dicrete form as a least squares problem: Find F ∈ Vh which minimizes
the objective function
ˆ Γ
J(F ) = ||U − U ||2 + µ||F ||2
Ω (14)
over Vh , where U ∈ Vh satisfies
(∇U, ∇v)Ω + (κU, v)Γ = (F, v)Ω for all v ∈ Vh (15)
and Vh the space of continuous piecewise linear functions on Ω of mesh
size h(x).
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8. Inverse Problems (cntd)
◮ Laplace Equation
∂u
Given q =
¯ on Γ1 , find f on Γ2 such that
∂n

 −∆u = 0 in Ω
u = 0 on Γ0 ∪ Γ1 (16)
u = f on Γ2

∂u
We define Bf = ∂n on Γ1 . As a least squares problem: Find f ∈ Γ2 which
minimizes the objective function
J(f ) = ||Bf − q ||2 1 + µ||f ||2 2
¯ Γ Γ (17)
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9. Applications
◮ Medical Imaging (Magnetic Resonance Imaging, fMRI, EEG, ECG, etc.)
◮ Non-destructive Testing
◮ Geophysics (Earthquake, petroleum, geothermal energy)
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10. Bibliography
1. A. Kirsch, An Introduction to the Mathematical Theory of Inverse
Problems.
2. V. Isakov, Inverse Problems for Partial Differential Equations.
3. F. Riesz and B. Sz-Nagy, Functional Analysis.
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11. Contact us
Delta Pi Systems
Optimization and Control of Processes and Systems
Thessaloniki, Greece
http://www.delta-pi-systems.eu
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