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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