Piecewise linear continuation

implicial Continuation

Simplicial Continuation, or Piecewise Linear Continuation (Allgower and Georg [1] , [3] ) is a one parameter continuation method which is well suited to small to medium embedding spaces. The algorithm has been generalized to compute higher dimensional manifolds by (Allgower and Gnutzman [2] ) and (Allgower and Schmidt [4] ).

The algorithm for drawing contours is a simplicial continuation algorithm, and since it is easy to visualize, it serves as a good introduction to the algorithm.

Contour Plotting

The contour plotting problem is to find the zeros (contours) of $f(x,y)=0,$ ($f(cdot),$ a smooth scalar valued function) in the square $0leq\; x\; leq\; 1,\; 0leq\; y\; leq\; 1,$,

The square is divided into small triangles, usually by introducing points at the corners of a regular square mesh $ih\_xleq\; xleq\; (i+1)h\_x,$, $jh\_yleq\; y\; leq\; (j+1)h\_y,$, making a table of the values of $f(x\_i,y\_j),$ at each corner $(i,j),$, and then dividing each square into two triangles. The value of $f(x\_i,y\_j),$ at the corners of the triangle defines a unique Piecewise Linear interpolant $lf(x,y),$ to $f(cdot),$ over each triangle. One way of writing this interpolant on the triangle with corners $(x\_0,y\_0),~(x\_1,y\_1),~(x\_2,y\_2),$ is as the set of equations:$(x,y)\; =\; (x\_0,y\_0)+(x\_1-x\_0,y\_1-y\_0)s+(x\_2-x\_0,y\_2-y\_0)t,$:$0leq\; s,$:$0leq\; t,$:$s+t\; leq\; 1,$:$lf(x,y)\; =\; f(x\_0,y\_0)+(f(x\_1,y\_1)-f(x\_0,y\_0))s+(f(x\_2,y\_2)-f(x\_0,y\_0))t,$The first four equations can be solved for $(s,t),$ (this maps the original triangle to a right unit triangle), then the remaining equation gives the interpolated value of $f(cdot),$. Over the whole mesh of triangles, this piecewise linear interpolant is continuous.

The contour of the interpolant on an individual triangle is a line segment (it is an interval on the intersection of two planes). The equation for the line can be found, however the points where the line crosses the edges of the triangle are the endpoints of the line segment.

The contour of the piecewise linear interpolant is a set of curves made up of these line segments. Any point on the edge connecting $(x\_0,y\_0),$ and $(x\_1,y\_1),$ can be written as:$(x,y)\; =\; (x\_0,y\_0)\; +\; t\; (x\_1-x\_0,y\_1-y\_0),,$with $t,$ in $(0,1),$, and the linear interpolant over the edge is:$f\; sim\; f\_0\; +\; t\; (f\_1-f\_0),$So setting $f\; =\; 0,$:$t\; =\; -f\_0/(f\_1-f\_0),$ and $(x,y)\; =\; (x\_0,y\_0)-f\_0*(x\_1-x\_0,y\_1-y\_0)/(f\_1-f\_0),$Since this only depends on values on the edge, every triangle which shares this edge will produce the same point, so the contour will be continuous. Each triangle can be tested independently, and if all are checked the entire set of contour curves can be found.

Piecewise Linear Continuation

Piecewise linear continuation is similar to contour plotting (Dobkin, Silvio, Thurston and Wilks [5] ), but in higher dimensions. The algorithm is based on the following results:

Lemma 1

References

[1] Eugene L. Allgower, K. Georg, Introduction to Numerical Continuation Methods, SIAM Classics in Applied Mathematics 45. 2003.

[2] Eugene L. Allgower, Stefan Gnutzmann, An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces. SIAM Journal on Numerical Analysis, Volume 24, Number 2, 452--469, 1987.

[3] E. L. Allgower, K. Georg, Simplicial and Continuation Methods for Approximations, Fixed Points and Solutions to Systems of Equations SIAM Review, Volume 22, 28--85, 1980.

[4] Eugene L. Allgower, Phillip H. Schmidt, An Algorithm for Piecewise-Linear Approximation of an Implicitly Defined Manifold SIAM Journal on Numerical Analysis, Volume 22, Number 2, 322--346, April 1985.

[5] David P. Dobkin, Silvio V. F. Levy, William P. Thurston and Allan R. Wilks, Contour Tracing by Piecewise Linear Approximations. ACM Transactions on Graphics, 9(4) 389-423, 1990.