General Problems of . . . Indirect Measurements How Constructive . . . Applied Constructive . . . Interval Computations as Why Intervals? Applied Constructive Interval Computations . . . Wiener Again Mathematics: from Shanin Standard Interval . . . Beyond Intervals to Wiener and Beyond Home Page Title Page Vladik Kreinovich Department of Computer Science ◭◭ ◮◮ University of Texas at El Paso ◭ ◮ El Paso, Texas 79968, USA vladik@utep.edu Page 1 of 25 Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 1. General Problems of Science and Engineering How Constructive . . . • The main objective of science is to understand the cur- Applied Constructive . . . rent state of the world and to predict its future state. Why Intervals? Interval Computations . . . • The main objective of engineering is to find controls Wiener Again and strategies that lead to a better future. Standard Interval . . . • The state of the world is usually described in terms of Beyond Intervals real numbers – values of physical quantities. Home Page • Some quantities we can measure directly: e.g., distance Title Page from here to our hotel. ◭◭ ◮◮ • Other quantities y we cannot measure directly: e.g., ◭ ◮ distance from here to a nearby star. Page 2 of 25 Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 2. Indirect Measurements How Constructive . . . • Since we cannot measure the quantity of interest y di- Applied Constructive . . . rectly, we measure it indirectly . Why Intervals? Interval Computations . . . • Namely, we measure related easier-to-measure quanti- Wiener Again ties x 1 , . . . , x n and get values � x i . Standard Interval . . . • Then, we use the known relation y = f ( x 1 , . . . , x n ) and Beyond Intervals known (approximate) values of x i to estimate y as Home Page y = f ( � � x 1 , . . . , � x n ) . Title Page ◭◭ ◮◮ ◭ ◮ Page 3 of 25 Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 3. How Constructive Mathematics Can Help How Constructive . . . • Ideally, we want to be able to estimate y with any given Applied Constructive . . . accuracy ε . Why Intervals? Interval Computations . . . • For this purpose, we need to have: Wiener Again – an algorithm that, given ε , computes the accuracy Standard Interval . . . δ with we should measure the inputs, and Beyond Intervals – an algorithm � f that, when applied to the measure- Home Page ment results � x i to get the desired estimate � y : Title Page x i − x i | ≤ δ ⇒ | � | � f ( � x 1 , . . . , � x n ) − f ( x 1 , . . . , x n ) | ≤ ε. ◭◭ ◮◮ ◭ ◮ • In a nutshell, this is what constructive mathematics is about – when limited to real numbers. Page 4 of 25 Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 4. Applied Constructive Mathematics How Constructive . . . • In practice, our ability to measure accurately is limited. Applied Constructive . . . Why Intervals? • So, we have measurement results � x i with some accura- Interval Computations . . . cies δ i : | � x i − x i | ≤ δ i . Wiener Again • The only information that we have about the actual Standard Interval . . . def value x i is that x i ∈ [ x i , x i ] = [ � x i − δ i , � x i + δ i ]. Beyond Intervals Home Page • What can we say about y = f ( x 1 , . . . , x n )? We can only conclude that Title Page def ◭◭ ◮◮ y ∈ [ y, y ] = { f ( x 1 , . . . , x n ) : x i ∈ [ x i , x i } . ◭ ◮ • Computing [ y, y ] is called interval computations. Page 5 of 25 • Yuri Matiyasevich called it applied constructive math- Go Back ematics. Full Screen Close Quit
General Problems of . . . Indirect Measurements 5. Why Intervals? How Constructive . . . • Usually, we do not just know the upper bound δ i on Applied Constructive . . . def the measurement error ∆ x i = � x i − x i : | ∆ x i | ≤ δ i . Why Intervals? Interval Computations . . . • We also know the probabilities of different values ∆ x i . Wiener Again • These probabilities come from comparing measurement Standard Interval . . . results with a standard (more accurate) instrument. Beyond Intervals Home Page • There are two situations when this is not possible: Title Page – state-of-the-art measurement, when we use the most accurate instrument; and ◭◭ ◮◮ – measurements on the shop floor, where we could ◭ ◮ calibrate everything, but it would cost too much. Page 6 of 25 • Then, all we have is an upper bound δ i on | ∆ x i | . Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 6. Why Wiener? A Brief History of Interval Com- How Constructive . . . putations Applied Constructive . . . • Origins: Archimedes (Ancient Greece), N. Wiener (1914) Why Intervals? Interval Computations . . . • Modern pioneers: Mieczyslaw Warmus (Poland), Teruo Wiener Again Sunaga (Japan), Ramon Moore (USA), 1956–59 Standard Interval . . . • First boom: early 1960s. Beyond Intervals • First challenge: taking interval uncertainty into ac- Home Page count when planning spaceflights to the Moon. Title Page • Current applications (sample): ◭◭ ◮◮ – design of elementary particle colliders: ◭ ◮ Martin Berz, Kyoko Makino (USA) Page 7 of 25 – will a comet hit the Earth: Go Back Martin Berz, Ramon Moore (USA) – robotics: L. Jaulin (France), A. Neumaier (Austria) Full Screen – chemical engineering: M. Stadtherr (USA) Close Quit
General Problems of . . . Indirect Measurements 7. Interval Computations – How? First Idea How Constructive . . . • In a computer, every computation is a sequence of el- Applied Constructive . . . ementary arithmetic operations. Why Intervals? Interval Computations . . . • In mathematical terms, this means that we consider Wiener Again compositions of simple arithmetic functions. Standard Interval . . . • So, a natural idea – known as straightforward interval Beyond Intervals computations – is to: Home Page – find interval analogues of simple arithmetic func- Title Page tions, and then ◭◭ ◮◮ – in the original algorithm, replace each arithmetic ◭ ◮ operation with the corresponding interval one. Page 8 of 25 Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 8. Interval Analogues of Simple Arithmetic Func- How Constructive . . . tions Applied Constructive . . . • When x 1 ∈ x 1 = [ x 1 , x 1 ] and x 2 ∈ x 2 = [ x 2 , x 2 ], then: Why Intervals? Interval Computations . . . – The range x 1 + x 2 for x 1 + x 2 is [ x 1 + x 2 , x 1 + x 2 ] . Wiener Again – The range x 1 − x 2 for x 1 − x 2 is [ x 1 − x 2 , x 1 − x 2 ] . Standard Interval . . . – The range x 1 · x 2 for x 1 · x 2 is [ y, y ], where Beyond Intervals Home Page y = min( x 1 · x 2 , x 1 · x 2 , x 1 · x 2 , x 1 · x 2 ); Title Page y = max( x 1 · x 2 , x 1 · x 2 , x 1 · x 2 , x 1 · x 2 ) . ◭◭ ◮◮ • The range 1 / x 1 for 1 /x 1 is [1 /x 1 , 1 /x 1 ] (if 0 �∈ x 1 ). ◭ ◮ • These operations are known as interval arithmetic. Page 9 of 25 Go Back Full Screen Close Quit
General Problems of . . . Indirect Measurements 9. Straightforward Interval Computations: Exam- How Constructive . . . ple and Limitations Applied Constructive . . . • Example: f ( x ) = ( x − 2) · ( x + 2), x ∈ [1 , 2]. Why Intervals? Interval Computations . . . • How will the computer compute it? Wiener Again • r 1 := x − 2; Standard Interval . . . • r 2 := x + 2; Beyond Intervals • r 3 := r 1 · r 2 . Home Page • Main idea: perform the same operations, but with in- Title Page tervals instead of numbers : ◭◭ ◮◮ • r 1 := [1 , 2] − [2 , 2] = [ − 1 , 0]; ◭ ◮ • r 2 := [1 , 2] + [2 , 2] = [3 , 4]; Page 10 of 25 • r 3 := [ − 1 , 0] · [3 , 4] = [ − 4 , 0]. Go Back • Actual range: f ( x ) = [ − 3 , 0] ⊂ [ − 4 , 0]. Full Screen • Comment: excess width (4 vs. 3) is unavoidable, since interval computations is NP-hard. Close Quit
General Problems of . . . Indirect Measurements 10. What Can We Do? How Constructive . . . • Representing an algorithm as a composition of elemen- Applied Constructive . . . tary arithmetic functions often does not work. Why Intervals? Interval Computations . . . • Idea: represent it as a composition of some other func- Wiener Again tions. Standard Interval . . . • What is the class of functions closed under composi- Beyond Intervals tion? Home Page • It is reasonable to require that this class is also closed Title Page under inversion. ◭◭ ◮◮ R n . • So, we are looking for group of transformations of I ◭ ◮ • The simplest such group is the group of all linear trans- Page 11 of 25 formations. Go Back • What are other such groups? Full Screen Close Quit
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