Testing Shock Absorbers: How This Problem Is . . . Towards a Faster - PowerPoint PPT Presentation

How to Describe . . . Need to Take . . . Formulation of the . . . Case of Fuzzy Uncertainty Testing Shock Absorbers: How This Problem Is . . . Towards a Faster Need for Parallelization Analysis of the . . . Parallelizable Algorithm Resulting Algorithm . . . Possibility of . . . Christian Servin Home Page Title Page Computational Sciences Program University of Texas at El Paso ◭◭ ◮◮ 500 W. University ◭ ◮ El Paso, TX 79968, USA christians@miners.utep.edu Page 1 of 22 Go Back Full Screen Close Quit

How to Describe . . . Need to Take . . . 1. How to Describe Shock Absorbers Formulation of the . . . • In general, we can use Newton’s law Case of Fuzzy Uncertainty m · d 2 x How This Problem Is . . . � x, dx � dt 2 = f ( t ) + F . Need for Parallelization dt Analysis of the . . . • In practice, the vertical displacement and the vertical Resulting Algorithm . . . velocity are relatively small. Possibility of . . . • Therefore, we can expand the dependence F in Taylor Home Page series and keep only linear terms: F = F 0 − b · x − a · dx dt . Title Page ◭◭ ◮◮ • On the ideally smooth road, with no vertical motion ( x = dx ◭ ◮ dt = 0), we should have F = 0, so F 0 = 0 and Page 2 of 22 m · d 2 x dt 2 + a · dx dt + b · x = f ( t ) . Go Back Full Screen • So, to describe the reaction of the shock absorbers to arbitrary road conditions, we must know m , a , and b . Close Quit

How to Describe . . . Need to Take . . . 2. How to Predict the Reaction of the Shock Ab- Formulation of the . . . sorber on the Given Force f ( t ) Case of Fuzzy Uncertainty • The equations are linear and time-shift-invariant: How This Problem Is . . . m · d 2 x Need for Parallelization dt 2 + a · dx dt + b · x = f ( t ) . Analysis of the . . . Resulting Algorithm . . . • So, the reaction x ( t ) to f ( t ) is also linear and time- Possibility of . . . � shift-invariant: x ( t ) = R ( t − s ) · f ( s ) ds. Home Page • From the above differential equation, we get Title Page R ( t ) = A · exp( − k · t ) · cos( ω · t + ϕ ) . ◭◭ ◮◮ ◭ ◮ • When we apply an impulse force, and measurement errors are negligible, we measure the following values: Page 3 of 22 x i = A · exp( − k · t i ) · cos( ω · t i + ϕ ) . Go Back Full Screen • How to find A , k , ω , and ϕ from these measurement results? Close Quit

How to Describe . . . Need to Take . . . 3. Analysis of the Problem Formulation of the . . . • General formula: reminder Case of Fuzzy Uncertainty How This Problem Is . . . x i = A · exp( − k · t i ) · cos( ω · t i + ϕ ) . Need for Parallelization Analysis of the . . . • There are many techniques for determining frequency ω – and the corresponding phase ϕ . Resulting Algorithm . . . Possibility of . . . • It is also relatively easy to find maxima and minima Home Page within each cycle, i.e., at the moments t i when Title Page cos( ω · t i + ϕ ) = ± 1 . ◭◭ ◮◮ • For these moments of time, we have ◭ ◮ Page 4 of 22 x i ≈ ± A · exp( − k · t i ) and | x i | ≈ A · exp( − k · t i ) . Go Back • Thus, the main remaining problem is to estimate the Full Screen values A and k based on the corresponding values x i . Close Quit

How to Describe . . . Need to Take . . . 4. How to Estimate the Values of A and k Formulation of the . . . • Reminder: we know values t i and x i for which Case of Fuzzy Uncertainty How This Problem Is . . . | x i | = A · exp( − k · t i ) . Need for Parallelization Analysis of the . . . • Problem: dependence on k is non-linear. Resulting Algorithm . . . • Idea: take logarithms: Possibility of . . . Home Page ln( | x i | ) ≈ ln( A ) − k · t i . Title Page • Result: this equation is linear in terms of unknowns k ◭◭ ◮◮ def and z = ln( A ): ◭ ◮ ln( | x i | ) ≈ z − k · t i . Page 5 of 22 • Solution: use the Least Squares Method to solve the Go Back corresponding system of linear equations. Full Screen Close Quit

How to Describe . . . Need to Take . . . 5. Need to Take Uncertainty into Account Formulation of the . . . • In practice, measurements are never 100% accurate. Case of Fuzzy Uncertainty How This Problem Is . . . • Often, we only know the upper bound ∆ i on the mea- Need for Parallelization surement error x i − x ( t i ). Analysis of the . . . • So, we only know that | x ( t i ) | = A · exp( − k · t i ) is in Resulting Algorithm . . . [ X i , X i ], where Possibility of . . . def def Home Page = max(0 , | x i | − ∆ i ); = | x i | + ∆ i . X i X i Title Page • Please note that we cut off the range at 0 from below, ◭◭ ◮◮ since the absolutely value is always non-negative. ◭ ◮ • In general, different values A and k are consistent with these inequalities. Page 6 of 22 • Our objective is to find the range of possible values of Go Back A and k . Full Screen • Thus, we arrive at the following problem. Close Quit

How to Describe . . . Need to Take . . . 6. Formulation of the Problem Formulation of the . . . • We are given the values x i and ∆ i . Case of Fuzzy Uncertainty How This Problem Is . . . • Based on these values, we compute Need for Parallelization X i = max(0 , | x i | − ∆ i ) and X i = | x i | + ∆ i . Analysis of the . . . Resulting Algorithm . . . • Our objective is to find: Possibility of . . . – the smallest A and the largest A values of A and Home Page – the smallest k and the largest k values of k ≥ 0 Title Page among all the values of A and k that, for all n mea- ◭◭ ◮◮ surements i = 1 , . . . , n , satisfy the constraints ◭ ◮ X i ≤ A · exp( − k · t i ) ≤ X i . Page 7 of 22 Go Back Full Screen Close Quit

How to Describe . . . Need to Take . . . 7. Case of Fuzzy Uncertainty Formulation of the . . . • We have guaranteed upper bounds ∆ i on the measure- Case of Fuzzy Uncertainty ment error x i − x ( t i ): | x i − x ( t i ) | ≤ ∆ i . How This Problem Is . . . Need for Parallelization • We often also have smaller bounds ∆ i ( α ) < ∆ i about Analysis of the . . . which we are not 100% certain. Resulting Algorithm . . . • So, for different α ∈ (0 , 1), we have an interval that Possibility of . . . contains x ( t i ) with this degree of uncertainty: Home Page [ x i − ∆ i ( α ) , x i + ∆ i ( α )] . Title Page ◭◭ ◮◮ • These intervals form a fuzzy set containing all this knowledge. ◭ ◮ Page 8 of 22 • In this case, we need to solve several interval problems – corresponding to different values α . Go Back • So, in the following text, we will concentrate on the Full Screen case of interval uncertainty. Close Quit

How to Describe . . . Need to Take . . . 8. Simplification of the Problem Formulation of the . . . • Idea: we apply logarithm to both sides of the inequality Case of Fuzzy Uncertainty How This Problem Is . . . X i ≤ A · exp( − k · t i ) ≤ X i . Need for Parallelization Analysis of the . . . • Result: constraints y i ≤ z − k · t i ≤ y i , where we denoted Resulting Algorithm . . . def def def = ln( X i ) , = ln( X i ) , and z = ln( A ) . y i y i Possibility of . . . Home Page • Problem: find the ranges [ z, z ] and [ k, k ] of values z Title Page and k for which, for all i , ◭◭ ◮◮ y i ≤ z − k · t i ≤ y i . ◭ ◮ • Once we have the bounds z and z , we compute Page 9 of 22 Go Back A = exp( z ) and A = exp( z ) . Full Screen Close Quit

How to Describe . . . Need to Take . . . 9. How This Problem Is Solved Now Formulation of the . . . • To find z , we minimize z under the constraints Case of Fuzzy Uncertainty How This Problem Is . . . y i ≤ z − k · t i ≤ y i . Need for Parallelization Analysis of the . . . • We also maximize z and minimize and maximize k . Resulting Algorithm . . . • In all these problems, we optimize a linear function Possibility of . . . under linear constraints. Home Page • There exist efficient algorithms for solving such linear Title Page programming problems. ◭◭ ◮◮ • These algorithms take time that grows with the prob- ◭ ◮ lem size as O ( n 3 . 5 ). Page 10 of 22 • While this dependence is polynomial, it still grows very Go Back fast for large n . Full Screen • It is therefore desirable to find faster algorithms. Close Quit

How to Describe . . . Need to Take . . . 10. Need for Parallelization Formulation of the . . . • Reminder: we need to find faster algorithms for solving Case of Fuzzy Uncertainty our problem. How This Problem Is . . . Need for Parallelization • In general: one way to speed up computations is to Analysis of the . . . parallelize them, i.e.: Resulting Algorithm . . . – to divide the computations Possibility of . . . – between several computers working in parallel. Home Page • Alas: linear programming is known to be the provably Title Page hardest problem to parallelize (to be precise, P-hard). ◭◭ ◮◮ • Thus: a new algorithm is needed. ◭ ◮ • What we do: in this paper, we propose a new faster and Page 11 of 22 easy-to-parallelize algorithm for solving our problem. Go Back Full Screen Close Quit