Why Linear Interpolation? y -Scale-Invariance Consistency Andrzej - PowerPoint PPT Presentation

Need for Interpolation Why Linear Interpolation? Reasonable Properties . . . x -Scale-Invariance Why Linear Interpolation? y -Scale-Invariance Consistency Andrzej Pownuk and Vladik Kreinovich Continuity Resulting Definition Computational Science Program University of Texas at El Paso Main Result 500 W. University Home Page El Paso, Texas 79968, USA Title Page ampownuk@utep.edu, vladik@utep.edu ◭◭ ◮◮ ◭ ◮ Page 1 of 18 Go Back Full Screen Close Quit

Need for Interpolation Why Linear Interpolation? 1. Need for Interpolation Reasonable Properties . . . • In many practical situations: x -Scale-Invariance y -Scale-Invariance – we know that the value of a quantity y is uniquely Consistency determined by the value of some other quantity x , Continuity – but we do not know the exact form of the corre- Resulting Definition sponding dependence y = f ( x ). Main Result • To find this dependence, we measure the values of x Home Page and y in different situations. Title Page • As a result, we get the values y i = f ( x i ) of the unknown ◭◭ ◮◮ function f ( x ) for several values x 1 , . . . , x n . ◭ ◮ • Based on this information, we would like to predict the Page 2 of 18 value f ( x ) for all other values x . Go Back • When x is between the smallest and the largest of the Full Screen values x i , this prediction is known as the interpolation . Close Quit

Need for Interpolation Why Linear Interpolation? 2. Why Linear Interpolation? Reasonable Properties . . . • Let’s consider the case n = 2. Let’s assume that f ( x ) x -Scale-Invariance is linear on [ x 1 , x 2 ]; then y -Scale-Invariance f ( x ) = x − x 1 · f ( x 2 ) + x 2 − x Consistency · f ( x 1 ) . x 2 − x 1 x 2 − x 1 Continuity Resulting Definition • This formula is known as linear interpolation . Main Result • The usual motivation for linear interpolation is sim- Home Page plicity: linear functions are the easiest to compute. Title Page • An interesting empirical fact is that in many practical ◭◭ ◮◮ situations, linear interpolation works reasonably well. ◭ ◮ • We know that in computational science, often very Page 3 of 18 complex computations are needed. Go Back • So we cannot claim that nature prefers simplicity. Full Screen • There should be another reason for the empirical fact that linear interpolation often works well. Close Quit

Need for Interpolation Why Linear Interpolation? 3. Reasonable Properties of Interpolation Reasonable Properties . . . • We want to be able, x -Scale-Invariance y -Scale-Invariance – given values y 1 and y 2 of the unknown function at Consistency points x 1 and x 2 , and a point x ∈ ( x 1 , x 2 ), Continuity – to provide an estimate for f ( x ). Resulting Definition • Let us denote this estimate by I ( x 1 , y 1 , x 2 , y 2 , x ); what Main Result are the reasonable properties of this function? Home Page • If y i = f ( x i ) ≤ y for both i , it is reasonable to expect Title Page that f ( x ) ≤ y . ◭◭ ◮◮ • In particular, for y = max( y 1 , y 2 ), we conclude that ◭ ◮ I ( x 1 , y 1 , x 2 , y 2 , x ) ≤ max( y 1 , y 2 ) . Page 4 of 18 • Similarly, if y ≤ y i for both i , it is reasonable to expect Go Back that y ≤ f ( x ). Full Screen • In particular, for y = min( y 1 , y 2 ), we conclude that Close min( y 1 , y 2 ) ≤ I ( x 1 , y 1 , x 2 , y 2 , x ) . Quit

Need for Interpolation Why Linear Interpolation? 4. x -Scale-Invariance Reasonable Properties . . . • The numerical value of a physical quantity depends: x -Scale-Invariance y -Scale-Invariance – on the choice of the measuring unit and Consistency – on the starting point. Continuity • If we change the starting point to the one which is b Resulting Definition units smaller, then b is added to all the values. Main Result • If we replace a measuring unit by a a > 0 times smaller Home Page one, then all the values are multiplied by a . Title Page • If we perform both changes, then each original value x ◭◭ ◮◮ is replaced by the new value x ′ = a · x + b . ◭ ◮ • For example, if we know the temperature x in C, then Page 5 of 18 the temperature x ′ in F is x ′ = 1 . 8 · x + 32. Go Back • The interpolation procedure should not change if we Full Screen simply re-scale: Close I ( a · x 1 + b, y 1 , a · x 2 + b, y 2 , a · x + b ) = I ( x 1 , y 1 , x 2 , y 2 , x ) . Quit

Need for Interpolation Why Linear Interpolation? 5. y -Scale-Invariance Reasonable Properties . . . • Similarly, we can consider different units for y . x -Scale-Invariance y -Scale-Invariance • The interpolation result should not change if we simply Consistency change the starting point and the measuring unit; so: Continuity – if we replace y 1 with a · y 1 + b and y 2 with a · y 2 + b , Resulting Definition – then the result of interpolation should be obtained Main Result by a similar transformation from the previous one: Home Page Title Page I ( x 1 , a · y 1 + b, x 2 , a · y 2 + b, x ) = a · I ( x 1 , y 1 , x 2 , y 2 , x ) + b. ◭◭ ◮◮ ◭ ◮ Page 6 of 18 Go Back Full Screen Close Quit

Need for Interpolation Why Linear Interpolation? 6. Consistency Reasonable Properties . . . • When x 1 ≤ x ′ 1 ≤ x ≤ x ′ 2 ≤ x 2 , the value f ( x ) can be x -Scale-Invariance estimated in two different ways. y -Scale-Invariance Consistency • We can interpolate directly from the values y 1 = f ( x 1 ) Continuity and y 2 = f ( x 2 ), getting I ( x 1 , y 1 , x 2 , y 2 , x ). Resulting Definition • Or we can: Main Result – first estimate the values f ( x ′ 1 ) = I ( x 1 , y 1 , x 2 , y 2 , x ′ 1 ) Home Page and f ( x ′ 2 ) = I ( x 1 , y 1 , x 2 , y 2 , x ′ 2 ), and Title Page – then use these two estimates to estimate f ( x ) as ◭◭ ◮◮ I ( x 1 , f ( x ′ 1 ) , x 2 , f ( x ′ 2 ) , x ) = ◭ ◮ I ( x ′ 1 , I ( x 1 , y 1 , x 2 , y 2 , x ′ 1 ) , x ′ 2 , I ( x 1 , y 1 , x 2 , y 2 , x ′ 2 ) , x ) . Page 7 of 18 • It is reasonable to require that these two ways lead to Go Back the same estimate for f ( x ): I ( x 1 , y 1 , x 2 , y 2 , x ) = Full Screen I ( x ′ 1 , I ( x 1 , y 1 , x 2 , y 2 , x ′ 1 ) , x ′ 2 , I ( x 1 , y 1 , x 2 , y 2 , x ′ 2 ) , x ) . Close Quit

Need for Interpolation Why Linear Interpolation? 7. Continuity Reasonable Properties . . . • Most physical dependencies are continuous. x -Scale-Invariance • Thus, when the two value x and x ′ are close, we expect y -Scale-Invariance Consistency the estimates for f ( x ) and f ( x ′ ) to be also close. Continuity • Thus, it is reasonable to require that: Resulting Definition – the interpolation function I ( x 1 , y 1 , x 2 , y 2 , x ) is con- Main Result Home Page tinuous in x , and – that for both i = 1 , 2, I ( x 1 , y 1 , x 2 , y 2 , x ) converges Title Page to f ( x i ) when x → x i . ◭◭ ◮◮ ◭ ◮ Page 8 of 18 Go Back Full Screen Close Quit

Need for Interpolation Why Linear Interpolation? 8. Resulting Definition Reasonable Properties . . . A function I ( x 1 , y 1 , x 2 , y 2 , x ) defined for x 1 < x < x 2 is x -Scale-Invariance called an interpolation function if: y -Scale-Invariance Consistency • min( y 1 , y 2 ) ≤ I ( x 1 , y 1 , x 2 , y 2 , x ) ≤ max( y 1 , y 2 ); Continuity • I ( a · x 1 + b, y 1 , a · x 2 + b, y 2 , a · x + b ) = I ( x 1 , y 1 , x 2 , y 2 , x ) Resulting Definition for all x i , y i , x , a > 0, and b ( x -scale-invariance); Main Result • I ( x 1 , a · y 1 + b, x 2 , a · y 2 + b, x ) = a · I ( x 1 , y 1 , x 2 , y 2 , x )+ b Home Page for all x i , y i , x , a > 0, and b ( y -scale invariance); Title Page • consistency: I ( x 1 , y 1 , x 2 , y 2 , x ) = ◭◭ ◮◮ I ( x ′ 1 , I ( x 1 , y 1 , x 2 , y 2 , x ′ 1 ) , x ′ 2 , I ( x 1 , y 1 , x 2 , y 2 , x ′ 2 ) , x ); ◭ ◮ • continuity: Page 9 of 18 – the expression I ( x 1 , y 1 , x 2 , y 2 , x ) is a continuous Go Back function of x , Full Screen – I ( x 1 , y 1 , x 2 , y 2 , x ) → y 1 when x → x 1 and Close I ( x 1 , y 1 , x 2 , y 2 , x ) → y 2 when x → x 2 . Quit

Need for Interpolation Why Linear Interpolation? 9. Main Result Reasonable Properties . . . • Result: The only interpolation function satisfying all x -Scale-Invariance the properties is the linear interpolation y -Scale-Invariance Consistency I ( x 1 , y 1 , x 2 , y 2 , x ) = x − x 1 · y 2 + x 2 − x · y 1 . Continuity x 2 − x 1 x 2 − x 1 Resulting Definition • Thus, we have indeed explained that linear interpola- Main Result tion follows from the fundamental principles. Home Page • This may explain its practical efficiency. Title Page ◭◭ ◮◮ ◭ ◮ Page 10 of 18 Go Back Full Screen Close Quit

Need for Interpolation Why Linear Interpolation? 10. Proof Reasonable Properties . . . • When y 1 = y 2 , the conservativeness property implies x -Scale-Invariance that I ( x 1 , y 1 , x 2 , y 1 , x ) = y 1 . y -Scale-Invariance Consistency • Thus, to complete the proof, it is sufficient to consider Continuity two remaining cases: when y 1 < y 2 and when y 2 < y 1 . Resulting Definition • We will consider the case when y 1 < y 2 . Main Result Home Page • The case when y 2 < y 1 is considered similarly. Title Page • So, in the following text, without losing generality, we assume that y 1 < y 2 . ◭◭ ◮◮ ◭ ◮ Page 11 of 18 Go Back Full Screen Close Quit