Towards Combining Interval Algorithm for the . . . and - - PowerPoint PPT Presentation

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 1 of 23

Towards Combining Interval and Probabilistic Uncertainty in Finite Element Methods

Vladik Kreinovich and Roberto Araiza

Department of Computer Science

Matthew G. Averill

Department of Geological Sciences University of Texas at El Paso El Paso, TX 79968, USA emails vladik@utep.edu raraiza@utep.edu, averill@geo.utep.edu

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 2 of 23

1. Uncertainty in FEM: Probabilistic Approach

• Traditional approach to FEM: we know the exact equa-

tions and the exact values of the parameters xi of these equations.

• In practice: we only know the approximate values of

the corresponding parameters.

• Question: estimate how the uncertainty in the param-

eters of the system can affect the result y of applying the FEM techniques.

• Probabilistic approach – assumption: we know the ex-

act probability distributions of all xi.

• Probabilistic approach – algorithm: e.g., use Monte-

Carlo simulations.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 23

2. Uncertainty in FEM: Interval Approach and Re- maining Problem

• Interval approach – assumption: sometimes, we only

know the lower bounds xi and the upper bounds xi on xi.

• Example: the probabilities of different values of Young

modulus may depend on the manufacturing process.

• Interval approach – objective: find the interval

[ y − ∆, y + ∆] that contains y.

• Interval approach – algorithms: presented in talks by

Muhanna, Nakao, Neumaier, Pownuk.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 4 of 23

3. Uncertainty in FEM: Remaining Problem

• In practice: we sometimes have both interval and prob-

abilistic uncertainty.

• Example:

– for manufacturing-related parameters, we may only know intervals of possible values; – for weather-related parameters, we also know the probabilities of different values (e.g., from the weather records).

• Objective: for different p ∈ [0, 1], find the value ∆(p)

that bound ∆y with probability p.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 5 of 23

4. General Algorithm

• Problem – reminder:

– for some xi, we know intervals [xi, xi]; – for some xj, we know the probability distribution; – we want to find the value ∆(p) that bound ∆y with probability p.

• Algorithm:

– use Monte-Carlo techniques to simulate parameters xj with known probability distributions; – for each such simulation, use interval FEM tech- niques to get an upper bound ∆ for |y − y|; – after several simulations, we get the resulting bounds distribution; – from this distribution, we can find the desired bound ∆(p).

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 6 of 23

5. Case Study: Determining Earth Structure

• Importance: civilization greatly depends on the things

we extract from the Earth: oil, gas, water.

• Need is growing, so we must find new resources.
• Problem: most easy-to-access mineral resources have

been discovered.

• Example: new oil fields are at large depths, under wa-

ter, in remote areas – so drilling is very expensive.

• Objective: predict resources before we invest in drilling.
• How: we know what structures are promising.
• Example: oil and gas concentrate near the top of (nat-

ural) underground domal structures.

• Conclusion: to find mineral resources, we must deter-

mine the structure at different depths z at different locations (x, y).

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 23

6. Data that We Can Use to Determine the Earth Structure

• Available measurement results: those obtained without

drilling boreholes.

• Examples:

– gravity and magnetic measurements; – travel-times ti of seismic ways through the earth.

• Need for active seismic data:

– passive data from earthquakes are rare; – to get more information, we make explosions, and measure how the resulting seismic waves propagate.

• Resulting seismic inverse problem:

– we know the travel times ti; – we want to reconstruct velocities at different depths.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 8 of 23

Hole Tomography Smashed Masked Velocity Models

• 40
• 30
• 20
• 10

Depth (km) xri_zpv7_do_sm.vel

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

V e l

• c

i t y ( k m / s )

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 9 of 23

7. Seismic Inverse Problem: Towards Mathematical Formulation

• General description: wave equation with unknown v(x).
• Difficulty: due to noise, we only know ti.
• Ray approximation: a seismic wave follows the shortest

path t = dℓ v → min; Eikonal equation |∇t| = 1 v.

• Discontinuity: v is only piece-wise continuous; Snell’s

law describes the transition sin(ϕ) v = sin(ϕ′) v′ .

• Ill-posed problem: a change in v outside paths does not

affect observed travel-times ti; hence, many drastically different v(x) are consistent with observations.

• Current solution:

– start with a meaningful first approximation; – use physically motivated iterations.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 23

8. Seismic Inverse Problem: FEM Approach

• v(x) = vj is constant within each element j.
• We know: travel-times ti between known points Ai, Bi.
• We want to find: velocities vj for which ti = ti(v)

def

= min

j

ℓij vj , where:

• min is taken over all paths between Ai and Bi, and
• ℓij is the length of the part of i-th path within ele-

ment j.

• Shortest path: straight inside each element, Snell’s law
• n each border.
• Simplification: use slownesses sj

def

= 1 vj ; ti =

j

ℓij · sj.

• Problem: system is under-determined.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 11 of 23

9. Existing Algorithm for the Seismic Inverse Prob- lem: General Description

• The most widely used: John Hole’s iterative algorithm.
• Starting point: reasonable initial slownesses s(0)

j .

• Starting an iteration: we use current (approximate)

slownesses s(k)

j

to: – find the shortest paths and – compute the corr. travel-times ti =

j

ℓij · s(k)

j .

• Fact: measured travel-times

ti are somewhat different: ∆ti

def

= ti − ti = 0.

• On each iteration:

– we find ∆sj for which

j

ℓij · (sj + ∆sj) = ti; – we take s(k+1)

j

= s(k)

j

+ ∆sj.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 12 of 23

10. Algorithm for the Inverse Problem: Details

• Objective (reminder): find ∆sj s.t. ℓij · ∆sj = ∆ti.
• Simplest case: one path.
• Specifics: under-determined system: 1 equation, many

unknowns ∆sj.

• Idea: no reason for ∆sj to be different: ∆sj ≈ ∆sj′.
• Formalization: minimize

j,j′(∆sj − ∆sj′)2 under the

constraint ℓij · ∆sj = ∆ti.

• Solution: ∆sj = ∆ti

Li for all j, where Li =

j

ℓij.

• Realistic case: several paths; we have ∆sij for different

paths i.

• Idea: least squares

i

(∆sj − ∆sij)2 → min.

• Solution: ∆sj is the average of ∆sij.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 13 of 23

11. Successes, Limitations, Need for Prior Knowledge

• Successes: the algorithm usually leads to reasonable

geophysical models.

• Limitations: often, the resulting velocity model is not

geophysically meaningful.

• Example: resulting velocities outside of the range of

reasonable velocities at this depth.

• What is currently done: trying different initial models

(hacking).

• Problem with this approach: there is no algorithm for

selecting a good starting model; often, dozens of tries are needed – and each try requires hours of computa- tions.

• It is desirable: to incorporate the expert knowledge

into the algorithm for solving the inverse problem.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 14 of 23

Hole Tomography Smashed Masked Velocity Models

• 40
• 30
• 20
• 10

Depth (km) xri_zpv7_do_sm.vel

2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0

V e l

• c

i t y ( k m / s )

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 15 of 23

12. Case of Interval Prior Knowledge

• Additional information: an interval [sj, sj] that con-

tains the (unknown) actual value sj.

• Problem: for ∆sij = ∆ti

Li , we may have s(k)

j

+ ∆sij ∈ [sj, sj].

• Idea: as s(k+1)

j

, take the value from [sj, sj] which is the closest to s(k)

j

+ ∆sij, i.e., cut off at sj (or at sj).

• Problem: since we decreased ∆sij, we have a remaining

discrepancy ∆t′

i def

= ∆ti −

j

ℓij · ∆sij.

• Solution: repeat the same process for ∆t′

i, etc.

• Problem: many iterations instead of one – increase in

computation time.

• Our result: a new linear time algorithm that computes

the final result of these additional iterations.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 16 of 23

13. A New Linear-Time Algorithm

• At each iteration, we have three sets:

– J− = {j : we know∆sij = ∆j

def

= sj − s(k)

j };

– J+ = {j : we know∆sij < ∆j}, – J = −(J− ∪ J+). and quantities A− def =

j∈J− ℓij · ∆j and L+ def

=

j∈J+ ℓij.

• We start with J− = J+ = ∅ and J = {1, . . . , c}.
• At each iteration:

– we compute the median m of the set J (median in terms of sorting by ∆j); – then, by analyzing the elements of the undecided set J one by one, we divide them into subsets P − def = {j : ∆j ≤ ∆m}, P + def = {j : ∆j > ∆m}.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 17 of 23

14. A New Linear-Time Algorithm (cont-d) – we compute a−

def

= A− +

j∈P − ℓij · ∆j and ℓ+ def

= L+ +

j∈P + ℓij;

– then, we compute ∆s = ∆i − a− ℓ+ ; also, among all the values from P +, we select the smallest value, which we will denote by ∆(p+1); – if ∆s > ∆(p+1), then we replace J− with J− ∪ P −, A− with a−, and J with P +; – if ∆s ≤ ∆m, then we replace J+ with J+ ∪ P +, L+ with ℓ+, and J with P −; – finally, if ∆m < ∆s ≤ ∆(p+1), then we replace J− with J− ∪ P −, J+ with J+ ∪ P +, and J with ∅.

• Iterations continue until J = ∅.
• Return ∆sij = ∆j when ∆j ≤ ∆m, else ∆sij = ∆s.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 18 of 23

15. Case of Probabilistic Prior Knowledge

• Description: from prior observations, we know

sj ≈ sj, and we know the st. dev. σj of this value.

• Minimize:

j,j′(∆sij −∆sij′)2 s.t. c

• j=1

ℓij ·∆sij = ∆ti and 1 n ·

c

• j=1

((s(k)

j

+ ∆sij) − sj)2 σ2

j

= 1.

• Solution (Lagrange multipliers): ∆s

def

= 1 n ·

c

• j=1

∆sij, 2 n ·∆sij − 2 n ·∆s+λ·ℓij + 2µ n · σ2

j

·(s(k)

j

+∆sij − sj) = 0.

• Fact: ∆sij is an explicit function of λ, µ, ∆s.
• Algorithm: solve 3 non-linear equations (above one +

2 constraints) with unknowns λ, µ, ∆s.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 19 of 23

16. Combination of Different Types of Prior Knowl- edge

• Need: we often have both:

– prior measurement results – i.e., probabilistic knowl- edge, and – expert estimates – i.e., interval knowledge.

• Minimize:

j,j′(∆sij − ∆sij′)2 s.t. c

• j=1

ℓij · ∆sij = ∆ti, 1 n ·

c

• j=1

((s(k)

j

+ ∆sij) − sj)2 σ2

j

≤ 1, and sj ≤ s(k)

j

+ ∆sij ≤ sj.

• Idea: we minimize a convex function under convex con-

straints; efficient algorithms are known.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 20 of 23

17. Combination of Different Types of Prior Knowl- edge: Algorithm

• Idea – method of alternating projections:

– first, add a correction that satisfy the first con- straint, – then, the additional correction that satisfies the second constraint, – etc.

• Specifics:

– first, add equal values ∆sij to minimize ∆ti; – restrict the values to the nearest points from [sj, sj], – find the extra corrections that satisfy the proba- bilistic constraint, – repeat until converges.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 21 of 23

18. Effects of Discretization: Technique

• Worst-case bound: ∞.
• Geophysical approach – “checkerboard” method:

– add a doubly periodic function ∆v(x) to the solu- tion v(x); – compute corresponding t′

i;

– reconstruct v′(x) from t′

i;

– compare v′(x) with v(x) + ∆v(x).

• Conclusion:

– For small-step ∆v(x), we will not see the difference – hence details of this size in v(x) are not reliable. – If for some step h, we see the difference, this means that details of size h are more reliable.

Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 22 of 23

19. Effects of Discretization: Mathematical Results

• Problem: how to select functions ∆vi(x)?
• Idea: we want a shift-invariant process.
• Analysis: in linear approximation, what matters is the

linear hull of the functions ∆vi(x).

• First result: for bounded ∆vi(x), we get

∆vi(x1, x2) = sin(a1 · x1 + b1) · sin(a2 · x2 + b2).

• Alternative formulation: we require that the family of

functions {∆vi(x)} is optimal w.r.t. a shift-invariant

• ptimality criterion.
• Second result: for each such criterion, we get the same

functions ∆vi(x1, x2) = sin(a1 · x1 + b1) · sin(a2 · x2 + b2).

Uncertainty in FEM: . . . General Algorithm Case Study: . . . Data that We Can Use . . . Seismic Inverse . . . Seismic Inverse . . . Existing Algorithm for . . . Algorithm for the . . . Successes, . . . Case of Interval Prior . . . A New Linear-Time . . . A New Linear-Time . . . Case of Probabilistic . . . Combination of . . . Combination of . . . Effects of . . . Effects of . . . Acknowledgments Title Page ◭◭ ◮◮ ◭ ◮ Page 23 of 23 Go Back Full Screen Close

20. Acknowledgments This work was supported in part by:

• NASA under cooperative agreement NCC5-209,
• NSF grants EAR-0225670 and DMS-0532645,
• Star Award from the University of Texas System, and
• Texas Department of Transportation grant No. 0-5453.