Measurement Under . . . Approximate Model: . . . How to Solve This . . . Towards Interval Techniques The Thermal . . . Thermal Challenge . . . for Model Validation Thermal Challenge . . . Results Jaime Nava and Vladik Kreinovich Additional Idea: How . . . Department of Computer Science Acknowledgments University of Texas at El Paso Title Page El Paso, Texas 79968, USA jenava@miners.utep.edu ◭◭ ◮◮ vladik@utep.edu ◭ ◮ Page 1 of 20 Go Back Full Screen Close Quit
Measurement Under . . . 1. Measurement Under Exact Model: Case Study Approximate Model: . . . How to Solve This . . . • Case study: find the unit vector � e k in the direction to The Thermal . . . a distant astronomical radio-source. Thermal Challenge . . . • Most accurate method: Very Large Baseline Interfer- Thermal Challenge . . . ometry (VLBI). Results • How it works: we measure the time delay τ i,j,k between Additional Idea: How . . . the signal observed by antennas i and j : Acknowledgments Title Page τ i,j,k = c − 1 · ( � b i − � b j ) · � e k + ∆ t i − ∆ t j , where ◭◭ ◮◮ • � b i is the location of the i -th antenna, and ◭ ◮ • ∆ t i is the bias of the clock on the i -th antenna. Page 2 of 20 • Ideal case: if we knew � b i and ∆ t i with high accuracy, Go Back we could easily find � e k . Full Screen • In practice: we only know � b i and ∆ t i approximately, Close with much lower accuracy than τ i,j,k . Quit
Measurement Under . . . 2. Measurement Under Exact Model: General Case Approximate Model: . . . How to Solve This . . . • In general: to find the desired values x 1 , . . . , x m ( � e k ), The Thermal . . . we use the measured values z 1 , . . . , z n ( τ i,j,k ). Thermal Challenge . . . • Fact: the values z i depend not only on x i , they also Thermal Challenge . . . depend on Results – the values s 1 , . . . , s p of known auxiliary quantities Additional Idea: How . . . (e.g., time of the experiment), and Acknowledgments Title Page – the values y 1 , . . . , y q of the auxiliary quantities which are only approximately known ( � b j and ∆ t i ). ◭◭ ◮◮ ◭ ◮ • In VLBI: we know the exact dependence z = f ( x, s, y ). Page 3 of 20 • Idea: we can determine both x i and y j if we measure each of several ( N ) different objects x in Go Back – several ( Q ) different settings y and Full Screen – several ( P ) different settings s . Close Quit
Measurement Under . . . 3. Measurement Under Exact Model: General Case Approximate Model: . . . (cont-d) How to Solve This . . . The Thermal . . . • Reminder: we measure n values z i for each of N objects Thermal Challenge . . . under Q different settings y and P different settings s . Thermal Challenge . . . • After these measurements: we get n · N · P · Q mea- Results surement results. Additional Idea: How . . . • Thus: we have n · N · P · Q equations z = f ( x, s, y ). Acknowledgments Title Page • We want: to determine N · m + Q · q unknown (to be ◭◭ ◮◮ more exact, approximately known) values x i and y j . ◭ ◮ • Fact: when N and Q are large, the number of equations exceeds the number of unknowns. Page 4 of 20 • Conclusion: we can find the x values for all the objects Go Back and and y values for all the settings. Full Screen Close Quit
Measurement Under . . . 4. Linearizable Case Approximate Model: . . . How to Solve This . . . • Usually, we know the approximate values � x and � y of The Thermal . . . the quantities x and y with accuracies ∆ x and ∆ y . Thermal Challenge . . . • In effect, we know the intervals [ � x − ∆ x , � x + ∆ x ] and Thermal Challenge . . . [ � y − ∆ y , � y + ∆ y ]. Results • These intervals contain the actual values x and y . Additional Idea: How . . . Acknowledgments • In this case, Title Page – we can linearize the system z = f ( x, s, y ) and ◭◭ ◮◮ – solve the resulting system of linear equations in ◭ ◮ def def terms of ∆ x = � x − x and ∆ y = � y − y : Page 5 of 20 q � n � ∂f ∂f � z = f ( � x, s, � y ) + · ∆ x i + · ∆ y j . Go Back ∂x i ∂y j i =1 j =1 Full Screen Close Quit
Measurement Under . . . 5. Approximate Model: Need for Model Validation Approximate Model: . . . How to Solve This . . . • In practice: we often only have an approximate model The Thermal . . . f ( x, s, y ) for the dependence of z on x , s , and y . Thermal Challenge . . . • In such situations: it is desirable to validate this model. Thermal Challenge . . . • Specifically: we want to supplement f ( x, s, y ) with a Results guaranteed accuracy ε > 0 of this approximate model. Additional Idea: How . . . Acknowledgments • Definition: a model is ε -correct if for every k , ∃ x i ∈ Title Page [ � x i − ∆ x , � x i +∆ x ] and ∃ y j ∈ [ � y j − ∆ y , � y j +∆ y ] for which ◭◭ ◮◮ z k ∈ [ f ( x, s, y ) − ε, f ( x, s, y ) + ε ] . � ◭ ◮ • Objective: to find the smallest possible ε > 0 for which Page 6 of 20 the given model is ε -correct. Go Back Full Screen Close Quit
Measurement Under . . . 6. How to Solve This Problem: Outline Approximate Model: . . . How to Solve This . . . • Fact: in the linearizable case, the above conditions of The Thermal . . . the type a ∈ a become linear inequalities. Thermal Challenge . . . • Hence: the problem of finding the smallest such ε be- Thermal Challenge . . . comes a linear programming problem. Results • In this talk: we describe the application of the resulting Additional Idea: How . . . techniques to a benchmark thermal problem. Acknowledgments Title Page • We need to find the accuracy of the approximate model ◭◭ ◮◮ describing the temperature inside the camera. ◭ ◮ • This constitutes a model with several approximately known parameters. Page 7 of 20 • The problem was presented at the 2006 Sandia Valida- Go Back tion Challenge Workshop. Full Screen Close Quit
Measurement Under . . . 7. The Thermal Challenge Problem: In Brief Approximate Model: . . . How to Solve This . . . • Situation: we need to analyze temperature response The Thermal . . . T ( x, t ) of a safety-critical device to a heat flux. Thermal Challenge . . . • Details: a slab of metal (or other material) of thickness Thermal Challenge . . . L = 1.90 cm is exposed to a heat flux q = 3500 W/m 2 . Results • We know: Additional Idea: How . . . Acknowledgments – thermal conductivity k , Title Page – volumetric heat capacity of the material ρC p , ◭◭ ◮◮ – the initial temperature T i = 25 C, ◭ ◮ – an approximate model: � ( k/ρC p ) · t Page 8 of 20 � x � 2 T ( x, t ) = T i + q · L + 1 3 − x L + 1 · 2 · − Go Back L 2 k L �� � � Full Screen � 6 � 2 1 − n 2 · π 2 · ( k/ρC p ) · t n · π · x π 2 · n 2 · exp · cos . Close L 2 L n =1 Quit
Measurement Under . . . 8. Thermal Challenge Problem: Seemingly Natural Approximate Model: . . . Approach How to Solve This . . . The Thermal . . . • Question: we do not know how accurate is the approx- Thermal Challenge . . . imate model. Thermal Challenge . . . • Task: estimate the accuracy ε of the model. Results • Ideal case: compare the model with measurement re- Additional Idea: How . . . sults. Acknowledgments Title Page • Difficulty: it is very difficult to measure temperatures ◭◭ ◮◮ for the desired flux (corr. to a strong fire). ◭ ◮ • Solution: we perform lab experiments with smaller flux values. Page 9 of 20 • Result: the accuracy of the model is low: ≈ 25 ◦ . Go Back Full Screen • Why this is not perfect: this makes predictions inaccu- rate. Close Quit
Measurement Under . . . 9. Thermal Challenge Problem: New Idea Approximate Model: . . . How to Solve This . . . • Reminder: The Thermal . . . – we estimate the model’s accuracy ε based on the Thermal Challenge . . . results of the lab experiments; Thermal Challenge . . . – the resulting accuracy is low ε ≈ 25 ◦ . Results Additional Idea: How . . . • Observation: in our computations, we assumed that Acknowledgments the given values of k and ρC p are exact. Title Page • In practice: ◭◭ ◮◮ – these values k and ρC p are only approximately known; ◭ ◮ – they may change from sample to sample. Page 10 of 20 • Idea: do not assume any value of these quantities, just Go Back assume that for each sample, there are some values. Full Screen Close Quit
Measurement Under . . . 10. Thermal Problem As Part of General Framework Approximate Model: . . . How to Solve This . . . • How this problem fits our general framework: The Thermal . . . – measured quantity z : temperature z = T ; Thermal Challenge . . . – known auxiliary quantity: time s 1 = t ; Thermal Challenge . . . – unknown auxiliary quantities: y 1 = k ; y 2 = ρC p ; Results Additional Idea: How . . . – we know the ≈ dependence z 1 ≈ f ( s 1 , y 1 , y 2 ). Acknowledgments • Additional complexity: the model is only approximate: Title Page | z ( k ) − f ( s ( k ) 1 , y ( k ) 1 , y ( k ) 2 ) | ≤ ε ◭◭ ◮◮ ◭ ◮ for some (unknown) accuracy ε . • Natural idea: once we know z ( k ) = T for different mo- Page 11 of 20 ments t = s ( k ) , find y 1 , y 2 for which ε → min, where: Go Back | z ( k ) − f ( s ( k ) Full Screen 1 , y 1 , y 2 ) | ≤ ε. Close Quit
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