1

INVERSE PROBLEMSTUTORIAL

H. T. BANKS and MARIE DAVIDIANN. C. STATE UNIVERSITY

MA/ST 810, Fall, 2009

2

Concepts for inverse problems/parameter estimation problemsillustrated by examples—Involves both deterministic and probabilistic/stochastic/statistical analysisIncludes:• Identifiability• Ill-posedness• Stability• Regularization• Approximation• Reduced order modeling {Proper Orthogonal Decomposition(POD)/Principal Component Analysis(PCA)}

3

SOME GENERAL REFERENCES:

1. G.Anger, Inverse Problems in Differential Equations, Plenum ,N.Y.,1990.

2. H.T.Banks and K.Kunisch, Estimation Techniques for DistributedParameter 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-posedProblems, Wiley, N.Y.,1985.

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:

4

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

5

FORWARD PROBLEMvs.

INVERSE PROBLEM

0 0

P

( , , ), z( )

arameter dependent dynamical system:

known,

z( ) , . ., z

,

( ) K

gdz g t z

t R

t z

i e tdt

is a vector

θθ= = ∈Θ

∈

0 0

0

: , z , z( ) : z( ) ,

Forward Given find t for t tInverse Given t for t t find

θθ

≥≥ ∈Θ

6

xm

kc

Mass-spring-dashpot system

F

2

2

0 0(0) (0)

d x dxm c kx Fdt dt

dxx x vdt

+ + =

= =

x equilibrium displacementof mass m=

0 0 0

0 0

: , , , , , , ( ) : ( ) , , , , ,

Forward Given m c k F x v find x t for t tInverse Given x t for t t v and F find m c and k

>≥

7

+

-

+

-

M

m

M

m

Electronic Polarization—electronic cloud displacement

= displacement of negative charge of mass m from equilibrium of electroniccloud center

2

2 ( )ad dm c k QE tdt dt

+ + =( )aE t

8

Usually are not given observations of all of system state z(t):Example(mass-spring-dashpot system):First, rewrite as first order vector system:

00

0

( )( )( ) , ( ) ( ) ( ), ( )

0 1 0( ) ( ) = ,( )

x t xdz tz t z t t zdx t vdtdt

k ctk c F t m mm m m

θ

θ θ

⎛ ⎞ ⎛ ⎞⎜ ⎟= = + = ⎜ ⎟⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞⎛ ⎞⎜ ⎟ ⎜ ⎟= ⎜ ⎟⎜ ⎟ ⎜ ⎟− − ⎝ ⎠⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

=

F

A F

A

9

( , ) ( , )( )( , ) ( )

=(0

1) ( , ) ( )

(1 0)

:

:

: :

, (

:

Observations

Laser vibrometer

Observation operatorProximi

f t z tdx tf t v

ty probe

tdt

f t x t

More likely discrete finitObservation operato

nr

e um

θ θ

θ

θ

=

= =

==

C

C

C

{ }1

) :

( , )n

j j jj

berobservations

y where y f t θ=

≈

10

2

1

model observations:

(

( ) " "

)= ( , )

nj jj

Can formulate as least squares fit ofto

where f is the model solution response or thatpart of the solution that we can observe orthat we car

J y f t

e

θ θ=

−

−∑

!about in design

11

“Model driven” vs. “data driven” inverse problems

:

( , )

( , ) ,

( ,

:

)

j j

j j j j

y f t

y f t is error

Depending on the error may need tointroduce variability int

Model d

o the mo

riven

delin

Data drive

gand analysis

n

θ

θ ε ε

=

= +

12

) / ( , "

)

" ) )

: ( , )j j

a design of spring shock system automotivesmart truck seats

b design of thermal

i System Design

ly conductive epoxies

Model driv

foruse

problem

e

in com

n y

e

s

t

r

f

put

θ=

) )

) (NDE) motherboards

a thermal interrogation of conductive structuresb eddy current based electromagneti

ii Nondestructive Evaluation problem

c damagedetec n

s

tio−

13

xm

kc

Mass-spring-dashpot system

F

2

2

0 0(0) (0)

d x dxm c kx Fdt dt

dxx x vdt

+ + =

= =

/ ( ," " )Designof spring shock systemautomotive smart truck seats

( , ) ( ) " " ( )

Choose k c to provide a given responsex t for a load m and perturbation F t

θ =⎧ ⎫⎨ ⎬⎩ ⎭

14

DESIGN OF THERMALLY CONDUCTIVE COMPOSITE ADHESIVES

GOALSDesign and analysis of thermally conductive composite adhesives (epoxies and gels filledwith highly conductive particles such as aluminum, diamond dust, and carbon)POTENTIAL AND SIGNIFICANCEDevelopment of enhanced thermally conductive adhesives for microelectronic devices,automotive and aeronautical components

, , ( )

( , )( ) ( ) ( ( ) ( , ))

p

p

Determine c and k all spatially varying in

u t xx c x k x u t xt

ufor a desired temperature u or flux u nn

on the boundary

ρ

ρ ∂= ∇ ∇

∂∂

= ∇ ⋅∂

15

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 CompositeAdhesives, 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.

16

DAMAGE DETECTION USING EDDY CURRENT TECHNIQUES

GOALSDevelop fast on-line computational methods for use with highly sensitive magnetic fluxsensors in detection of subsurface damages POTENTIAL AND SIGNIFICANCEDevelopment of portable, real time scanning devices for damage detection in conductivematerials. Potential for fast scanners for nondestructive evaluation in aging aircraft,spacecraft and other structures

( )( )

,

1 ( , ) ( , ) ( , ) ( , ) ( , )

( ,

(

( , ) ,

)

0)

cs

Using measurements of the magnetic vector potentialdetermine any voids in the material characteriz

x y x y i x y i x yx y

I x

for x y

ed by wh

y i

ere

σ ωε ω φμ

σ ωε

σ

⎛ ⎞∇× ∇× = + − −∇⎜ ⎟

⎝ ⎠

=

∈Ω

= +

A

A A

( )( )( , ) ( , ) ( , )cs

x y i x y d for x ya csω φ− − ∈∇ ⋅∫ A n

17

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 usinga 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 computationalmethodology 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 interrogationtechniques 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 forEddy Current Based Nondestructive Evaluation Techniques, Ph.D. Thesis, NCSU,June, 2001.

18

: ( , ) , j j j jData driven y f t is errorθ ε ε= +

Many (most!) of examples lead to the introduction ofvariability into both the modeling and the analysis!!

i) Physiologically Based Pharmacokinetic (PBPK) modeling in toxicokinetics

ii) Modeling of HIV pathogenesis

19

PBPK Models forTCE in Fat Cells

Millions of cells withvarying size, residencetime, vasculature, geometry:“Axial-dispersion” typeadipose tissue compartmentsto embody uncertainphysiological heterogeneitiesin single organism (rat) =intra-individual variability

Inter-individual variability treated with parameters (including dispersionparameters) as random variables –estimate distributions from aggregatedata (multiple rat data) which also contains uncertainty (noise)

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( ) ( )( ) ( ) ( )( )

( )( ) ( )( )

2

0 02 2

,

/

sinsin

v br k l m tv f B br k l m t c v

br k l m t

a c v p c c p b

brbr br a br br

B B B BB B I BI I I B B A BA A A B B

I

dC t Q Q Q Q QV Q C t C t C t C t C t C t Q C tdt P P P P P

C t Q C t Q C t Q Q P

dC tV Q C t C t P

dtC V D CV vC f C f C f C f C

r r

V

π ε

φ λ μ θ λ μ θφ φ φ φ

= − + + + + + −

= + +

= −

⎡ ⎤⎛ ⎞∂ ∂∂= − + − + −⎢ ⎥⎜ ⎟∂ ∂ ∂⎝ ⎠⎣ ⎦

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( )

0

0

2

2 2 21

2

2 2 20

1 1 sinsin sin

1 1 sinsin sin

I I I I IB I BI B B I I IA A A I I

A A A A AA B A BA B B A A IA I I A A

kk k a

C V D C C f C f C f C f Ct r

C V D C CV f C f C f C f Ct r

dC tV Q C t C

dt

θ

θ

φ δ θ χ φ λ μ μφ θ φ φ φ

φ δ θ χ φ λ μ μφ θ φ φ φ

⎡ ⎤⎛ ⎞∂ ∂ ∂∂= + + − + −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

⎡ ⎤⎛ ⎞∂ ∂ ∂∂= + + − + −⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠⎣ ⎦

= − ( )( )( ) ( ) ( ) ( ) ( )

( ) ( ) ( )( )( ) ( ) ( )( )

max

/

/

/

k k

l l l ll l a M

l ll

mm m a m m

tt t a t t

t P

dC t C t C t C tV Q C t v k

P Pdt P

dC tV Q C t C t P

dtdC t

V Q C t C t Pdt

⎛ ⎞ ⎛ ⎞ ⎛ ⎞= − − +⎜ ⎟ ⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠

= −

= −Plus boundary conditionsand initial conditions

Whole-body system of equations

21

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 systemictransport 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, CRSC-TR02-27,NCSU,Sept.2002; Math. Biosciences, submitted.

22

MODELING OF HIV PATHOGENESIS

GOALSDEVELOPMENT OF DYNAMIC MODELS INVOLVING INTRA-AND INTER-INDIVIDUAL VARIABILITY TO AID IN UNDERSTANDING OF FUNDAMENTAL MECHANISMSOF INFECTION AND SPREAD OF DISEASE-AGGREGATEDATA ACROSS POPULATIONS

POTENTIAL AND SIGNIFICANCEPOPULATION LEVEL ESTIMATION OF SPREAD RATES AND EFFICACY IN TREATMENT PROGRAMS FOR EXPOSURE

23

Involves systems of equations of the form (generally nonlinear)

τwhere is a production delay (distributed across the population of cells). That is, one should write

where k is a probability density to be estimated from aggregate data.

Even if k is given, these systems are nontrivial to simulate—requiredevelopment of fundamental techniques.

( ) ( ) ( ) ( ) ( )a c vtdV cV t n A t n C t n V t T tdt

τ= − + − + −

0

( ) ( ) ( ) ( ) ( ) ( )a c vtdV cV t n A t k d n C t n V t T tdt

τ τ τ∞

= − + − + −∫

24

10

20

20

( ) ( ) ( ) ( ) ( ) ( , )

( ) ( ( )) ( ) ( ) ( ) ( , )

( ) ( ( )) ( ) ( ) ( )

( ) ( ( )) ( ) ( , )

r

A C

r

v A

r

v C

u u

V t cV t n A t d n C t p V T

A t r X t A t A t d p V T

C t r X t C t A t d

T t r X t T t p V T S

τ π τ

δ δ γ τ π τ

δ δ γ τ π τ

δ δ

= − + − + −

= − − − − +

= − − + −

= − − − +

∫

∫

∫

HIV Model:

{ }

{ }

2 20

1 10

1

2

( ) ( ; ) ( ; ) ( ) , acute cells

( ) ( ) ( ), ( ) ( ; ) ( ; ) ( )

delay from acute infection to viral production delay from acute infection to chronic infection

r

r

A C A A A

C t C t C t d A

V t V t V t V t V t V t d

T

τ τ π τ

τ τ π τ

ππ

= Ε = =

= + = Ε =

↔↔=

∫

∫

target cells, to tal (infected+uninfected) cellsX =

where

25

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 contextof 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 InfectionUsing Functional Differential Equations, Ph. D. Thesis, NCSU, August, 2002.

26

The problems above are ( as are most others) notoriouslyill-posed!! This concept is difficult to explain in the contextof the problems outlined above—so we turn to some exceedingly simple examples to illustrate the ideas behindwell-posedness! Simplest case:

1

( ) ( )

one observation y for f and need to find preimagef y for a given y

θ

θ ∗ −

− −

=

θ ∗ yf

ΘY1f −

27

Well-posedness:

i. ExistenceIdentifiability

i. Uniqueness

ii. Continuous dependence of solutions on observations

“stability” of inverse problem

}

28

2( ) 1f θ θ= −

1θ2θ1θ 2θ

( ) ( ) 1, 2 j j jy f f jθ θ= = =

θ

1 11 2 1 2

1 2

small ( ) ( )

= small

y y f y f y

θ θ

− −− ⇒ −

−

Lack of continuity of inverse map:Non-uniqueness:Non-existence:

3y−

−−

3 3 3 ( )No such that f yθ θ =

y

1y2y

29

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!!)

21 1 ( ) ( )

!

!

Define J y f for a given yand then apply a standard iterative method to obtaina solution

θ θ= −

Iterative methods:1) Direct search (simplex, Nelder-Mead,………)2) Gradient based (Newton, steepest descent,

conjugate gradient,………)

k+1 k k 1 k. ., : [ ( )] ( )e g Newton J Jθ θ θ θ−′= −

30

0 1 0

0 1

21 1

0

( ) ( ) , ( ) 2( ( ))( ( ))

( ) 2( )( ) 0, , .

( ) 2( )( ) 0, ,

.

For J

J etc

J etc

y f J y f f

θ

θ θ

θ

θ θ θ

θ θ θ

θ′ = − − − < ⇒ >

′ = − − + > ⇒ <

′ ′= − = − −

−

θ∗0θ

0( )f θ′•

k+1 k k 1 k[ ( )] ( ) J Jθ θ θ θ−′= −

θ ∗ 0θ

•

0( )f θ′

1y

31

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(compacti) constrained optimization constraint sets)

implicit(Lagrange multipliers)

a) Tikhonov regularization(1963)ii) regularization (compactification, convexification)

b) regularization by discretization{

{

32

Tikhonov regularization

2 21 0

21: ( ) ( ) ,

" "

( ) ( )

" " !

Idea Problem for J y f is ill posedso replace it by a near by problem for

where is a regularization parameter to beappropri

J y f

ately chosen

β

θ θ

θ θ

β

β θ θ= − −

−

+

= −

−

!PRO: When done correctly, provides convexity and compactnessin 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 done so!!

33

EXAMPLE:

0 1

1 0

1 0

( ) 1 sin( ), 0 100 0, .01,...,1.0,...,10,..., 40,...,80, 100,

, ,

1) =1, 1.5, 0 (tik)2) =.5, .8, 0 (tik1)3) =.

*

5,

f ranging from tothru values

several values of and y

yyy

θ α πθ β β

α θ

α θα θα

= + =

= == =

1 0

1 0

1 0

1 0

1 0

1.6 ( ), 0 (tik2)4) =1, 1.5, 1.0 (tik4)5) 1, 1.5, 1.8 (tik6)6) 1, 1.5, .5 (tik7)7) 1, 1.5, .5

*

**

* (tik8)

not in range of fyyyy

θα θα θα θα θ

= == =

= = == = == = = −

( / )alt tab

34

RUN MOVIE EXAMPLES

-2 -1 0 1 20

0.5

1

1.5

2

θ

f( θ)

f(θ) = 1+sin(π*θ)

f(θ)yhat

-2 -1 0 1 20

0.5

1

1.5

2

2.5

θJ β

( θ)

Jβ(θ) = | yd - f(θ) |2 + β |θ|2

β = 0 β = 0.01

35

-2 -1 0 1 20

0.5

1

1.5

2

θ

f( θ)

f(θ) = 1+sin(π*θ)

f(θ)yhat

-2 -1 0 1 20

0.5

1

1.5

2

2.5

θJ β

( θ)

Jβ(θ) = | yd - f(θ) |2 + β |θ|2

β = 0 β = 0.1

36

-2 -1 0 1 20

0.5

1

1.5

2

θ

f( θ)

f(θ) = 1+sin(π*θ)

f(θ)yhat

-2 -1 0 1 20

1

2

3

4

5

θJ β

( θ)

Jβ(θ) = | yd - f(θ) |2 + β |θ|2

β = 0β = 1

37

-2 -1 0 1 20

0.5

1

1.5

2

θ

f( θ)

f(θ) = 1+sin(π*θ)

f(θ)yhat

-2 -1 0 1 20

5

10

15

20

θJ β

( θ)

Jβ(θ) = | yd - f(θ) |2 + β |θ|2

β = 0β = 6

38

-2 -1 0 1 20

0.5

1

1.5

2

θ

f( θ)

f(θ) = 1+sin(π*θ)

f(θ)yhat

-2 -1 0 1 20

20

40

60

80

100

120

θJ β

( θ)

Jβ(θ) = | yd - f(θ) |2 + β |θ|2

β = 0 β = 40

39

SENSITIVITY

21

k+1

1

(

( , ) ( , )

( ) 2( ( ))) ( ) ?

(?

( ))

How does f t z t change with respect to andhow does this affect the effort to minimize

Recall that J y f f and NewtonJ y fθ θ

θ θ θ

θ θ θ

θ

=

′ ′−

=

= −

−

C

k k 1 k

1 2

[ ( [ ,

)] ( ) ]

stalls for initial valueJ sn

Ji

θ θ θθ θ

−′= −

1θ 2θ

( )f θ( ) 0J θ′ =

40

(

se

:

nsitivity theory:

( )

(

, , )

,

, ( )) ( )

dz g t zd

f z

s t

z t

So w e are interested in

w hich is obtained from general

Exam ple For w e find

satisfies

w here

tds t g gs t

d z

g

t

z

θ θ

θθ

θ

θ

∗

∗

∗ ∗

=

∂ ∂⎛ ⎞ ⎛ ⎞= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

∂ ∂=

∂ ∂

≡

∂∂

∂⎛ ⎞ =⎜ ⎟∂⎝ ⎠

C

( , ( , ), ),

( , ( , ), )

g t z tz

g g t z t

θ θ

θ θθ θ

∗ ∗

∗∗ ∗

∂∂

∂ ∂⎛ ⎞ =⎜ ⎟∂ ∂⎝ ⎠

41

APPROXIMATION/COMPUTATIONAL ISSUES

,

. ,

( , ) ( , ) , As we have noted most observations have the form

where z is the solution of an ordinary or partialdifferential equation In gene

f t

ral one cannot obtain

z tθ θ=C

.

.

these solutions in closed form even if is givenThus one approximationscomputationa

mul so

st turn tluti

o anons

dθ

42

Nk

,

( , , ),

finite difference techniques ,

For example in the case of z satisfying an ODE

one can apply to d

dz g t z

iscretizethe system obtaining an algebraic system for

dt

z

θ=

≈

N N N N Nk+1 0 1

k

k

( )

( , ,..., , ).

. ., , ,

z

z g z z z

e g Runge Kutta predicto

tgive

r corrector stiff methods

n

Ge

b

of ar

yθ=

− −

43

N Nk k

2nN Nj jj=1

N

N

,

ˆ

( ) ( )

( ) ( )

ˆ ˆ

.

:

???,

Thus one must use

in

which yields sol

f z

J y f

What is relationship of

utions

QuestionConvergence preservation of st

to

θ θ

θ θ

θ

θ

θ

=

= −∑

C

, , , ., ,

???

abilitysensitivity well posedness etc of problemssolutions

44

,

. (" ": )

,

In the case of partial differential equation systemsone can introduce finite difference or finite elementapproximations

linear elExample Finite elementsdispersion equation

ements inheat os p−

( , ) ( , )

,

( ) ( , )

, .pulation dispersal

molecular diffusiu t x u t xx F t x

t x x

on etc

θ∂ ∂ ∂⎛ ⎞= +⎜ ⎟∂ ∂ ∂⎝ ⎠

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{ }

NN N Nk kk=1

NNk k

N N N1 2 N

=1

:

,

( ) ( ( ), ( ),...

( ( )

,

, ) ( )

Idea

u t x z t x

Look for approximate solutions of the form

for a given set of basis elements leading

to a system for z t z t z t z=

= Ψ

Ψ

∑

N

N N N

( ))

( , ) ( , ).

t to be

used in f t z tθ θ=C

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Linear ElementsNk ( )xΨ

Nkx N

k+1xNk-1x

x

( )N Ni j

NN N N

N ( ) ( ) ( )

( ) ( ) (

) ( )

( ) x x x dx

dz t z t tdt

leads to finite dimensional system

where

θ

θ

θ ′ ′Ψ Ψ

= +

= ∫

A

A

F

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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 reductiontechniques 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 REMARKS1. Two classes of problems (model/design driven-no data,

and data driven)2. In both classes, may need to introduce variability/un-

certainty (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 develop-ment 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 problems!