Scientific Computing, Spring 1999 (http://www.math.nyu.edu/faculty/goodman/teaching/sci comp 99/)

Assignment 4 Multidimensional approximation, pure theory

Given February 16, due March 2.

Local polynomial approximation approximation in more complicated in more than one dimension. Let x = (x_{1}, x_{2}) be a point in two dimensional space. The unit square^{1} is

Q = {x : 0 ≤ x_{1} ≤ 1 , 0 ≤ x_{2} ≤ 1} .

We have a smooth function, f, defined on Q. We have values of f at n points^{2} x_{1}, …, x_{n}, and wish to find an “interpolating” function f̂, to estimate f at other points inside Q. In order to do this, we will suppose that the points are the vertices of a triangulation of Q, and use piecewise linear interpolation. This is the simplest kind of interpolation in 2D, just as piecewise linear interpolation is the simplest kind of interpolation in 1D. Unfortunately, that does not make it simple.

A triangulation of Q is a covering of Q with a collection of non overlapping triangles. A triangle has a boundary, which consists of the three vertices and the three edges connecting them, and an interior, which is the inside excluding the boundary. The triangles in a triangulation should be non overlapping in that no point is in the interior of more than one triangle. Any point in the interior of Q and on the boundary of a triangle must be on the boundary of at least one other triangle.

Suppose that a single triangle, T, has vertices A, B, and C. We say that f̂ is affine in T if there is a formula,

f̂(x) = a + b_{1}x_{1} + b_{2}x_{2} ,

which holds for all x ∈ T. The linear interpolation problem from the vertices of a triangle is to find f̂ in T that is affine in T and satisfies the interpolation conditions:

f̂(A) = f_{A} , f̂(B) = f_{B} , f̂(C) = f_{C} ,  (1)

where the values f_{A}, f_{B}, and f_{C} are given.

1. Show that the linear interpolation problem on any triangle is solvable whenever the area of T is not zero.

Higher order polynomial interpolation in more than one dimension gets complicated fast. The function f̂ is quadratic in T if there is a formula,

f̂(x) = a + b_{1}x_{1} + b_{2}x_{2} + c_{0}x_{1}^{2} + c_{1}x_{1}x_{2} + c_{2}x_{2}^{2} ,  (2)

which holds for all x ∈ T. Let us denote the midpoint of the segment from A to B by \overline{AB}. This notation reminds us that the midpoint is the average of A and B, and is not supposed to represent the segment from A to B. For quadratic interpolation, we need more interpolation conditions. One often used set of conditions is interpolation at the midpoints of the edges:

1“Q” is for Quadrat, the German word for square.

2It should be clear whether x_{1} is the first point or the first component of a point x, any less ambiguous notation seems much more clumsy.

f̂(\overline{AB}) = f\overline{AB} , f̂(\overline{AC}) = f\overline{AC} , f̂(\overline{BC}) = f\overline{BC} ,  (3)

where again the values f\overline{AB}, f\overline{AC}, and f\overline{BC} are given.

2. Show that there is a unique f̂ that is quadratic in T and satisfies the interpolation conditions (1) and (3), as long as the area of T is not zero.

Suppose that the points x_{1}, …, x_{n} are all the vertices of the triangles in a triangulation of Q. We want a function, f̂, defined in Q, that is affine in each triangle and satisfies interpolation conditions

f̂(x_{k}) = f_{k} for k = 1, …, n.  (4)

3. Show that this is possible if no two vertices coincide, and if no vertex is on the segment connecting two other vertices.

To analyze the accuracy of piecewise linear interpolation^{3} on a triangulation, let h be the length of the longest edge among all the edges of all the triangles in the triangulation of Q.

4. Show that if f_{k} = f(x_{k}) for k = 1, …, n, and if f is a smooth function on Q, then there is a constant, C, depending only on f (i.e., not on the triangulation or x) so that

|f̂(x) − f(x)| ≤ Ch^{2} for all x ∈ Q.  (5)

Hint: show that the Taylor series expansion of f about the center of a triangle, T, gives a linear function that is both a good approximation to f in T and almost satisfies the interpolation conditions at the vertices of T.

Now suppose that f̂ is piecewise quadratic on a triangulation of Q and interpolates f at the vertices and at the midpoints of the edges.

5. Show that

|f̂(x) − f(x)| ≤ Ch^{3} for all x ∈ Q.

6. In either case (piecewise linear or quadratic interpolation), show that f̂ is continuous. Hint: what this really means is that if e is a common edge of triangles T and T′, then the values of f̂ defined by interpolation on T and interpolation on T′ agree on e.

7. The inequality (5) shows that f̂ is a good approximation to f when h is small no matter what triangulation is used. In the also true that ▽f̂is a good approximation to ▽f under the same circumstances? Note: we showed the answer is “yes” for one dimensional interpolation.

3It is common to call this piecewise linear interpolation even though it should be called piecewise affine.