Numerical Solution of Periodic Fredholm Integral Equations of the Second Kind by Means of Attenuation Factors

Numerical solution of boundary integral equations by means of attenuation factors

“…Indeed, the presence of the sign function in ρ m makes them all quite different from the corresponding expressions for the general Fredholm equation in Berrut & Reifenberg (1997). The same is true also for the evaluation of the approximate solution in the following paragraph.…”

“…In Berrut & Reifenberg (1997) we solved Fredholm integral equations of the second kind x + Hx = f by approximating the operator H :…”

“…REMARK 3.3 Since the corresponding matrices are large and dense, in practice we do not solve the linear system (3.11) by Gaussian elimination, but we directly apply the fixedpoint iteration described in Berrut & Reifenberg (1997). This means that at the ( j + 1)th iteration step, j = 0, 1, .…”

“…Further interesting examples of approximation operators which can be used with our method are operators which are only defined in Fourier spaces as linear and translation invariant smoothing operators like cosine-, Cesáro-, Lanczos-and spline-smoothing, see Berrut & Reifenberg (1997).…”

Periodic smoothing splines for FFT-based solvers

“…Indeed data smoothing has been initially developed to separate smooth patterns in data from the rough sequence corresponding to (experimental) noise. A great number of techniques has been developed to perform data smoothing, based on kernel regression [26], polynomial regression [27], attenuation factors [28] and polynomial smoothing splines [29][30][31], among others. In the context of FFT-based solvers applied to field dislocation mechanics, it is worth noting that Brenner et al [8] proposed to spread the dislocation density corresponding to pixel-wise single dislocations on a surface of 3 × 3 pixels.…”