optimal design of groundwater remediation systems with
TRANSCRIPT
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Optimal Design of GroundwaterRemediation Systems with Sampling
MethodsC. T. Kelley
Department of Mathematics
Center for Research in Scientific Computation
North Carolina State University
Raleigh, North Carolina, USA
Chinese University of Hong Kong
Hong Kong, January 25, 2006
Supported by NSF, ARO, DOEd.
C. T. Kelley – p.1
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Outline
• Collaborators
• General formulation and example problem• Formulation• Optimization Landscapes• Results
• Implicit Filtering
• Community problems
• Conclusions
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Collaborators
• IFFCO developers from NCSU Math:Tony Choi, Owen Eslinger, Paul Gilmore,Alton Patrick, Vincent Bannister
• NCSU Math: Corey Winton, Dan Finkel, Jörg Gablonsky,Katie Fowler , Chris Kees, Jill Reese, Todd Coffey
• Other Places:
• Boeing: Andrew Booker, John Dennis
• UNC: Casey Miller, Matt Farthing, Glenn Williams
• ERDC: Stacy Howington
• Univ. Trier: Astrid Battermann
• Mich Tech: Alex Mayer
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What’s the problem.
• Control flow of contaminants in groundwater.• Keep plume on site.• Keep concentrations at acceptable levels.• Minimize cost, mass of contaminant,
contaminant concentration . . .
• Control flow and pressure.• Municipal water supplies.• Agriculture.
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Many approaches
• Tightly coupled simulation/optimization (Shoemaker)
• GAs (Mayer, Pinder, Minkser, Yeh . . . )
• Surrogates: response surface, neural nets
Our long-term objectives:
•• Examine many formulation, simulator, optimizercombinations in a portable way.
• Build testbed for both groundwater and optimizationcommunities.
• Design new approaches.
Today: one problem/simulator/optimizer triple.C. T. Kelley – p.5
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What we do.
• Black-box optimization:Use accepted, widely-used, production 3D simulators.• Improved portability/documentation relative to
research codes.• Community might listen to us.• No guarantee of differentiability wrt design
variables.
• Put problems/solutions on the web.http://www4.ncsu.edu/˜ctk/community.html
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Flow in the saturated zone
Ss∂h∂ t
= ∇ · (K∇h)+S ,
Data:
• BC, IC, spatial domain Ω• Ss (specific storage coefficient)
• K (hydraulic conductivity)
• S is the souce/sink term,computed from the design variables.
Output: h (hydraulic head)Typical simulators: ADH, FEMWATER, MODFLOW.
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Species Transport
∂θC∂ t
= ∇ · (θD ·∇C)−∇ · (θvC)+SC.
Data: porosity θ , interphaseDesign: S C mass sources/sinks
• C is concentration, solution of PDE;
• v is velocity, computed from h;
• D is the dispersion tensor, computed from h.
Typical simulator: MT3D
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Computing the fluid velocity v
Darcy’s law says
θv =kµ
(∇p+ρg∇z)
• p = ρg(h− z): fluid pressure
• k: intrinsic permeability; µ : dynamic viscosity
• ρ : density; g: gravitational acceleration
• ∇z: vector in vertical direction
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What’s D
Di j = δi jαt |v|+(αl−αt)viv j
|v|+δi jτD∗
• αl, αt : longitudinal/transverse dispersivities
• τ : tortuosity of the porous medium
• D∗: free liquid diffusivity.
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Design variables
Number and location of wells, pumping rates.Pumping rates and well locations go in the source term forflow
∫
ΩS (t)dΩ =
n
∑i=1
Qi
and for concentration∫
ΩS
C(t)dΩ =n
∑i=1
C(xi)Qi.
Examples:
• Sum of δ functions at well locations.
• Well model with well diameter, well type, ...
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Example: Hydraulic Capture
Minimize total cost:
f T (Q) =n
∑i=1
c0db0i + ∑
Qi<−10−6
c1|Qim|b1(zgs−hmin)b2
︸ ︷︷ ︸
f c
+
∫ t f
0
(
∑i,Qi<−10−6
c2Qi(hi− zgs)+ ∑i,Qi>10−6
c3Qi
)
dt
︸ ︷︷ ︸
f o
,
to keep a contaminant inside a “capture zone”.Ω = [0,1000]× [0,1000]
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Notation
• (xi,yi) are well locations.
• Qi is pumping rate(> 0 for injection, < 0 for extraction.
• di is depth of well i
• hi is head at well i (MODFLOW)
• zgs is elevation of ground surface
• Qm is design pumping rate.
• hmin is minimum allowable pumping rate.
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Boundary conditions: Unconfined aquifer
∂h∂x
∣∣∣∣x=0
=∂h∂y
∣∣∣∣y=0
=∂h∂ z
∣∣∣∣z=0
= 0, t > 0
K∂h∂ z
(x,y,z = h, t > 0) =−1.903×10−8 (m/s).
h(1000,y,z, t > 0) = 20−0.001y(m),h(x,1000,z, t > 0) = 20−0.001x(m),h(x,y,z,0) = hs.
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Constraints I
Simple bounds:
Qemax ≤ Qi ≤ Qimax, i = 1, ...,n
Limits on the pumps.Simple linear inequality:
∑i
Qi ≥ QmaxT ,
limit on total net extraction rate.
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Constraints II
Keep wells away from Dirichlet boundary
0≤ xi,yi ≤ 800.
Bounds on h
hmin ≤ hi ≤ hmax, i = 1, ...,n
No dry holes.Velocity Highly nonlinear function of well locations.50×50×10 grid.
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Formulation Decisions I
• Contain plume: constrain velocity at zone boundary.Test velocity at five downstream locations.Approximate velocity with difference of h.Five new constraints.Need only flow code. Better simulations in progress.
• Implicit filtering deals with bounds naturally.
• Treat constraints as yes/no for sampling method• Stratify by cost.• Avoid simulator if infeasible wrt cheap (linear)
constraints.
• Well is de-installed if pumping rate is suff small.
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Formulation Decisions II
• Discontinuous objective.• 50×50×10 grid. Wells must be on grid nodes.
Move to nearest.• Remove well from array (di = 0) if pumping rate is
too small.
• Treat head constraint and linear constraintsas hidden or yes-no.
• Initial iterate: two extraction, two injection
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Initial iterate
No wells
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
24
23
22.5
22
21.5
21
20.5
20
19.5
Four well configuration
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
20
20.5
21
22
22.5
22
23
21
20.5
22
22.5
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Initial/Final Plumes
Initial Plume Plume after 5 years
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Results
Optimal configuration
0 100 200 300 400 500 600 700 800 900 10000
100
200
300
400
500
600
700
800
900
1000
22
20.5
19.5
20 19.5
20
Cost of Optimization
0 50 100 150 200 250 300 350 4002
3
4
5
6
7
8
9x 10
4
Calls to MODFLOW
Fun
ctio
n V
alue
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Landscapes
Vary (x1,y1) near initial iterate
400450
500550
600650
700
100
150
200
250
300
350
4007.88
7.9
7.92
7.94
7.96
7.98
8
8.02
x 104
xy
Fun
ctio
n V
alue
Vary pumping rate initial iterate
−1.5−1
−0.50
0.51
1.5
x 10−3
−1.5
−1
−0.5
0
0.5
1
1.5
x 10−33
4
5
6
7
8
x 104
well 4
Well 1F
unct
ion
Val
ue
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Other Approaches
100
101
102
103
2
3
4
5
6
7
8
9
10
11
12x 10
4
Calls to MODFLOW
Fun
ctio
n V
alue
HC
IFFCO DIRECT−L Nomad2N Nomad2N+1GA DE APPS
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Wait a minute!
• Optimal point has one well, we start with four.Was this fair?
• How does performance depend on initial iterate?
• Do some methods benefit from special choices?
• How can you construct a “rich” set of initial iterates fortesting?
• We’re trying:• Use DIRECT to find feasible points.• Use statistics to identify clusters.• Sample wisely within the clusters.
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Optimization strategy
minx∈D
f (x)
• Conventional gradient-based methods can fail if f is• multi-modal,• non-convex,• discontinuous,• non-deterministic, or if
• D is not determined by smooth inequalities.
Sampling methods attempt to address these problems.
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Stencil-based sampling methods
• Begin with a base point x.
• Examine points on a stencil;reject or adjust points not in D .
• Determine location of next stencil.
• If f (x) is smallest, shrink the stencil.
Examples: Coordinate Search, Nelder-Mead,Hooke-Jeeves, (P)MDS, GPS, Implicit Filtering
This is not global optimization.
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Example: coordinate search
Sample f at x on a stencil centered at x, scale=h
S(x,h) = x±hei
• Move to the best point.
• If x is the best point, reduce h.
Necessary Conditions: No legal direction points downhill(which is why you reduce h).
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What if x is the best point?Smooth Objective
If f (x)≤minz∈S(x,h) f (z) (stencil failure)then‖∇ f (x)‖= O(h)
So, if (xn,hn) are the points/scales generated by coordinatesearch and f has bounded level sets, then
• hn→ 0 (finitely many grid points/level) and therefore
• any limit point of xn is a critical point of f .
Not a method for smooth problems.
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Model Problemmotivated by the landscapes.
minRN
f
f = fs +φ
• fs smooth, easy to minimize; φ noise
• N is small, f is typically costly to evaluate.
• f has multiple local minimawhich trap most gradient-based algorithms.
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Convergence?
Stencil failure implies that
‖∇ fs(xn)‖= O
(
hn +‖φ‖S(xn,hn)
hn
)
where‖φ‖S(x,h) = max
z∈S|φ(z)|.
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Bottom line
So, if (xn,hn) are the points/scales generated by coordinatesearch, f has bounded level sets, and
limn→∞
(hn +h−1n ‖φ‖S(x,hn)) = 0
then
• hn→ 0 (finitely many grid points/level) and therefore
• any limit point of xn is a critical point of f .
Analysis for Hooke-Jeeves, MPS, GPS is similar.Nelder-Mead is different.
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Implicit Filtering
Accelerate coordinate search with a quasi-Newton method.imfilter(x, f , pmax,τ ,hn,amax)
for k = 0, . . . dofdquasi(x, f , pmax,τ ,hn,amax)
end forpmax, τ , amax are termination parameters
fdquasi = finite difference quasi-Newton method using acentral difference gradient ∇h f .
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fdquasi(x, f , pmax,τ ,h,amax)
p = 1
while p≤ pmax and ‖∇h f (x)‖ ≥ τh docompute f and ∇h f
terminate with success on stencil failure
update the model Hessian H if appropriate; solve
Hd =−∇h f (x)
use a backtracking line search, with at most amax backtracks,
to find a step length λterminate with failure on > amax backtracks
x← x+λd; p← p+1
end whileif p > pmax report iteration count failure
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Implicit Filtering: Start
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Implicit Filtering: Move
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Implicit Filtering: Move
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Implicit Filtering: Stencil Failure
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Implicit Filtering: Shrink/Move
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Implicit Filtering: Termination
−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1−0.2
0
0.2
0.4
0.6
0.8
1
1.2
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Basic Convergence Theorem
Let (xn,hn) be the sequence from implicit filtering.If
• ∇ fs is Lipschitz continuous.
• limn→∞(hn +h−1n ‖φ‖S(x,hn)) = 0
• fdquasi terminates with success for infinitely many n.
then any limit point of xn is a critical point of fs.Convergence rates need more.
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Hidden Constraints
A hidden constraint is violated if the call to f fails.One can (we do) treat all but bound constraints as hidden.What to do?
• Assign a large value.
• Assign a value of infinity and reject the sample. OK forHJ/MDS, bad for IF.
• Assign a value a bit higher than the nearby points.
• Reject and use least squares to compute ∇h f .• Coming in new version.
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NCSU fortran implementation: IFFCO
• Naturally parallel; but watch out for load balancing.
• Use best value in stencil + quasi-Newton search.
• Quasi-Newton model Hessian essential in practice.
• Termination• fdquasi: stencil failure, small gradient, amax, pmax• overall: list of scales, budget, target
• parameters: IFFCO has reasonable defaults
• hidden constraints: f does not return a valueIFFCO is prepared
• MATLAB: imfil.m new version coming soon
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How to get the software
• IFFCO: Implicit Filtering For Constrained Optimization
• New version released May, 2001MPI/PVM/Serial
• ftp to ftp.math.ncsu.edu inFTP/kelley/iffco/IFFCO.tar.gz or email [email protected]://www4.ncsu.edu/˜ctkhttp://www4.ncsu.edu/˜ctk/iffco.html
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Community Problems
• Suite of problems in groundwater remediation3D, flow+transport, varying difficulty.
• We provide or point to simulators/optimization codesthat will produce a formulation and a solution.
• No pretense that formulation or solution is bestpossible.
• Portable, good testbed for optimization codes.
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How to get the Community Problems
• Constantly updated onhttp://www4.ncsu.edu/˜ctk/community.html
• Packages include problems, makefiles, IFFCOexample.You need to get the simulators; we tell you how.
• Tested on• g77: Solaris, Red Hat 7.3,8.0, MAC OSX, IBM-SP• MPI: IBM-SP, several linux clusters
• Three problems in place (only MODFLOW).
• New problems under construction.
• Massive comparison in progressGA, NOMAD, Boeing DE, DIRECT, APPS
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Conclusions
• Optimal design of groundwater remediation problems• Formulation: constraints
specification of problemchoice of simulators
• Community problems• Solution: we like sampling methods
• Sampling methods• Variants of coordinate search• Implicit filtering
C. T. Kelley – p.41