Tikhonov regularisation in standard form for polynomial basis conversion

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Polynomial basis conversion is an important industrial requirement for high-integrity data exchange between computer-aided design systems. These conversions may be numerically ill-conditioned, and this paper considers the use of Tikhonov regularisation in standard form for their stabilisation. It is shown that the method yields a good approximation to the theoretically exact solution if it is known a priori that the desired solution satisfies the discrete Picard condition. An example of the use of Tikhonov regularisation for the stabilisation of the conversion between the Bernstein and B-spline polynomial bases is given. This basis transformation can also be performed by the Oslo algorithm, but it is difficult to determine whether this algorithm is unstable for given knot vectors. However when this basis transformation is performed by Tikhonov regularisation, it is readily determined whether the linear algebraic equation is unstable, and if it is, regularisation follows immediately if the discrete Picard condition is satisfied. © 1997 by Elsevier Science Inc.




Winkler, J. R. (1997). Tikhonov regularisation in standard form for polynomial basis conversion. Applied Mathematical Modelling, 21(10), 651–662. https://doi.org/10.1016/S0307-904X(97)00081-4

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