Volumetric heat Kernel: Padé-Chebyshev approximation, convergence, and computation

•Discretization and spectrum-free computation of the volumetric heat kernel.•Simple scale selection as compromise between approximation accuracy and smoothness.•Higher approximation accuracy with respect to previous work.•Convergence results as the polynomial degree increases.•Volumetric heat kernel is independent of evaluation the Laplacian spectrum.

This paper proposes an accurate and computationally efficient solver of the heat equation (∂t+Δ)F(·,t)=0(∂t+Δ)F(·,t)=0, F(·,0)=fF(·,0)=f, on a volumetric domain, through the (r,r  )-degree Padé-Chebyshev rational approximation of the exponential representation F(·,t)=exp(−tΔ)fF(·,t)=exp(−tΔ)f of the solution. To this end, the heat diffusion problem is converted to a set of r   differential equations, which involve only the Laplace–Beltrami operator, and whose solution converges to F(·,t)F(·,t), as r→+∞r→+∞. The discrete heat equation is equivalent to r sparse, symmetric linear systems and is independent of the volume discretization as a tetrahedral mesh or a regular grid, the evaluation of the Laplacian spectrum, and the selection of a subset of eigenpairs. Our approach has a super-linear computational cost, is free of user-defined parameters, and has an approximation accuracy lower than 10−r. Finally, we propose a simple criterion to select the time value that provides the best compromise between approximation accuracy and smoothness of the solution.

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