Peter Alfeld, --- Department of Mathematics, --- College of Science --- University of Utah

Bibliography


  1. Peter Alfeld, A Survey of Zadunaisky's Device Applied to Ordinary Differential Equations, M.Sc. Thesis, The University Dundee, 1975, pp 1-132.
  2. Peter Alfeld, Correction in the Dominant Space: A New Technique for the Numerical Solution of Certain Stiff Initial Value Problems, Ph.D. Thesis, The University Dundee, 1977, pp 1-344.
  3. Peter Alfeld and J.D. Lambert, Correction in the Dominant Space: A numerical technique for a certain class of stiff initial value problems, Math. Comp. 31 (1977), pp 922-938.
  4. Peter Alfeld, Inverse Linear Multistep Methods for the Numerical Solution of Initial Value Problems of Ordinary Differential Equations, Math. Comp. 33 (1979), pp 111-124.
  5. Peter Alfeld, An Improved Version of the Reduction to Scalar CDS Method for the Numerical Solution of Separably Stiff Initial Value Problems, Math. Comp. 33 (1979), pp 535-539.
  6. Peter Alfeld, A Special Class of Explicit Linear Multistep Methods for the Correction in the Dominant Space technique, Math. Comp. 33 (1979), pp 1195-1212.
  7. Peter Alfeld, Two Devices for Improving the Efficiency of Stiff ODE Solvers, in Proceedings of the 1979 SIGNUM Meeting on Numerical Ordinary Differential Equations, U. Ill. Dept. of Comp. Science Rep. 79-1710, pp 24-1 to 24-3.
  8. Peter Alfeld, A Method of Skipping the Transient Phase in the Solution of Separably Stiff Ordinary Initial Value Problems, Math. Comp. 35 (1980), pp 1173-1176.
  9. Peter Alfeld, Fixed Point Iteration With Inexact Function Values, Math. Comp. 38 (1982), pp 87-98.
  10. Peter Alfeld, A Reduce Algorithm for the symbolic computation of Padé approximants, manuscript.
  11. F.C. Hoppensteadt, Peter Alfeld, and R.C. Aiken, Numerical Treatment of Chemical Kinetics by Perturbation and Projection Methods, in Modelling of Chemical Reaction Systems, K.Ebert, P.Deuflhard, W. Jäger, eds., Springer-Verlag, 1981, pp 31-38.
  12. K. Furlong, D. Chapman, and Peter Alfeld, Thermal Constraints on the Geometry of Subduction: Tectonic Implications, Journal of Geophysical Research, v 87, 1982, pp 1786-1802.
  13. F.C. Hoppensteadt and Peter Alfeld, Explosion Mode Analysis of an H_2-O_2 Reaction, in R.C. Aiken (ed.) Proceedings of The International Conference on Stiff Systems, Park City, April 12-14, 1982.
  14. Peter Alfeld, Least Squares and Number Theory, manuscript. I periodically assign a term project based on this idea.
  15. Peter Alfeld and R.E. Barnhill, A Transfinite C^2 Interpolant over Triangles, Rocky Mountain Journal of Mathematics, v 14 (1984), pp 17-39.
  16. Peter Alfeld, Two Discrete C^2 Interpolants, Appendix of above reference.
  17. Peter Alfeld and Bill Harris, MICROSCOPE:, A Software System for Multivariate Analysis. MRC Technical Summary Report #2701, Mathematics Research Center, University of Wisconsin-Madison, 1984, plus a portable FORTRAN software package of approximately 6000 lines of code, available from netlib or my web page.
  18. Peter Alfeld, A discrete C^1 interpolant for tetrahedral data, Rocky Mountain Journal of Mathematics, v 14 (1984), pp 5-16.
  19. Peter Alfeld, Multivariate Perpendicular Interpolation, SIAM Journal on Numerical Analysis, v 22 (1985), pp 95-106.
  20. Peter Alfeld, Derivative Generation from Multivariate Scattered Data by Functional Minimization , Computer Aided Geometric Design J., v 2 (1985), pp 281-296. You may view a dvi file or postscript file.
  21. Peter Alfeld, A Trivariate Clough-Tocher Interpolation Scheme, Computer Aided Geometric Design J., v 1 (1984), pp 169-181.
  22. Peter Alfeld, A Bivariate C^2 Clough-Tocher Scheme , Computer Aided Geometric Design J., v 1 (1984), pp 257-267.
  23. Peter Alfeld, Triangular Extrapolation, MRC Technical Summary Report #2707, Mathematics Research Center, University of Wisconsin-Madison, 1984.
  24. P. Alfeld, On the Dimension of Piecewise Polynomial Functions, in D.E. Griffiths and G.A.Watson (ed.) Numerical Aalysis, Pitman Research Notes in Mathematics Series, No. 140, pp. 1-23, Proceedings of the Biennial Dundee Conference on Numerical Analysis, June 25-28, 1985, Langman Scientific and Technical. You may view a dvi file or postscript file.
  25. Peter Alfeld, A Case Study of Multivariate Piecewise Polynomials, in ``Geometric Modeling'', G. Farin (ed.), SIAM publication, 1987, pp. 149-160. (This paper is revised periodically to provide a record of a growing set of examples. This is the the newest version.
  26. Peter Alfeld, Trivariate Adaptive Cubature, Proceedings of the Fifth International Symposium on Approximatioon Theory, College Station, Texas, January 12-17, 1986, C. Chui, L.L. Schumaker and J.D. Ward (ed.), Academic Press, 1986, pp. 231-234.
  27. Peter Alfeld, Bruce Piper, and L.L. Schumaker, Minimally Supported Bases for Spaces of Bivariate Piecewise Polynomials of Smoothness r and Degree d> =4r+1, Computer Aided Geometric Design J 4 (1987), pp. 105-124.
  28. Peter Alfeld and L.L. Schumaker, The Dimension of Bivariate Spline Spaces of Smoothness r for Degree d> =4r+1, J. Construct. Approx. Theory, Springer Verlag, 1987, pp. 189-197.
  29. Peter Alfeld, Bruce Piper, and L.L. Schumaker, An Explicit Basis for C^1 Quartic Bivariate Splines, SIAM J. Num.Anal. 24 (1987), pp. 891-911.
  30. Peter Alfeld, Bruce Piper, and L.L. Schumaker, Spaces of Bivariate Splines on Triangulations with Holes , J. Approx. its Appl., v. 3 (1987), pp. 1-10.
  31. Peter Alfeld, The Multivariate Spline Newsletter , Published privately: Issue 1 (9/2/87), Issue 2 (2/21/88), Issue 3 (10/18/88).
  32. Alfeld, P., Scattered Data Interpolation in Three or More Variables, in Tom Lyche and Larry L. Schumaker (eds), ``Mathematical Methods in Computer Aided Geometric Design'', Academic Press, 1989, 1-34.
  33. Alfeld, P., and Eyre, D.J, Algorithm 701, Goliath , A Software System for the Exact Analysis of Rectangular Rank-Deficient Sparse Rational Linear Systems, ACM TOMS, 17 No. 4, December 1991, 519-532.
  34. Alfeld, P., L.L. Schumaker, and M. Sirvent, On Dimension and Existence of Local Bases for Multivariate Spline Spaces, Journal of Approximation Theory, 70 (1992), pp. 243-264.
  35. Peter Alfeld, David J. Eyre, and Larry L. Schumaker, Machine-Aided Investigation of Multivariate Spline Spaces, in C.K. Chui, L.L. Schumaker, and J.D. Ward (eds), Approximation VI, Academic Press, 1989, 1-4.
  36. Alfeld, P., and Sirvent, M., A Recursion Formula for the Dimension of Superspline Spaces of Smoothness r and Degree d > r2^k, W. Schempp and K. Zeller (eds), Approximation Theory V, Proceedings of the Oberwolfach Meeting, February 12-18, 1989, Birkhä user Verlag, pp. 1-8.
  37. Alfeld, P., and Schumaker, L.L., 1989, On the Dimension of Bivariate Spline Spaces of Smoothness r and Degree d=3r+1, Numer. Math. 57, 651-661 (1990).
  38. Alfeld, P., and David Eyre, The Exact Analysis of Sparse Rectangular Linear Systems, ACM TOMS, 17 No. 4, December 1991, 502-518.
  39. Alfeld, P., and Sirvent, M.,, The Structure of Multivariate Superspline Spaces of High degree, Math. Comp. 57 (1991), pp 299-308.
  40. Alfeld, P., L. L. Schumaker, and W. Whiteley, The generic dimension of the space of C^1 splines of degree d>= 8 on tetrahedral decompositions, SIAM JNA, v. 30, pp. 889-920, 1993. You may view a dvi file or postscript file.
  41. Alfeld, P., Upper and Lower Bounds on the Dimension of Multivariate Spline Spaces, SIAM JNA, v.~33, No. 2, pp. 571--588, April 1996. You may view a dvi file or postscript file.
  42. Alfeld, P., M. Neamtu, and L.L. Schumaker, Bernstein-Bézier Polynomials on Spheres and Sphere-Like Surfaces., CAGD Journal 13 (1996), 333--349. You may view a dvi file or postscript file. Click here to view otherwise unpublished graphs of the Bernstein-Bézier Polynomials on the Sphere.
  43. Johnson, C., and Alfeld, P., Computational Engineering and Science at the University of Utah, IEEE Computational Science and Engineering, Fall 1994, pp. 7-10.
  44. Alfeld, P., M. Neamtu, and L.L. Schumaker, Dimension and Local Bases of Homogeneous Spline Spaces , SIAM J. Mathematical Analysis, v. 27, No. 5, pp. 1482-1501, September 1996. You may view a dvi file or postscript file.
  45. Alfeld, P., M. Neamtu, and L.L. Schumaker, Circular Bernstein-Bézier Polynomials, in Mathematical Methods in CAGD, M. Daehlen, T. Lyche, and L. L. Schumaker (eds), Vanderbilt University Press, 1995, 1--10. You may view a dvi file or postscript file.
  46. Alfeld, P., M. Neamtu, and L.L. Schumaker, Fitting Scattered Data on Sphere-Like Surfaces using Spherical Splines, Journal of Computational and Applied Mathematics, 73 (1996), 5--43. You may view a dvi file or postscript file. Click here to view otherwise unpublished examples of our interpolants generated by some 16,000 lines of code whose development took the three of us about 2 years.
  47. Alfeld, P., and L.L. Schumaker Non-existence of Star-supported Spline Bases SIAM J. Math. Anal. 31 (2000), 455-465.
  48. Alfeld, P, Bivariate Splines and Minimal Determining Sets, Journal of Computational and Applied Mathematics, 119 (2000), 13--27.
  49. Alfeld, P., and L.L. Schumaker, Smooth Macro-Elements Based on Clough-Tocher Triangle Splits, Numer. Math. 90 (2002), 597--616.
  50. Alfeld, P., and L.L. Schumaker, Smooth Macro-Elements Based on Powell-Sabin Triangle Splits, Adv. Comp. Math., 16 (2002), 29-46.
  51. Alfeld, P., and L.L.~Schumaker, Upper and Lower Bounds on the Dimension of Superspline Spaces, Constructive Approximation 19 (2003), 145-161.
  52. Alfeld, P., and L.L.~Schumaker, A C2 Trivariate Macro-Element Based on the Clough-Tocher Split of a Tetrahedron, CAGD journal, 22 (2005), pp. 710-721.
  53. Alfeld, P., and L.L.~Schumaker,A C2 Trivariate Macro-Element Based on the Worsey-Farin Split of a Tetrahedron, SIAM Journal on Numerical Analysis, 43 (2005), No. 4, pp. 1750-1756.
  54. Alfeld, P., and L.L.~Schumaker, A Trivariate Double-Clough-Tocher Macro-Element, in Approximation Theory XI: Gatlinburg 2004, C. Chui, M. Neamtu, and L. L. Schumaker (eds), Nashboro Press (Brentwood), 2005, 1--14.
  55. Alfeld, P., and L.L.~Schumaker, Bounds on the Dimensions of Trivariate Spline Spaces, submitted for publication, Advances in Computational Mathematics, Springer Verlag, DOI 10.1007/s10444-007-9051-6, 2007.
  56. Alfeld, P., and T. Sorokina; Two Tetrahedral C1 Cubic Macro Elements; Journal of Approximation Theory, 157 (2009), 53.69, DOI: 10.1016/j.jat.2008.07.001.
  57. Alfeld, P., L.L. Schumaker, and T. Sorokina; Two Condensed Macro-Elements with Full Approximation Power ; Advances in Comp. Math., 32 (2010), pp. 381-391.
  58. [58] Alfeld, P.; Many Formulas; Journal of the Oughtred Society, v. 18, No. 2, 2009, pp. 18-21.

Keywords for this page: multivariate splines, spline spaces, dimensions, interpolation, approximation, interpolation on the sphere, homogeneous splines, triangulations, finite elements, spherical splines, circular splines, sphere-like surfaces, tetrahedra, tetrahedral decompositions, spline, dimension, tetrahedron, Bernstein Polynomials, Bernstein-Bezier form, ODEs, stiff ODEs, separably stiff ODEs, explosion mode analysis, nested iteration, fixed point iteration.

Last revised: [19-Feb-2012]

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