Propagating Range Propagating Range . . . (Uncertainty) and - PowerPoint PPT Presentation

How to Describe . . . Functional . . . Need to Take . . . Propagating Range . . . Propagating Range Propagating Range . . . (Uncertainty) and Importance of . . . Propagating . . . Continuity Information Main Result Examples Through Computations: Home Page From Real-Valued Intervals Title Page to General Sets ◭◭ ◮◮ ◭ ◮ Vladik Kreinovich Page 1 of 28 Department of Computer Science University of Texas at El Paso Go Back University, El Paso, TX 79968, USA vladik@utep.edu Full Screen Close Quit

How to Describe . . . Functional . . . 1. How to Describe Quantities: From Real Values Need to Take . . . to General Sets Propagating Range . . . • Usually, the values of physical quantities are described Propagating Range . . . by real numbers. Importance of . . . Propagating . . . • However, some physical quantities require a more com- Main Result plex description: Examples – some quantities are characterized by a vector (e.g., Home Page force or velocity), Title Page – some by a function (e.g., a current value of a field) ◭◭ ◮◮ or by a geometric shape. ◭ ◮ • In view of this possibility, we will assume that the set S of possible values of each quantity: Page 2 of 28 Go Back – is not necessarily a set of real numbers, – it can be a general set. Full Screen Close Quit

How to Describe . . . Functional . . . 2. Functional Dependencies are Ubiquitous and Need to Take . . . Can Be Complex Propagating Range . . . • In many practical situations, quantities are dependent Propagating Range . . . on each other. Importance of . . . Propagating . . . • Often, we know a function y = f ( x 1 , . . . , x n ) that re- lates quantities x 1 , . . . , x n with a quantity y . Main Result Examples • In simple cases, we have an explicit expression relating Home Page x i and y . Title Page • In more complex cases, we have a sequence of such ◭◭ ◮◮ expressions ◭ ◮ – we first determine some intermediate quantities z j in terms of x i , Page 3 of 28 – then other intermediate quantities z k in terms of z j , Go Back – . . . Full Screen – finally, y in terms of the the intermediate quantities Close z j (and maybe also in terms of x i ). Quit

How to Describe . . . Functional . . . 3. Definition Need to Take . . . • Let n and N be natural numbers, and let S 1 , . . . , S n be Propagating Range . . . sets. Propagating Range . . . Importance of . . . • A computation scheme f of length N w/ n inputs is a Propagating . . . seq. of tuples t n + j ( j = 1 , . . . , N ) each of which has: Main Result – a set S n + j ; Examples – a finite sequence of positive integers Home Page a ( j, 1) < . . . < a ( j, k ( j )) < n + j ; and Title Page ◭◭ ◮◮ – a function f n + j : S a ( j, 1) × . . . × S a ( j,k ( j )) → S n + j . ◭ ◮ • Let us select a sequence x 1 ∈ S 1 , . . . , x n ∈ S n . Page 4 of 28 • Once the values x 1 , . . . , x n + j − 1 are defined, the next value x n + j is defined as f n + j ( x a ( j, 1) , . . . , x a ( j,k ( j )) ) . Go Back • The value x n + N is called the result f ( x 1 , . . . , x n ) of ap- Full Screen plying f to x i . Close Quit

How to Describe . . . Functional . . . 4. Example Need to Take . . . • The expression f ( x 1 ) = x 1 · (1 − x 1 ) can be described Propagating Range . . . by the following computation scheme: Propagating Range . . . Importance of . . . – first, we compute x 2 = 1 − x 1 , Propagating . . . – then we compute y = x 3 = x 1 · x 2 . Main Result • In this case: Examples Home Page – S 1 = S 2 = S 3 = I R, Title Page – on the first intermediate step, we have a function of one variable f 2 ( a ) = 1 − a ; ◭◭ ◮◮ – on the second computation step, we have a function ◭ ◮ of two variables f 3 ( a, b ) = a · b . Page 5 of 28 Go Back Full Screen Close Quit

How to Describe . . . Functional . . . 5. Intermediate Results as Functions of the Inputs Need to Take . . . • The result of each intermediate step is a function of Propagating Range . . . the inputs: x n + j = g n + j ( x 1 , . . . , x n ). Propagating Range . . . Importance of . . . • Then, g n + N ( x 1 , . . . , x n ) = f ( x 1 , . . . , x n ). Propagating . . . • The function g n + j appears if we “truncate” the original Main Result computation scheme on the j -th step. Examples Home Page • The original values x 1 , . . . , x n can also be viewed as functions of the n input variables x 1 , . . . , x n : Title Page g i ( x 1 , . . . , x i − 1 , x i , x i +1 , . . . , x n ) = x i . ◭◭ ◮◮ ◭ ◮ • In terms of these functions, each computation step Page 6 of 28 takes the form Go Back x n + j = g n + j ( x 1 , . . . , x n ) = Full Screen f n + j ( g a ( j, 1) ( x 1 , . . . , x n ) , . . . , g a ( j,k ( j )) ( x 1 , . . . , x n )) . Close Quit

How to Describe . . . Functional . . . 6. Need to Take Uncertainty into Account Need to Take . . . • In practice, we only have partial information about the Propagating Range . . . inputs x i . Propagating Range . . . Importance of . . . • For each i , there is a whole set X i of values which are Propagating . . . consistent with our knowledge. Main Result • In general, different values x i ∈ X i lead to different Examples values y = f ( x 1 , . . . , x n ). Home Page • It is therefore desirable to find the range of possible Title Page values, i.e., the set ◭◭ ◮◮ def f ( X 1 , . . . , X n ) = { f ( x 1 , . . . , x n ) : x 1 ∈ X 1 , . . . , x n ∈ X n } . ◭ ◮ Page 7 of 28 • If it is difficult to compute the range, we need at least an enclosure Y ⊇ f ( X 1 , . . . , X n ) for this range. Go Back Full Screen Close Quit

How to Describe . . . Functional . . . 7. Types of Sets for Describing Uncertainty Need to Take . . . • In interval computations, we usually assume: Propagating Range . . . Propagating Range . . . – that the set S i is the set of real numbers, and Importance of . . . – that the set X i is an interval. Propagating . . . • However, it is also possible that the set X i is more Main Result general. Examples Home Page • The set X i may be a multi-interval: a union of finitely many intervals. Title Page ◭◭ ◮◮ • When S i is a multi-dimensional Euclidean space, the set X i can be: ◭ ◮ – a box (rectangular parallelepiped), Page 8 of 28 – an ellipsoid, or Go Back – a more general (convex or non-convex) set. Full Screen Close Quit

How to Describe . . . Functional . . . 8. Propagating Range Through Computations: Idea Need to Take . . . • We follow the computations of f ( x 1 , . . . , x n ) step-by- Propagating Range . . . step: Propagating Range . . . Importance of . . . – we start with ranges X 1 , . . . , X n of the inputs, Propagating . . . – we sequentially compute the enclosures X n + j for Main Result the ranges of all intermediate results, Examples – finally, on the last computation step, we get the Home Page desired enclosure Y = X n + N . Title Page • On each intermediate step, we have a procedure ◭◭ ◮◮ G ( Y 1 , . . . , Y m ) that transforms: ◭ ◮ – enclosures Y i for the ranges g a ( j,k ) ( X 1 , . . . , X n ) Page 9 of 28 – into an enclosure for the range of the result. Go Back • Requirement: if Y i ⊇ Z i , then Full Screen G ( Y 1 , . . . , Y m ) ⊇ g ( Z 1 , . . . , X n ) . Close Quit

How to Describe . . . Functional . . . 9. Propagating Range Through Computations: In- Need to Take . . . terval Computations as an Example Propagating Range . . . • Parsing: inside the computer, every algorithm consists Propagating Range . . . of elementary operations (+, − , · , min, max, etc.). Importance of . . . Propagating . . . • Interval arithmetic: for each elementary operation f ( a, b ), Main Result – if we know the intervals a and b , Examples – we can compute the exact range f ( a , b ): Home Page [ a, a ]+[ b, b ] = [ a + b, a + b ]; [ a, a ] − [ b, b ] = [ a − b, a − b ]; Title Page [ a, a ] · [ b, b ] = [min( a · b, a · b, a · b, a · b ) , max( a · b, a · b, a · b, a · b )]; ◭◭ ◮◮ � 1 � 1 a, 1 [ a, a ] 1 [ a, a ] = if 0 �∈ [ a, a ]; [ b, b ] = [ a, a ] · [ b, b ] . ◭ ◮ a Page 10 of 28 • Main idea: replace each elementary operation in f by Go Back the corresponding operation of interval arithmetic. Full Screen • Known result: we get an enclosure Y ⊇ y for the de- sired range. Close Quit

How to Describe . . . Functional . . . 10. Interval Computations: toy example Need to Take . . . • The expression f ( x 1 ) = x 1 · (1 − x 1 ) can be described Propagating Range . . . by the following computation scheme: Propagating Range . . . Importance of . . . – first, we compute x 2 = 1 − x 1 , Propagating . . . – then we compute y = x 3 = x 1 · x 2 . Main Result • The range y = f ( x 1 ) of the function f ( x 1 ) = x 1 · (1 − x 1 ) Examples over the interval x 1 = [0 , 1] is y = [0 , 0 . 25]. Home Page • Straightforward interval computations: Title Page – compute ◭◭ ◮◮ x 2 = 1 − [0 , 1] = [1 , 1] − [0 , 1] = [1 − 1 , 1 − 0] = [0 , 1] , ◭ ◮ – then compute Page 11 of 28 Y = x 3 = x 1 · x 2 = [0 , 1] · [0 , 1] = Go Back [min(0 · 0 , 0 · 1 , 1 · 0 , 1 · 1) , max(0 · 0 , 0 · 1 , 1 · 0 , 1 · 1)] = Full Screen [0 , 1] . Close Quit