INVERSE PROBLEMS
TUTORIAL
H. T. BANKS and MARIE DAVIDIAN
N. C. STATE UNIVERSITY
MA/ST 810, Fall, 2009
1
Concepts for inverse problems/parameter estimation problems
illustrated by examples—Involves both deterministic and
probabilistic/stochastic/statistical analysis
Includes:
• Identifiability
• Ill-posedness
• Stability
• Regularization
• Approximation
• Reduced order modeling {Proper Orthogonal
Decomposition(POD)/Principal Component Analysis(PCA)}
2
SOME GENERAL REFERENCES:
JOURNALS:
Inverse Problems, Institute of Physics Pub. ,(18 Vol thru 2002)
J. Inverse and Ill-Posed Problems, VSP, (9 Vol thru 2001)
SIAM (J.Control and J.Appl.Math)
BOOKS:
1. G.Anger, Inverse Problems in Differential Equations, Plenum ,N.Y.,1990.
2. H.T.Banks and K.Kunisch, Estimation Techniques for Distributed
Parameter Systems, Birkhauser,Boston,1989.
3. H.T.Banks,M.W.Buksas,and T.Lin, Electromagnetic Material
Interrogation Using Conductive Interfaces and Acoustic Wavefronts, SIAM
FR 21,Philadelphia,2002.
4. J. Baumeister, Stable Solutions of Inverse Problems, Vieweg,Braunschweig,
1987.
5. J.V.Beck,B.Blackwell and C.St.Clair, Inverse Heat Conduction: Ill-posed
3
Problems, Wiley, N.Y.,1985.
6.H.W.Engl and C.W.Groetsch(eds.), Inverse and Ill-posed Problems,
Academic,Orlando,1987.
7. C.W.Groetsch, Inverse Problems in the Mathematical Sciences, Vieweg,
Braunschweig,1993.
8. C.W.Groetsch, The Theory of Tikhonov Regularization for Fredholm
Equations of the First Kind, Pitman,London,1984.
9. B.Hoffman, Regularization for Applied Inverse and Ill-posed
Problems,Teubner,Leipzig,1986.
10. A.N.Tikhonov and V.Y.Arsenin, Solutions of Ill-posed Problems, Winston
and Sons,Washington,1977.
11. C.R.Vogel, Computational Methods for Inverse Problems, SIAM FR23,
Philadelphia,2002
4
FORWARD PROBLEM
vs.
INVERSE PROBLEM
Parameter dependent dynamical system:
dz
= g (t , z , θ ), z(t0 ) = z0 , g known, θ ∈ Θ
dt
K
z(t ) ∈ R , i.e., z(t ) is a vector
Forward : Given θ , z 0 , find z(t ) for t ≥ t0
Inverse : Given z(t ) for t ≥ t0 , find θ ∈ Θ
5
Mass-spring-dashpot system
d 2x
dx
m
+c
+ kx = F
2
dt
dt
dx
x (0 ) = x 0
(0 ) = v 0
dt
x = equilibrium displacement x
of mass m
c
k
m
F
Forward : Given m, c, k, F, x0 , v0 , find x(t) for t > t0
Inverse : Given x(t) for t ≥ t0 , v0 , and F, find m, c, and 6k
Electronic Polarization—electronic cloud displacement
= displacement of negative charge of
mass m from equilibrium of electronic
cloud center
+M
-
+
M
m
d A
dA
m 2 +c
+ k A = QE a (t )
dt
dt
2
-
m
Ea(t)
7
Usually are not given observations of all of system state z(t):
Example(mass-spring-dashpot system):
First, rewrite as first order vector system:
⎛ x(t ) ⎞
⎛ x0 ⎞
dz (t )
⎜
⎟
= A (θ ) z (t ) + F (t ), z0 = ⎜ ⎟
z (t ) = dx(t ) ,
⎜⎜
⎟⎟
dt
⎝ v0 ⎠
⎝ dt ⎠
⎛ 0
A (θ ) = ⎜ k
⎜⎜ −
⎝ m
1 ⎞
⎛ 0 ⎞
⎟ F (t ) = ⎜
⎟ θ=⎛ k , c ⎞
c⎟
F (t ) ⎟
⎜
⎟
⎜
m
m
− ⎟
⎝
⎠
⎜
⎟
m⎠
⎝ m ⎠
8
Observations : f (t ,θ ) = C z (t ,θ )
dx(t )
Laser vibrometer : f (t ,θ ) = v(t ) =
dt
Observation operator : C =( 0 1)
Proximity probe : f (t ,θ ) = x(t )
Observation operator : C = (1 0)
More likely, discrete ( finite number )
observations :
{ y }
j
n
j =1
where y j ≈ f (t j ,θ )
9
Can formulate as least − squares fit of
model to observations:
J (θ )=∑ j=1 y j − f (t j ,θ )
n
2
where f is the model solution(response) or that
part of the solution that we can "observe" or
that we care about in design!
10
“Model driven” vs. “data driven” inverse problems
Model driven : y j = f (t j ,θ )
Data driven : y j = f (t j ,θ ) + ε j , ε j is error
( Depending on the error , may need to
introduce variability into the modeling
and analysis )
11
Model driven : y j = f (t j , θ )
i ) System Design problem s
a ) design of spring / shock system ( automotive,
"smart " truck seats )
b ) design of thermally conductive epoxies for
use in com puter motherboards
ii ) Nondestructive Evaluation (NDE) problem s
a ) thermal interrogation of conductive structures
b ) eddy current − based electromagnetic damage
detection
12
Design of spring / shock system
Mass-spring-dashpot system
(automotive,"smart " truck seats)
d 2x
dx
m
+c
+ kx = F
2
dt
dt
dx
x (0 ) = x 0
(0 ) = v 0
dt
c
k
x
m
F
⎧Choose θ = (k, c) to provide a given response⎫
⎨
⎬
⎩x(t) for a "load " m and perturbation F(t) ⎭
13
DESIGN OF THERMALLY CONDUCTIVE COMPOSITE ADHESIVES
GOALS
Design and analysis of thermally conductive
composite adhesives (epoxies and gels filled
with highly conductive particles such as
aluminum, diamond dust, and carbon)
POTENTIAL AND SIGNIFICANCE
Development of enhanced thermally
conductive adhesives for microelectronic
devices,automotive and aeronautical
components
Determine ρ , c p , and k ( all spatially varying ) in
G
G
G ∂ u (t , x )
G
G
ρ ( x )c p ( x )
= ∇ ( k ( x ) ∇ u ( t , x ))
∂t
G
∂u
for a desired temperature u or flux
= ∇u ⋅ n
∂n
on the boundary
14
References:
1) H.T.Banks and K.L.Bihari, Modeling and estimating uncertainty
in parameter estimation, CRSC-TR99-40, NCSU, Dec.,1999;
Inverse Problems 17(2001),1-17.
2) K.L.Bihari, Analysis of Thermal Conductivity in Composite
Adhesives, Ph.D. Thesis, NCSU, August, 2001.
3) H.T.Banks and K.L.Bihari, Analysis of thermal conductivity in
composite adhesives, CRSC-TR01-20, NCSU, August, 2001;
Numerical Functional Analysis and Optimization, Dec.,2002,
to appear.
15
DAMAGE DETECTION USING EDDY CURRENT TECHNIQUES
GOALS
Develop fast on-line computational methods
for use with highly sensitive magnetic flux
sensors in detection of subsurface damages
POTENTIAL AND SIGNIFICANCE
Development of portable, real time scanning
devices for damage detection in conductive
materials. Potential for fast scanners for
nondestructive evaluation in aging aircraft,
spacecraft and other structures
Using measurements of the magnetic vector potential A,
determine any voids in the material (characterized by σ = 0) where
⎛ 1
⎞
∇×⎜
∇ × A( x, y ) ⎟ = (σ ( x, y ) + iωε ( x, y ) )( −iω A( x, y ) − ∇φ )
⎝ μ ( x, y )
⎠
for ( x, y ) ∈ Ω,
I cs = ∫ (σ ( x, y ) + iωε ( x, y ) )( −iω A( x, y ) − ∇φ ) ⋅ nda for ( x, y ) ∈ cs
cs
16
References:
1) H.T.Banks,M.L.Joyner,B.Wincheski,and W.P.Winfree, Evaluation of material integrity
using reduced order computational methodology, CRSC-TR99-30, NCSU, August, 1999.
2) H.T.Banks,M.L.Joyner,B.Wincheski,and W.P.Winfree, Nondestructive evaluation using
a reduced-order computational methodology, ICASE Tech Rep 2000-10, NASA LaRC,
March 2000; Inverse Problems 16(2000),929-945.
3) H.T.Banks,M.L.Joyner,B.Wincheski,and W.P.Winfree, A reduced order computational
methodology for damage detection in structures, in Nondestructive Evaluation of Ageing
Aircraft, Airports and Aerospace Hardware (A.K.Mal,ed.) SPIE 3994(2000),10-17.
4) H.T.Banks,M.L.Joyner,B.Wincheski,and W.P.Winfree, Electromagnetic interrogation
techniques for damage detection,CRSC-TR01-15,NCSU, June,2001; Proceedings ENDE01
(Kobe, Japan, May,2001), to appear.
5) H.T.Banks,M.L.Joyner,B.Wincheski,and W.P.Winfree, Real time computational
algorithms for eddy current based damage detection, CRSC-TR01-16,NCSU,June,2001;
Inverse Problems, to appear.
6) M.L.Joyner, An Application of a Reduced Order Computational Methodolgy for
Eddy Current Based Nondestructive Evaluation Techniques, Ph.D. Thesis, NCSU,
17
June, 2001.
Data driven : y j = f (t j ,θ ) + ε j , ε j is error
Many (most!) of examples lead to the introduction of
variability into both the modeling and the analysis!!
i) Physiologically Based Pharmacokinetic
(PBPK) modeling in toxicokinetics
ii) Modeling of HIV pathogenesis
18
PBPK Models for
TCE in Fat Cells
Millions of cells with
varying size, residence
time, vasculature,
geometry:
“Axial-dispersion” type
adipose tissue compartments
to embody uncertain
physiological heterogeneities
in single organism (rat) =
intra-individual variability
Inter-individual variability treated with parameters (including dispersion
parameters) as random variables –estimate distributions from aggregate
data (multiple rat data) which also contains uncertainty (noise)
19
Whole-body system of equations
Vv
dCv ( t )
Q
Q
Q
Q
Q
= Qf CB ( t, π − ε 2 ) + br Cbr ( t ) + k Ck ( t ) + l Cl ( t ) + m Cm ( t ) + t Ct ( t ) − QcCv ( t )
dt
Pbr
Pk
Pl
Pm
Pt
Ca ( t ) = ( QcCv ( t ) + QpCc ( t ) ) ( Qc + Qp Pb )
dCbr ( t )
= Qbr ( Ca ( t ) − Cbr ( t ) / Pbr )
dt
⎛ DB ∂CB
⎞⎤
VB ∂ ⎡
∂C
VB B =
vC
sin
φ
−
⎢
⎜
B ⎟ ⎥ + λI μBI ( f I CI (θ0 ) − f BCB ) + λA μBA ( f ACA (θ0 ) − f BCB )
∂φ r2 sin φ ∂φ ⎣
⎝ r2 ∂φ
⎠⎦
Vbr
∂C V D
VI I = I 2 I
r1
∂t
⎡ 1 ∂2CI
∂CI ⎞⎤
1 ∂ ⎛
sin
φ
+
⎢ 2
⎜
⎟⎥ + δθ0 (θ ) χ B (φ ) λI μBI ( f BCB − f I CI ) + μIA ( f ACA − f I CI )
2
sin
sin
φ
θ
φ
φ
φ
∂
∂
∂
⎝
⎠⎦
⎣
∂CA VA DA ⎡ 1 ∂2CA
∂CA ⎞⎤
1 ∂ ⎛
sin
φ
VA
= 2 ⎢ 2
+
⎜
⎟⎥ + δθ0 (θ ) χB (φ ) λAμBA ( f BCB − f ACA ) + μIA ( f I CI − f ACA )
r0 ⎣ sin φ ∂θ 2 sin φ ∂φ ⎝
∂t
∂φ ⎠⎦
dCk ( t )
= Qk ( Ca ( t ) − Ck ( t ) / Pk )
dt
dC ( t )
C (t ) ⎞ ⎛
⎛
C (t ) ⎞ ⎛
C (t ) ⎞
= Ql ⎜ Ca ( t ) − l ⎟ − ⎜ vmax l ⎟ ⎜ kM + l ⎟
Vl l
Pl ⎠ ⎝
Pl ⎠
dt
Pl ⎠ ⎝
⎝
dC ( t )
Vm m = Qm ( Ca ( t ) − Cm ( t ) / Pm )
dt
Plus boundary conditions
dCt ( t )
Vt
= Qt ( Ca ( t ) − Ct ( t ) / Pt )
dt
and initial conditions
Vk
References:
1)R.A.Albanese,H.T.Banks,M.V.Evans,and L.K.Potter, PBPK models for the
transport of trichloroethylene in adipose tissue,CRSC-TR01-03,NCSU,Jan.2001;
Bull. Math Biology 64(2002), 97-131
2)H.T.Banks and L.K.Potter,Well-posedness results for a class of toxicokinetic
models,CRSR-TR01-18,NCSU,July,2001; Discrete and Continuous Dynamical
Systems,submitted
3)L.K.Potter,Physiologically based pharmacokinetic models for the systemic
transport of Trichloroethylene, Ph.D. Thesis,NCSU, August, 2001
4)H.T.Banks and L.K. Potter, Model predictions and comparisions for three
Toxicokinetic models for the systemic transport of TCE,CRSC-TR01-23,NCSU,
August,2001; Mathematical and Computer Modeling 35(2002), 1007-1032
5)H.T.Banks and L.K.Potter, Probabilistic methods for addressing uncertainty
and variability in biological models: Application to a toxicokinetic model, CRSCTR02-27,NCSU,Sept.2002; Math. Biosciences, submitted.
21
MODELING OF HIV PATHOGENESIS
GOALS
DEVELOPMENT OF DYNAMIC
MODELS INVOLVING INTRAAND INTER-INDIVIDUAL
VARIABILITY TO AID IN
UNDERSTANDING OF
FUNDAMENTAL MECHANISMS
OF INFECTION AND SPREAD
OF DISEASE-AGGREGATE
DATA ACROSS POPULATIONS
POTENTIAL AND SIGNIFICANCE
POPULATION LEVEL ESTIMATION OF
SPREAD RATES AND EFFICACY IN
TREATMENT PROGRAMS FOR
EXPOSURE
22
Involves systems of equations of the form (generally nonlinear)
dV
= −cV (t ) + na A(t − τ ) + nc C (t ) − nvtV (t )T (t )
dt
where τ is a production delay (distributed across the population
of cells). That is, one should write
∞
dV
= −cV (t ) + na ∫ A(t − τ )k (τ )dτ + ncC (t ) − nvtV (t )T (t )
dt
0
where k is a probability density to be estimated from aggregate
data.
Even if k is given, these systems are nontrivial to simulate—require
development of fundamental techniques.
23
HIV Model:
V (t ) = −cV (t ) + n
r
A
∫ A(t − τ )dπ (τ ) + n C (t ) − p(V , T )
C
1
0
r
A (t ) = (rv − δ A − δ X (t )) A(t ) − γ ∫ A(t − τ )dπ 2 (τ ) + p (V , T )
0
r
C (t ) = (rv − δ C − δ X (t ))C (t ) + γ ∫ A(t − τ )dπ 2 (τ )
0
T (t ) = (ru − δ u − δ X (t ))T (t ) − p(V , T ) + S
where C ( t ) = Ε 2 {C ( t ; τ )} =
r
∫ C (t ;τ ) d π
2
(τ ) , A = acu te cells
0
r
V ( t ) = V A ( t ) + V C ( t ), V A ( t ) = Ε 1 {V A ( t ; τ )} = ∫ V A ( t ; τ ) d π 1 (τ )
0
π 1 ↔ d elay fro m acu te in fectio n to viral p ro d u ctio n
π 2 ↔ d elay fro m acu te in fectio n to ch ro n ic in fectio n
T = targ et cells, X = to tal (in fected + u n in fected ) cells
24
References:
1) D. Bortz, R. Guy, J. Hood, K. Kirkpatrick, V. Nguyen, and V. Shimanovich,
Modeling HIV infection dynamics using delay equations, in 6th CRSC Industrial
Math Modeling Workshop for Graduate Students, NCSU(July,2000), CRSC
TR00-24, NCSU, Oct, 2000
2) H. T. Banks, D. M. Bortz, and S. E. Holte, Incorporation of variability into the
modeling of viral delays in HIV infection dynamics, CRSC-TR01-25, Sept, 2001;
Math Biosciences, submitted.
3) H.T.Banks and D.M.Bortz, A parameter sensitivity methodology in the context
of HIV delay equation models, CRSC-TR02-24, August, 2002; J. Math. Biology,
submitted
4) D.M.Bortz, Modeling, Analysis,and Estimation of an In Vitro HIV Infection
Using Functional Differential Equations, Ph. D. Thesis, NCSU, August, 2002.
25
The problems above are ( as are most others) notoriously
ill-posed!! This concept is difficult to explain in the context
of the problems outlined above—so we turn to some
exceedingly simple examples to illustrate the ideas behind
well-posedness! Simplest case:
one observation − y for f (θ ) − and need to find preimage
θ = f ( y ) for a given y
∗
−1
θ
Θ
∗
f
f
y
−1
Y
26
Well-posedness:
i. Existence
i. Uniqueness
}
Identifiability
ii. Continuous dependence of solutions on observations
“stability” of inverse problem
27
y
− y3
f (θ ) = 1 − θ 2
y2 −
−y1
θ1 θ2
θ2 θ 1
θ
Non-existence:
No θ3 such that f (θ3 ) = y3
Non-uniqueness:
y j = f (θ j ) = f (θ j ) j = 1, 2
Lack of continuity of inverse map:
y1 − y2 small ⇒ f −1( y1 ) − f −1 ( y2 )
= θ1 −θ2 small
28
Why is this so important???
Why not just apply a good numerical algorithm for a
least squares (for example) fit to try to find the “best”
possible solution???? (Seldom expect zero residual!!)
2
Define J (θ ) = y1 − f (θ )
for a given y1
and then apply a standard iterative method to obtain
a solution !!
Iterative methods:
1) Direct search (simplex, Nelder-Mead,………)
2) Gradient based (Newton, steepest descent,
conjugate gradient,………)
e.g., Newton : θ
k+1
k −1
k
′
= θ −[J (θ )] J (θ )
k
29
θ
k+1
k −1
k
′
= θ −[J (θ )] J (θ )
k
For J (θ ) = y1 − f (θ )
2
, J ′(θ ) = 2( y1 − f (θ ))(− f ′(θ ))
J ′(θ 0 ) = 2(−)(− −) < 0, ⇒ θ 1 > θ 0 , etc.
0
1
0
′
J (θ ) = 2(−)(− +) > 0, ⇒ θ < θ , etc.
f ′(θ )
0
•
0
′
f (θ )
•
−
y1
∗
θ
θ
0
θ
0
θ
∗
30
This behavior is not the fault of steepest descent
algorithms, but is a manifestation of the inherent
“ill-posedness” of the problem!!
How to fix this is the subject of much research over
the past 40 years!! Among topics are:
explicit(compact
i) constrained optimization
constraint sets)
implicit(Lagrange
multipliers)
a) Tikhonov regularization(1963)
ii) regularization
(compactification, convexification)
b) regularization by discretization
{
{
31
Tikhonov regularization
Idea : Problem for J (θ ) = y1 − f (θ ) is ill − posed ,
2
so replace it by a " near − by " problem for
J β (θ ) = y1 − f (θ ) + β θ − θ0
2
2
where β is a regularization parameter to be
" appropriately chosen " !!
PRO: When done correctly, provides convexity and compactness
in the problem!
CON: Even when done correctly, it changes the problem and
solutions to the new problems may not be close to those of original!
Moreover, it is not easy to do correctly or even to know if you have
32
done so!!
EXAMPLE:
f (θ ) = 1 + α sin(πθ ),
β ranging from β = 0 to
100 thru values 0, .01,...,1.0,...,10,..., 40,...,80, 100,
several values of α , θ 0 , and y1
1) α =1, y1 = 1.5, θ 0 = 0 (tik)*
2) α =.5, y1 = .8, θ 0 = 0 (tik1)
3) α =.5, y1 = 1.6 (not in range of f ), θ 0 = 0 (tik2)*
4) α =1, y1 = 1.5, θ 0 = 1.0
(tik4)
5) α = 1, y1 = 1.5, θ 0 = 1.8
(tik6)*
6) α = 1, y1 = 1.5, θ 0 = .5
7) α = 1, y1 = 1.5, θ 0 = −.5
(tik7)*
(tik8)*
( alt / tab )
33
β=0
β = 0.01
f(θ)
y hat
2
2
J (θ) = | yd - f(θ) | + β |θ|
β
f(θ) = 1+sin(π *θ)
2
2.5
2
1.5
1
J (θ)
β
f(θ)
1.5
RUN MOVIE EXAMPLES
0.5
0
-2
1
0.5
-1
0
θ
1
2
0
-2
-1
0
1
2
θ
34
β=0
β = 0.1
f(θ)
y hat
2
2
J (θ) = | yd - f(θ) | + β |θ|
β
f(θ) = 1+sin(π *θ)
2
2.5
2
1.5
J (θ)
β
f(θ)
1.5
1
0.5
0
-2
1
0.5
-1
0
θ
1
2
0
-2
-1
0
1
2
θ
35
β=0
β=1
f(θ)
y hat
2
2
J (θ) = | yd - f(θ) | + β |θ|
β
f(θ) = 1+sin(π *θ)
2
5
4
1.5
J (θ)
β
f(θ)
3
1
2
0.5
0
-2
1
-1
0
θ
1
2
0
-2
-1
0
1
2
θ
36
β=0
β=6
f(θ)
y hat
2
2
20
1.5
15
J (θ)
β
f(θ)
f(θ) = 1+sin(π *θ)
1
10
0.5
0
-2
2
J (θ) = | yd - f(θ) | + β |θ|
β
5
-1
0
θ
1
2
0
-2
-1
0
1
2
θ
37
β=0
β = 40
f(θ)
y hat
2
2
J (θ) = | yd - f(θ) | + β |θ|
β
f(θ) = 1+sin(π *θ)
2
120
100
1.5
J (θ)
β
f(θ)
80
1
60
40
0.5
20
0
-2
-1
0
θ
1
2
0
-2
-1
0
1
2
θ
38
SENSITIVITY
How does f (t,θ ) = C z(t,θ ) change with respect to θ and
how does this affect the effort to minimize
J (θ ) = y1 − f (θ ) ??
2
Recall that J ′(θ ) = 2( y1 − f (θ ))(− f ′(θ )) and Newton
θ k+1 = θ k − [ J ′(θ k )]−1 J (θ k ) stalls for initial values
in [θ1 ,θ2 ]
f (θ)
J′(θ) = 0
θ1
θ2
39
∂f
∂z
So w e are interested in
=C
∂θ
∂θ
w hich is obtained from general
se nsitivity theory:
dz
E xam ple : F or
= g ( t , z , θ ) , w e find s ( t ) ≡
dt
∗
ds ( t ) ⎛ ∂ g ⎞
∂ z (t , θ )
⎛ ∂g ⎞
satisfies
=⎜
⎟ s (t ) + ⎜
⎟
dt
∂θ
⎝ ∂z ⎠
⎝ ∂θ ⎠
∗
∗
∗
∂g
⎛ ∂g ⎞
∗
∗
w here ⎜
(
t
,
z
(
t
,
θ
),
θ
),
=
⎟
∂z
⎝ ∂z ⎠
∗
∂g
⎛ ∂g ⎞
∗
∗
t
z
t
=
(
,
(
,
θ
),
θ
)
⎜
⎟
∂θ
⎝ ∂θ ⎠
40
APPROXIMATION/COMPUTATIONAL ISSUES
As we have noted , most observations have the form
f (t,θ ) = C z(t,θ ),
where z is the solution of an ordinary or partial
differential equation. In general, one cannot obtain
these solutions in closed form even if θ is given.
Thus one must turn to approximations and
computational solutions.
41
For example, in the case of z satisfying an ODE
dz
= g(t, z,θ ),
dt
one can apply finite difference techniques to discretize
the system, obtaining an algebraic system for zkN ≈ z(tk )
given by
N
k+1
z
= g (z , z ,..., z ,θ ).
N
N
0
N
1
N
k
e.g., Runge − Kutta, predictor − corrector, stiff methods
of Gear
42
Thus, one must use
f (θ ) = C z (θ )
N
k
N
k
in
J (θ ) = ∑ j=1 y j − f j (θ )
N
n
N
2
N
ˆ
which yields solutions θ .
N
ˆ
Question : What is relationship of θ to θˆ ???
Convergence, preservation of stability,
sensitivity, well posedness, etc., of problems,
solutions ???
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In the case of partial differential equation systems,
one can introduce finite difference or finite element
approximations.
Example : Finite elements ("linear elements ") in
dispersion equations − heat, population dispersal,
molecular diffusion, etc.
∂u(t, x) ∂ ⎛
∂u(t, x) ⎞
= ⎜θ (x)
⎟ + F(t, x)
∂t
∂x ⎝
∂x ⎠
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Idea : Look for approximate solutions of the form
u (t , x) = ∑
N
N
z (t )Ψ ( x)
N
k=1 k
for a given set of basis elements {Ψ
N
k
}
N N
k k=1
, leading
to a system for z (t ) = ( z (t ), z (t ),..., z (t )) to be
N
N
1
N
2
N
N
used in f (t ,θ ) = C z (t ,θ ).
N
N N
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Linear Elements
Ψ kN ( x)
x
N
N
xk-1
xkN xk+1
leads to finite dimensional system
dz N (t )
= A N (θ ) z N (t ) + F N (t )
dt
where
A (θ ) =
N
(∫
N
′
θ ( x ) Ψ ( x ) Ψ j ( x ) ′ dx
N
i
)
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Finite elements generally result in large
(dimension ~ 10,000-20,000) approximating
systems!! These can be extremely time
consuming in inverse problem calculations.
So there is great interest in model reduction
techniques that will result in substantial
reduction in time! To illustrate one such
technique (Proper Orthogonal Decomposition),
we return to the eddy current based NDE
example.
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SUMMARY REMARKS
1. Two classes of problems (model/design driven-no data,
and data driven)
2. In both classes, may need to introduce variability/uncertainty (recall PBPK, HIV examples ) even when
considering simple case of a single individual
3. If design/model driven efforts are successful (recall eddy
current NDE example), most likely will lead to
validation experiments, data, and necessitate development of statistical models
4. There are significant issues, challenges, and
methodology ( well-posedness, regularization,
approximation/computation, model reduction, etc.)
that are important to consider in both classes of
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problems!