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A fast collocation method for the radiosity equation, based on the hierarchical algorithm of Hanrahan and Salzman: the 1D case. (English) Zbl 1072.65165

The author studies a modified collocation method for the solution of the planar radiosity equation \[ u(x)=E(x)+\frac{\rho(x)}{2}\int_LV(x, y)\frac{n(x)\cdot(y-x)\,n(y)\cdot(x-y)}{\| x-y\|}\,u(y)\,dy,\quad x\in L, \tag{1} \] where \(L\) is a curve in \(\mathbb R^2\), \(n(\cdot)\) is the curve normal which should exist almost everywhere, \(E(x)\) is the emissivity, \(\rho:L\to[0,1]\) is the reflection coefficient and \(V(x,y)\) is visibility function. Only the unoccluded case \(V = 1\) is considered. For the approximate solution of (1) the author uses midpoint collocation with piecewise constant trial functions.
The complicated regularity structure of the solution of the radiosity equation makes the efficient use of higher order trial functions very difficult. So the author considers a modified collocation method in which the matrix of the collocation method is approximated by a method developed by P. M. Hanrahan, D. Salzman and L. Aupperle [A rapid hierarchical radiosity algorithm, in: SIGGRAPH ’91 Proceedings, Computer Graphics 25, No. 4, 197–206 (1991)]. The author proves that the modified collocation method results in a reduction of work while the order of convergence stays the same. Numerical examples demonstrate the theoretical results for two model problems.

MSC:

65R20 Numerical methods for integral equations
65Y20 Complexity and performance of numerical algorithms
45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)

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