Constrained Regularization for Lagrangian Actinometry Eric Cox - - PowerPoint PPT Presentation

Constrained Regularization for Lagrangian Actinometry

Eric Cox

Department of Computer Science Purdue University emcox@purdue.edu

September 21, 2010

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UV Irradiation and Disinfection: UV Reactors

http://water-techology.net/projects/sharjah

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UV Irradiation and Disinfection

http://em.wikipedia.org/wiki/Pyrimidine dimers

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UV Dose: Master Variable (Lagrangian = Particle Specific)

Integral Dose = t I(t) · dt Discrete Dose ≈

n

• j=1

Ij(R, z) · ∆tj

◮ Exposure Time ◮ Intensity Field ◮ Intensity History ◮ Particle Trajectory

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UV Dose Distributions: CFD-I Models

◮

Chiu, et al. [1] (Particle Tracking)

◮

Lyn and Blatchley [2] (CFD models for UV disinfection)

◮

• J. Ducoste et al. [3] (Lagrangian vs. Eulerian)

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Dose Distribution (Chiu et al. 1999 [1])

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Lagrangian Actinometry (LA): Dyed Microspheres

◮ Blatchley et al. [4]

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Lagrangian Actinometry (LA): Dose-Response Calibration

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Lagrangian Actinometry (LA): Flow Cytometry [5]

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Lagrangian Actinometry (LA): Dose-Response Calibration

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Lagrangian Actinometry (LA): UV Reactor Experiment

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Lagrangian Actinometry (LA): UV Reactor FI Distributions

350 400 450 500 550 600 650 700 750 10 20 30 40 50 60 70 80 FI (AU) Abundance A B C

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Lagrangian Actinometry (LA): Linear Equations

Linear Combination [4] ki1x1 + ki2x2 + · · · + kinxn = yi (1) with, i = 1, 2, · · · , m, FI channels. UV dose j = 1, 2, · · · , n

350 400 450 500 550 600 650 700 750 10 20 30 40 50 60 70 80 FI (AU) Abundance A B C 13 / 44

Lagrangian Actinometry: Linear Equations

Linear Least-Squares Problem (linear model) y = Kx + ǫ (2) with y ∈ Rm, measured reactor FI distribution, K ∈ Rm×n, dose-response calibration matrix, x ∈ Rn dose distribution, ǫ ∼ N(0, S2) vector of mea- surement errors. y =                                     K =                                    

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Historical Method Regularization Method Truncated SVD Constrained Regularization Application to Large-scale UV reactors

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Constrained Minimization Method (FMINCON)

Constrained Minimization Problem (FMINCON) min

x ϕ(x) = y − Kx2 2

subject to

• Bx = d →

i xi = 1

0 ≤ x ≤ 1 (3)

50 100 150 200 250 300 0.02 0.04 0.06 0.08

ν x(ν)

Trojan102308 (FMINCON) 1A 1B 1C

◮ As of 2006, This Summarizes the Extent of Knowledge on

Numerical Methods for LA.

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Constrained Minimization Test Problem

Objective: Determine if Solution is Stable Under Small Perturbations to ytr = Kx∗ Definitions

◮ ytr = Kx∗, with x∗ is a “true solution” ◮ y = Kx∗ + ǫ, with ǫ being the perturbation to ytr

K matrix

200 300 400 500 600 700 800 0.005 0.01 0.015 0.02 K(f,ν) f K(f,0) K(f,48) K(f,118) K(f,248)

x∗ “true solution”

50 100 150 200 250 300 0.005 0.01 0.015 0.02 0.025 0.03 0.035 x(ν) ν

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Constrained Minimization Test Problem: Right-Hand Side

200 400 600 800 1000 1200 1 2 3 4 5 6 7 x 10

−3

f ytr(f) + ε

Data Generation y = Kx∗ + ǫ (4) ǫ ∼ N(0, S2) (5) with, S2 = diag(s2

1, s2 2, · · · , s2 m)

si = 10−5√ytr,i (6)

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Scaled Solutions

Constrained Weighted Least-Squares min

x≥0 ϕ = S−1(y − Kx)2 2,

wTx = 1 (7) Computed Solution, ˆ x

50 100 150 200 250 300 0.02 0.04 0.06 0.08 0.1 ν x(ν) and xhat xhat x*

Residuals

200 400 600 800 1000 −4 −3 −2 −1 1 2 3 4 t (b − Ax)i ||r||2 = 1.17E+03

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Singular Value Decomposition (SVD) Characteristics

Characteristics of Ill-Posed Problems (SVD) [10], [11]

• 1. The right singular vectors vj become more oscillatory as j increases.
• 2. The singular values σj of A gradually decay to zero without a

noticeable gap.

• 3. The discrete Picard condition occurs.

◮ For problems with SVD characteristics above, truncated SVD is

effective

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Singular Value Decomposition: SVD

SVD A = U ˆ ΣV T (8) A = U

• Σ
• V T,

Σ = diag(σ1, σ2, . . . , σn) (9) A ∈ Rm×n, U ∈ Rm×m, Σ ∈ Rm×n, and V ∈ Rn×n. SVD Properties

◮ Σ = diag(σ1, σ2, . . . , σn), σ1 ≥ σ2 ≥ · · · ≥ σn, and ◮

UTU = Im = UUT, V TV = In = VV T (10) SVD Least-Squares Solution min

x∈Rn b − Ax2 2 = min x∈Rn

• UT

1

UT

2

• b −
• Σ
• V Tx
• 2

2

, (11) Note: U1 ∈ Rn×m, U2 ∈ Rm−n×m. V Tˆ x = Σ−1UT

1 b,

∴ ˆ x = V z (12)

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Right Singular Vectors, V

20 40 60 80 100 120 140 160 −0.14 −0.12 −0.1 −0.08 −0.06 −0.04 −0.02 j v(j)

(a) v1

20 40 60 80 100 120 140 160 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 j v(j)

(b) v20

20 40 60 80 100 120 140 160 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 v(j) j

(c) v40

20 40 60 80 100 120 140 160 −0.2 −0.15 −0.1 −0.05 0.05 0.1 0.15 0.2 v(j) j

(d) v120

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Singular Values, σj

20 40 60 80 100 120 140 160 10

−5

10

−4

10

−3

10

−2

10

−1

10

log10 σi i

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The Discrete Picard Condition

50 100 150 200 10

−10

10

−8

10

−6

10

−4

10

−2

10

log10 σi and |ui

Ty*|

i

σi |ui

Ty*|

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Truncated Solution V Tx = z (Rust and O’Leary, [8], [9])

SVD Least-Squares Solution From SVD solution, V Tˆ x = Σ−1UT

1 b,

∴ ˆ x = V z (13) Truncated SVD Equation (V T˜ x)i =

• (uT

i b)

σi ,

if |uT

i b| > τ

0,

• therwise,

(14) i = 1, 2, . . . , n Truncated SVD Solution V T˜ x = Σ−1 U1

Tb

(15)

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Truncating |UTb|

20 40 60 80 100 120 140 160 10

−7

10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

10

i log10 σi(K) and |ui

Ty|

τ = 2.0 x 10−3

σi |ui

Ty|

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TSVD for Test Problem

K matrix

200 300 400 500 600 700 800 0.005 0.01 0.015 0.02 K(f,ν) f K(f,0) K(f,48) K(f,118) K(f,248)

x∗ “true solution”

50 100 150 200 250 300 0.005 0.01 0.015 0.02 0.025 0.03 0.035 x(ν) ν

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Truncated SVD vs Constrained Minimization (FMINCON) solutions

TSVD solution

50 100 150 200 250 300 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 ν x(ν) and xTSVD xTSVD x*

FMINCON Solution

50 100 150 200 250 300 0.02 0.04 0.06 0.08 0.1 ν x(ν) and xhat xhat x*

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Residual Comparison for truncated SVD and FMINCON Solns.

TSVD Residuals

200 400 600 800 1000 −4 −3 −2 −1 1 2 3 4 f (b − Ax)i ||r||2 = 1.0124E+03

FMINCON Residuals

200 400 600 800 1000 −4 −3 −2 −1 1 2 3 4 t (b − Ax)i ||r||2 = 1.17E+03

Recall: m = 1024, b − Aˆ x2

2 ∈ [978.7, 1069.2]

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Reactor Data

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TSVD vs Full SVD Solution: UV Reactor Data

SVD Least-Squares Solution V Tˆ x = Σ−1UT

1 b,

∴ ˆ x = V z (16)

50 100 150 200 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 ν x(ν) TSVD, k = 6 50 100 150 −20 −15 −10 −5 5 10 15 20 25 30 ν x(ν) 31 / 44

Background for Constrained TSVD

After Truncation V T˜ x = Σ−1 UT

1 b,

(17) if ˜ z := Σ−1 UT

1 b then one obtains an n × n linear system,

V T˜ x = ˜ z (18) New Minimization Problem ϕ = min

¯ x≥0 ˜

z − V T¯ x2

2,

subject to eT¯ x = 1, (19) Since Eqn. was solved by FMINCON, Constrained TSVD scheme is TSVD-FMINCON

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TSVD-FMINCON Solution

20 40 60 80 100 120 140 160 180 200 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 ν x(ν) TSVD, k = 6, τ = 2.0E−03 TSVD−FMINCON TSVD

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Bench Scale Reactor: TSVD-FMINCON vs. CFD-I and FMINCON

50 100 150 200 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 UV dose (mJ/cm2) distribution fraction TSVD, k = 6, τ = 2.0E−03 TSVD−FMINCON CFD FMINCON 50 100 150 200 −0.2 0.2 0.4 0.6 0.8 1 1.2 UV Dose (mJ/cm2)

• cum. fraction
• Cum. Density Function

TSVD−FMINCON CFD FMINCON

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Large Scale Reactors Tested

Matrix Operating Conditions, y TROJAN 102308 1 (A, B, C) - 9 (A, B, C) WEDECO 111307 1 (A, B, C) - 5 (A, B, C) Trojan Reactor Wedeco Reactor

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TROJAN102308: TSVD-FMINCON vs. FMINCON Dose Distributions

50 100 150 200 250 300 0.01 0.02 0.03 0.04 0.05 0.06 0.07 UV dose (mJ/cm2) distribution fraction TSVD−FMINCON 1A 1B 1C 2A 2B 2C 50 100 150 200 250 300 −0.01 0.01 0.02 0.03 0.04 0.05 0.06 UV dose (mJ/cm2) distribution fraction FMINCON 1A 1B 1C 2A 2B 2C

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WEDECO111307: TSVD-FMINCON vs. FMINCON

50 100 150 200 250 300 0.005 0.01 0.015 0.02 0.025 0.03 UV dose (mJ/cm2) distribution fraction TSVD−FMINCON 1A 1B 1C 2A 2B 2C 50 100 150 200 250 300 −0.005 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 UV dose (mJ/cm2) distribution fraction FMINCON 1A 1B 1C 2A 2B 2C 37 / 44

LA: Prediction of Microbial Inactivation

◮ For disinfection purposes microbial inactivation predictions are

used...

5 10 15 20 25 30 35 40 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 UV Dose [=] mJ/cm2 (hypothetical) P(D)

N N0

• Reactor

≈

n

• j=1

N N0

• batch,j

· Pj(Dj) · ∆Dj

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TROJAN102308: Log inactivation predictions MS2

1 2 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5 3 3.5 4 −log10 (N/N0)

• perating condition

TSVD−FMINCON 1 2 3 4 5 6 7 8 9 0.5 1 1.5 2 2.5 3 3.5 4 −log10 (N/N0)

• perating condition

FMINCON

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WEDECO111307: Log inactivation predictions MS-2

1 2 3 4 5 1 2 3 4 5 −log10 (N/N0)

• perating condition

TSVD−FMINCON 1 2 3 4 5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 −log10 (N/N0) flow number WEDECO/111307 FMINCON

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Summary

◮ SVD leads to the verification that the LA problem shared

characteristics common to ill-posed problems.

◮ The constrained truncated SVD (TSVD-FMINCON) scheme reduced

noise in dose distributions, spurious zero-dose contributions, and for most reactor tests it provided better microbial inactivation predictions when compared to the “historical” method.

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Acknowledgments

◮ Partial Funding: NYSERDA, WRF ◮ Data provided by HydroQual, Inc. UV Validation Research Center

(C. Shen, and K. Scheible)

Questions

emcox@purdue.edu http://www.cs.purdue.edu/homes/emcox/

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References I

Chiu, et al. 1999, J. Env. Engr. 125(1):7-16

• D. Lyn and E. R. Blatchley

2005, J. Env. Engr. 131(6):838-849

• J. Ducoste et al.

2005, J. of Env. Engr. 131(10):1393-1403 Blatchley et al. 2006 J. Env. Engr 132(11):1390-1430.

• H. Shapiro

Practical Flow Cytometry, 2003

• A. Tikhonov

Numerical Methods for The Solution of Ill-Posed Problems, 1995 O’Leary and Rust 1986 SIAM J. Sci. Stat. Comput., 7(2):473-489

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References II

Rust and O’Leary 2008 Inverse Problems, 034005(24):1-30 Rust 1998, NIST Tech. Report. P.C. Hansen 1994 BIT 30(4):658-672

• M. Espa˜

nol Ph.D. Dissertation, Tufts University, 2009 G.H. Golub and C.F.Van Loan. Matrix Computations. 3rd edition.

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