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 1 / 44

UV Irradiation and Disinfection: UV Reactors http://water-techology.net/projects/sharjah 2 / 44

UV Irradiation and Disinfection http://em.wikipedia.org/wiki/Pyrimidine dimers 3 / 44

UV Dose: Master Variable (Lagrangian = Particle Specific) Integral ◮ Exposure Time � t Dose = I ( t ) · dt ◮ Intensity Field 0 ◮ Intensity History Discrete ◮ Particle Trajectory n � Dose ≈ I j ( R , z ) · ∆ t j j =1 4 / 44

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) 5 / 44

Dose Distribution (Chiu et al. 1999 [1]) 6 / 44

Lagrangian Actinometry (LA): Dyed Microspheres ◮ Blatchley et al. [4] 7 / 44

Lagrangian Actinometry (LA): Dose-Response Calibration 8 / 44

Lagrangian Actinometry (LA): Flow Cytometry [5] 9 / 44

Lagrangian Actinometry (LA): Dose-Response Calibration 10 / 44

Lagrangian Actinometry (LA): UV Reactor Experiment 11 / 44

Lagrangian Actinometry (LA): UV Reactor FI Distributions 80 A B 70 C 60 50 Abundance 40 30 20 10 0 350 400 450 500 550 600 650 700 750 FI (AU) 12 / 44

Lagrangian Actinometry (LA): Linear Equations Linear Combination [4] k i 1 x 1 + k i 2 x 2 + · · · + k in x n = y i (1) with, i = 1 , 2 , · · · , m , FI channels. UV dose j = 1 , 2 , · · · , n 80 A B 70 C 60 50 Abundance 40 30 20 10 0 350 400 450 500 550 600 650 700 750 FI (AU) 13 / 44

Lagrangian Actinometry: Linear Equations Linear Least-Squares Problem (linear model) y = K x + ǫ (2) with y ∈ R m , measured reactor FI distribution, K ∈ R m × n , dose-response calibration matrix, x ∈ R n dose distribution, ǫ ∼ N (0 , S 2 ) vector of mea- surement errors.                                         y = K =                                 14 / 44

Historical Method Regularization Method Truncated SVD Constrained Regularization Application to Large-scale UV reactors 15 / 44

Constrained Minimization Method (FMINCON) Constrained Minimization Problem (FMINCON) � B x = d → � i x i = 1 x ϕ ( x ) = � y − K x � 2 min subject to (3) 2 0 ≤ x ≤ 1 Trojan102308 (FMINCON) 0.08 1A 1B 1C 0.06 x( ν ) 0.04 0.02 0 0 50 100 150 200 250 300 ν ◮ As of 2006, This Summarizes the Extent of Knowledge on Numerical Methods for LA. 16 / 44

Constrained Minimization Test Problem Objective: Determine if Solution is Stable Under Small Perturbations to y tr = K x ∗ Definitions ◮ y tr = K x ∗ , with x ∗ is a “true solution” ◮ y = K x ∗ + ǫ , with ǫ being the perturbation to y tr x ∗ “true solution” K matrix 0.035 0.02 K(f,0) K(f,48) 0.03 K(f,118) 0.015 K(f,248) 0.025 0.02 K(f, ν ) x( ν ) 0.01 0.015 0.01 0.005 0.005 0 0 200 300 400 500 600 700 800 0 50 100 150 200 250 300 f ν 17 / 44

Constrained Minimization Test Problem: Right-Hand Side −3 7 x 10 6 5 4 y tr (f) + ε 3 2 1 0 0 200 400 600 800 1000 1200 f Data Generation y = K x ∗ + ǫ (4) ǫ ∼ N (0 , S 2 ) (5) with, S 2 = diag( s 2 1 , s 2 2 , · · · , s 2 m ) s i = 10 − 5 √ y tr , i (6) 18 / 44

Scaled Solutions Constrained Weighted Least-Squares x ≥ 0 ϕ = � S − 1 ( y − K x ) � 2 w T x = 1 min (7) 2 , Computed Solution, ˆ x Residuals ||r|| 2 = 1.17E+03 0.1 x hat 4 x * 3 0.08 2 x( ν ) and x hat 1 0.06 (b − Ax) i 0 0.04 −1 −2 0.02 −3 −4 0 0 200 400 600 800 1000 0 50 100 150 200 250 300 t ν 19 / 44

Singular Value Decomposition (SVD) Characteristics Characteristics of Ill-Posed Problems (SVD) [10], [11] 1. The right singular vectors v j become more oscillatory as j increases. 2. The singular values σ j 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 20 / 44

Singular Value Decomposition: SVD SVD A = U ˆ Σ V T (8) � � Σ V T , A = U Σ = diag( σ 1 , σ 2 , . . . , σ n ) (9) 0 A ∈ R m × n , U ∈ R m × m , Σ ∈ R m × n , and V ∈ R n × n . SVD Properties ◮ Σ = diag( σ 1 , σ 2 , . . . , σ n ), σ 1 ≥ σ 2 ≥ · · · ≥ σ n , and ◮ U T U = I m = UU T , V T V = I n = VV T (10) SVD Least-Squares Solution � � � � � � 2 � � U T Σ � � x ∈ R n � b − A x � 2 1 V T x min 2 = min b − (11) � � , U T 0 � � x ∈ R n 2 2 Note: U 1 ∈ R n × m , U 2 ∈ R m − n × m . V T ˆ x = Σ − 1 U T 1 b , ∴ ˆ x = V z (12) 21 / 44

Right Singular Vectors, V 0 0.2 0.15 −0.02 0.1 −0.04 0.05 −0.06 v(j) v(j) 0 −0.08 −0.05 −0.1 −0.1 −0.12 −0.15 −0.14 −0.2 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 j j (a) v 1 (b) v 20 0.2 0.2 0.15 0.15 0.1 0.1 0.05 0.05 v(j) 0 v(j) 0 −0.05 −0.05 −0.1 −0.1 −0.15 −0.15 −0.2 −0.2 0 20 40 60 80 100 120 140 160 0 20 40 60 80 100 120 140 160 j j (c) v 40 (d) v 120 22 / 44

Singular Values, σ j 0 10 −1 10 −2 10 log 10 σ i −3 10 −4 10 −5 10 0 20 40 60 80 100 120 140 160 i 23 / 44

The Discrete Picard Condition 0 10 σ i T y * | |u i −2 10 T y * | log 10 σ i and |u i −4 10 −6 10 −8 10 −10 10 0 50 100 150 200 i 24 / 44

Truncated Solution V T x = z (Rust and O’Leary, [8], [9]) SVD Least-Squares Solution From SVD solution, x = Σ − 1 U T V T ˆ ∴ ˆ x = V z (13) 1 b , Truncated SVD Equation � ( u T i b ) | u T if i b | > τ σ i , ( V T ˜ x ) i = (14) 0 , otherwise, i = 1 , 2 , . . . , n Truncated SVD Solution T b x = Σ − 1 � V T ˜ U 1 (15) 25 / 44

Truncating | U T b | 0 10 σ i T y| |u i −1 10 T y| −2 10 log 10 σ i (K) and |u i τ = 2.0 x 10 −3 −3 10 −4 10 −5 10 −6 10 −7 10 0 20 40 60 80 100 120 140 160 i 26 / 44

TSVD for Test Problem x ∗ “true solution” K matrix 0.035 0.02 K(f,0) K(f,48) 0.03 K(f,118) 0.015 K(f,248) 0.025 0.02 K(f, ν ) x( ν ) 0.01 0.015 0.01 0.005 0.005 0 0 200 300 400 500 600 700 800 0 50 100 150 200 250 300 f ν 27 / 44

Truncated SVD vs Constrained Minimization (FMINCON) solutions FMINCON Solution TSVD solution 0.1 0.035 x hat x TSVD 0.03 x * x * 0.08 0.025 x( ν ) and x TSVD x( ν ) and x hat 0.02 0.06 0.015 0.04 0.01 0.005 0.02 0 −0.005 0 0 50 100 150 200 250 300 0 50 100 150 200 250 300 ν ν 28 / 44

Residual Comparison for truncated SVD and FMINCON Solns. TSVD Residuals FMINCON Residuals ||r|| 2 = 1.0124E+03 ||r|| 2 = 1.17E+03 4 4 3 3 2 2 1 1 (b − Ax) i (b − Ax) i 0 0 −1 −1 −2 −2 −3 −3 −4 −4 0 200 400 600 800 1000 0 200 400 600 800 1000 t f x � 2 Recall: m = 1024 , � b − A ˆ 2 ∈ [978 . 7 , 1069 . 2] 29 / 44

Reactor Data 30 / 44

TSVD vs Full SVD Solution: UV Reactor Data SVD Least-Squares Solution V T ˆ x = Σ − 1 U T 1 b , ∴ ˆ x = V z (16) TSVD, k = 6 30 0.035 25 0.03 20 0.025 15 0.02 10 0.015 x( ν ) x( ν ) 5 0.01 0 0.005 −5 0 −10 −0.005 −15 −20 −0.01 0 50 100 150 200 0 50 100 150 ν ν 31 / 44

Background for Constrained TSVD After Truncation x = Σ − 1 � V T ˜ U T 1 b , (17) z := Σ − 1 � U T if ˜ 1 b then one obtains an n × n linear system, V T ˜ x = ˜ z (18) New Minimization Problem x � 2 z − V T ¯ e T ¯ ϕ = min x ≥ 0 � ˜ subject to x = 1 , (19) 2 , ¯ Since Eqn. was solved by FMINCON, Constrained TSVD scheme is TSVD-FMINCON 32 / 44

TSVD-FMINCON Solution TSVD, k = 6, τ = 2.0E−03 0.035 TSVD−FMINCON TSVD 0.03 0.025 0.02 x( ν ) 0.015 0.01 0.005 0 −0.005 0 20 40 60 80 100 120 140 160 180 200 ν 33 / 44

Bench Scale Reactor: TSVD-FMINCON vs. CFD-I and FMINCON TSVD, k = 6, τ = 2.0E−03 Cum. Density Function 0.035 1.2 TSVD−FMINCON 0.03 CFD 1 FMINCON 0.025 distribution fraction 0.8 cum. fraction 0.02 0.6 0.015 0.4 0.01 0.2 0.005 TSVD−FMINCON 0 0 CFD FMINCON −0.005 −0.2 0 50 100 150 200 0 50 100 150 200 UV dose (mJ/cm 2 ) UV Dose (mJ/cm 2 ) 34 / 44

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 35 / 44

TROJAN102308: TSVD-FMINCON vs. FMINCON Dose Distributions TSVD−FMINCON FMINCON 0.07 0.06 1A 1A 1B 1B 0.06 0.05 1C 1C 2A 2A distribution fraction 0.05 distribution fraction 0.04 2B 2B 2C 2C 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 −0.01 0 50 100 150 200 250 300 0 50 100 150 200 250 300 UV dose (mJ/cm 2 ) UV dose (mJ/cm 2 ) 36 / 44