Computational . . . Transportation . . . General Problem of . . . Interval Computations Interval . . . Linearization and their Possible Use Interval Arithmetic: . . . Solving Systems of . . . in Estimating Accuracy of the Optimization: . . . Acknowledgments Optimization-Based Reconstruction Case Study: Chip Design of the Early Universe Case Study: . . . Title Page Vladik Kreinovich ◭◭ ◮◮ Department of Computer Science, University of Texas at El Paso, ◭ ◮ El Paso, TX 79968, USA Page 1 of 65 vladik@utep.edu http://www.cs.utep.edu/vladik Go Back Interval computations website: Full Screen http://www.cs.utep.edu/interval-comp Close Quit
Computational . . . Transportation . . . 1. Computational Problems of Cosmology General Problem of . . . Interval . . . • One of the main objectives of physics: predict mea- Linearization surement results. Interval Arithmetic: . . . • Specifics of cosmology: prediction may take billions of Solving Systems of . . . years to check :-( Optimization: . . . • Objective modified for cosmology: to be able to use Acknowledgments some observable values to correctly reconstruct others. Case Study: Chip Design Case Study: . . . • General idea of prediction – Laplace determinism: Title Page – we know the current values of the quantities; ◭◭ ◮◮ – we know the dynamical equations; ◭ ◮ – we can solve the corresponding Cauchy problem Page 2 of 65 and predict the future (and/or past) values. Go Back Full Screen Close Quit
Computational . . . Transportation . . . 2. Computational Problems of Cosmology: Challenges General Problem of . . . Interval . . . • Known practical limitation of Laplace determinism: Linearization – we only know the current values with uncertainty Interval Arithmetic: . . . (measurement inaccuracy, etc.); Solving Systems of . . . – uncertainty in the initial values gets drastically in- Optimization: . . . creased; Acknowledgments – as a result, after some time, prediction is not prac- Case Study: Chip Design tically possible. Case Study: . . . Title Page • Known example: weather prediction is only possible for a few days. ◭◭ ◮◮ • Cosmology is even more challenging: we need predic- ◭ ◮ tions over billion years: Page 3 of 65 – the Universe started in an almost homogeneous state Go Back (as seen from 3K), and Full Screen – now we have large local inhomogeneities. Close Quit
Computational . . . Transportation . . . 3. Transportation Problem Approach: Important Break- General Problem of . . . through Interval . . . Linearization • Reminder: due to uncertainty, it is not possible to start Interval Arithmetic: . . . w/the Big Bang & predict the present. Solving Systems of . . . • Similarly: it is not possible to start w/the present & Optimization: . . . predict the Big Bang density. Acknowledgments • Important discovery (Brenier, Frisch, H´ enon, Lopere, Case Study: Chip Design Matarrese, Mohayaee, Sobolevski): Case Study: . . . Title Page – we know the current and the initial densities; ◭◭ ◮◮ – based on these densities, we can reconstruct large- scale motions; ◭ ◮ – thus, we can predict current velocities of different Page 4 of 65 large-scale structures. Go Back Full Screen Close Quit
Computational . . . Transportation . . . 4. Transportation Problem Approach: Details General Problem of . . . Interval . . . • Reminder: Linearization – based on current ρ cur ( y ) and the initial ρ init ( x ) den- Interval Arithmetic: . . . sities, Solving Systems of . . . – we can predict current velocities of different large- Optimization: . . . scale structures. Acknowledgments • Detail: we predict the amount of matter ρ ( x, y ) moved Case Study: Chip Design from location x to location y . Case Study: . . . Title Page • Prediction method: transportation problem ◭◭ ◮◮ � ρ ( x, y ) · d 2 ( x, y ) dx dy Minimize ◭ ◮ Page 5 of 65 under the constraints � � Go Back ρ ( x, y ) dy = ρ init ( x ) and ρ ( x, y ) dx = ρ cur ( y ) . Full Screen Close Quit
Computational . . . Transportation . . . 5. Transportation Problem Approach: Results General Problem of . . . Interval . . . • Reminder: based on ρ cur ( y ) and ρ init ( x ), we can predict Linearization velocities � v ( x ) of different large-scale structures. Interval Arithmetic: . . . • Empirical test: 60% of velocities are reconstructed cor- Solving Systems of . . . rectly. Optimization: . . . • Possible reason why not 100%: we only known densities Acknowledgments with uncertainty. Case Study: Chip Design Case Study: . . . • Result: due to data uncertainty, we can only predict v Title Page with uncertainty: v ≈ � v ± ∆. ◭◭ ◮◮ • Fact: in 40% of the cases, observed velocities v are ◭ ◮ different from the predicted values � v . Page 6 of 65 • Natural question: are the observed velocities v within the range [ � v − ∆ , � v + ∆]? Go Back Full Screen Close Quit
Computational . . . Transportation . . . 6. Transportation Problem Approach: Related Com- General Problem of . . . putational Challenge Interval . . . Linearization • Reminder: Interval Arithmetic: . . . – based on ρ cur ( y ) and ρ init ( x ), Solving Systems of . . . – we can predict velocities � v ( x ) of different large-scale Optimization: . . . structures. Acknowledgments • Related challenge: describe how Case Study: Chip Design Case Study: . . . – uncertainty in ρ cur ( y ) and ρ init ( x ) Title Page – propagates via the prediction algorithm. ◭◭ ◮◮ • Plan for this talk: overview algorithms for uncertainty ◭ ◮ propagation. Page 7 of 65 Go Back Full Screen Close Quit
Computational . . . Transportation . . . 7. General Problem of Data Processing under Uncer- General Problem of . . . tainty Interval . . . Linearization • Indirect measurements: way to measure y that are are Interval Arithmetic: . . . difficult (or even impossible) to measure directly. Solving Systems of . . . • Idea: y = f ( x 1 , . . . , x n ) Optimization: . . . Acknowledgments x 1 � ✲ Case Study: Chip Design x 2 � y = f ( � x 1 , . . . , � x n ) � f ✲ Case Study: . . . ✲ · · · Title Page � x n ✲ ◭◭ ◮◮ ◭ ◮ • Problem: measurements are never 100% accurate: � x i � = Page 8 of 65 x i (∆ x i � = 0) hence Go Back y = f ( � x 1 , . . . , � x n ) � = y = f ( x 1 , . . . , x n ) . � Full Screen def What are bounds on ∆ y = � y − y ? Close Quit
Computational . . . Transportation . . . 8. Probabilistic and Interval Uncertainty General Problem of . . . Interval . . . ∆ x 1 Linearization ✲ ∆ x 2 ∆ y Interval Arithmetic: . . . f ✲ ✲ . . . Solving Systems of . . . ∆ x n Optimization: . . . ✲ Acknowledgments Case Study: Chip Design • Traditional approach: we know probability distribution Case Study: . . . for ∆ x i (usually Gaussian). Title Page • Where it comes from: calibration using standard MI. ◭◭ ◮◮ • Problem: calibration is not possible in fundamental sci- ◭ ◮ ence like cosmology. Page 9 of 65 • Natural solution: assume upper bounds ∆ i on | ∆ x i | Go Back hence x i ∈ [ � x i − ∆ i , � x i + ∆ i ] . Full Screen Close Quit
Computational . . . Transportation . . . 9. Interval Computations: A Problem General Problem of . . . Interval . . . x 1 Linearization ✲ x 2 y = f ( x 1 , . . . , x n ) Interval Arithmetic: . . . f ✲ · · · ✲ Solving Systems of . . . x n Optimization: . . . ✲ Acknowledgments Case Study: Chip Design • Given: an algorithm y = f ( x 1 , . . . , x n ) and n intervals Case Study: . . . x i = [ x i , x i ]. Title Page • Compute: the corresponding range of y : ◭◭ ◮◮ [ y, y ] = { f ( x 1 , . . . , x n ) | x 1 ∈ [ x 1 , x 1 ] , . . . , x n ∈ [ x n , x n ] } . ◭ ◮ • Fact: NP-hard even for quadratic f . Page 10 of 65 • Challenge: when are feasible algorithm possible? Go Back • Challenge: when computing y = [ y, y ] is not feasible, Full Screen find a good approximation Y ⊇ y . Close Quit
Computational . . . Transportation . . . 10. Interval Computations as a Particular Case of Global General Problem of . . . Optimization Interval . . . Linearization • Given: an algorithm y = f ( x 1 , . . . , x n ) and n intervals Interval Arithmetic: . . . x i = [ x i , x i ]. Solving Systems of . . . • Compute: the corresponding range of y : Optimization: . . . [ y, y ] = { f ( x 1 , . . . , x n ) | x 1 ∈ [ x 1 , x 1 ] , . . . , x n ∈ [ x n , x n ] } . Acknowledgments Case Study: Chip Design • Reduction to optimization: in the general case, y ( y ): Case Study: . . . Title Page Minimize (Maximize) f ( x 1 , . . . , x n ) ◭◭ ◮◮ where f is directly computable, under the constraints ◭ ◮ x 1 ≤ x 1 ≤ x 1 , . . . , x n ≤ x n ≤ x n . Page 11 of 65 • Cosmological case: f is not directly computable: Go Back def Full Screen f ( x 1 , . . . , x n ) = argmin F ( x 1 , . . . , x n , y 1 , . . . , y m ) . Close Quit
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