Download m(x)dx=d(s)

G(m)=d mathematical model
d
data
m
model
G
operator
d=G(mtrue)+ = dtrue + 
Forward problem: find d given m
Inverse problem (discrete parameter estimation):
find m given d
Discrete linear inverse problem: Gm=d
Continuous inverse problem:
b
 g(s,x)m(x)dx=d(s)
a
g is the kernel
Convolution equation:
b
 g(s-x)m(x)dx=d(s)
a
Example: linear regression for ballistic trajectory
y(t)=m1+m2t-0.5m3t2
o
Y(t)
1
t1 -0.5t12
m1
y1
1
t2 -0.5t22
m2
y2
o
o
1 t3 -0.5t32
o
…..
t
1 tm -0.5tm2
m1 initial altitude, m2 initial vertical velocity,
m3 effective gravitational acceleration
m3 = y3
.
ym
Earthquake location
m=[x ]
G(m)=t
ti=||S.,i-x||2/c+
(arrival time of wave at station i)
Nonlinear problem!


j
Traveltime tomography
T = ∫ 1/v(s)ds = ∫u(s)ds
Tj = ∑ Gij ui
i=1
j-th ray
Gravity
d(s)
h
(x)
∞
∞
2
2
3/2
d(s) =G  h m(x) dx / [(x-s) +h ] =  g(x-s) m(x) dx
-∞
-∞
d(s)
h(x)

∞
d(s) = G -∞m(x)  dx / [(x-s)2+m(x)2]3/2
nonlinear in m(x)
Existence: maybe no model that fits data (bad model, noisy data)
Uniqueness: maybe several (infinite?) number of models that fit data
Instability: small change in data leading to large change in estimate
Analysis:
What are the data?
Discrete or continuous data?
Sources of noise?
What is the mathematical model?
Discrete or continuous model?
What physical laws determine G?
Is G linear or nonlinear?
Any issues of existence, uniqueness, or instability?