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We consider the interpolation problem with the inverse multiquadric radial basis function. The problem usually produces a large dense linear system that has to be solved by iterative methods. The efficiency of such methods is strictly related to the computational cost of the multiplication between the coefficient matrix and the vectors computed by the solver at each iteration. We propose an efficient technique for the calculation of the product of the coefficient matrix and a generic vector. This computation is mainly based on the well-known spectral decomposition in spherical coordinates of the Green's function of the Laplacian operator. We also show the efficiency of the proposed method through numerical simulations.

The design of heat exchanger fields is a key phase to ensure the long-term sustainability of such renewable energy systems. This task has to be accomplished by modelling the relevant processes in the complex system made up of different exchangers, where the heat transfer must be considered within exchangers and outside exchangers. We propose a mathematical model for the study of the heat conduction into the soil as consequence of the presence of exchangers. Such a problem is formulated and solved with an analytical approach. On the basis of such analytical solution, we propose an optimisation procedure to compute the best position of the exchangers by minimising the adverse effects of neighbouring devices. Some numerical experiments are used to show the effectiveness of the proposed method also by taking into account a reference approximation procedure of the problem based on a finite difference method.