A Class of Algorithms for Recovery of Continuous Relaxation Spectrum from Stress Relaxation Test Data Using Orthonormal Functions

. 2023 Feb 15;15(4):958. doi: 10.3390/polym15040958

$G (t)$	$linear relaxation modulus, Equation (1), P a$
$G (t_{i})$	$relaxation modulus obtained in stress relaxation test, i = 1, \dots, N$ $, P a$
$\bar{G} (t_{i})$	noise corrupted $measurement of the modulus G (t_{i})$ $, i = 1, \dots, N$ $, P a$
$G_{K} (t)$	relaxation modulus model corresponding to $H_{K}^{M} (v)$ $, Equation (8), P a$
${\bar{G}}_{N}$	$N$ dimensional vector of the measurements $\bar{G} (t_{i})$ , Equation (11)
$G_{N}$	$N$ dimensional vector of noise-free measurements $G (t_{i})$
$g_{k}$	parameters of the $model H_{K}^{M} (v)$ $, k = 0, 1, \dots K - 1$ , $Equation (7), P a \cdot s^{1 / 2}$
$g_{K}$	$vector of parameters g_{k}$ $of the models H_{K}^{M} (v)$ $(7) and G_{K} (t)$ (8)
${\bar{g}}_{K}^{λ}$	solution of regularized task (14) given for regularization parameter $λ$ by (15) or (19)
${\tilde{g}}_{K}^{λ}$	solution of regularized task (14) for ideal measurements, Equation (25)
${\bar{g}}_{K}^{λ_{G C V}}$	vector of optimal model parameters determined by GCV method
${\bar{g}}_{K}^{N}$ $, {\tilde{g}}_{K}^{N}$	normal solutions of least-squares problem (13) for noise and noise-free measurements of the relaxation modulus
$H (v)$	continuous spectrum of relaxation frequencies $v$ $, Equation (3), P a$
$H^{M} (v)$	$modified relaxation spectrum, Equation (4), P a \cdot s$
$H_{K}^{M} (v)$	$model of the modified relaxation spectrum H^{M} (v)$ $, Equation (7), P a \cdot s$
${\bar{H}}_{K}^{M} (v)$	model of $modified spectrum H^{M} (v)$ $for ‘ regularized ’ parameter {\bar{g}}_{K}^{λ}$ $, Equation (22), P a \cdot s$
${\bar{H}}_{K} (v)$	model of $spectrum H (v)$ $for ‘ regularized ’ parameter {\bar{g}}_{K}^{λ}$ $, Equation (23), P a$
${\tilde{H}}_{K}^{M} (v)$	model of $H^{M} (v)$ obtained for ideal measurements, $Equation (24), P a \cdot s$
${\bar{H}}_{K}^{N} (v)$ $, {\tilde{H}}_{K}^{N} (v)$	models of $H^{M} (v)$ obtained for the normal solutions ${\bar{g}}_{K}^{N}$ $, {\tilde{g}}_{K}^{N}$ $described by Equations (26) and (27), P a$
$h_{k} (v)$	basis functions of the $model H_{K}^{M} (v)$ $, k = 0, 1, \dots K - 1$ , $Equation (7), P a \cdot s$
$I_{K, K}$	$K$ dimensional identity matrix
$K$	number of $model H_{K}^{M} (v)$ summands, Equation (7), –
$N$	number of measurements in the stress relaxation test, –
$Q_{N} (g_{K})$	square identification index defined by (10), ${P a}^{2}$
$r$	number of non-zero singular values of $Φ_{N, K}$ , –
$t_{i}$	$sampling instants used in the stress relaxation test, i = 1, \dots, N$ $, s$
$T$	time interval from which the sampling instants $t_{i}$ were generated
$V$ $, U$	orthogonal matrices of singular value decomposition of $Φ_{N, K}$ , Equation (18)
$V_{G C V} (λ)$	the GCV functional defined by (16) and expressed by (21)
$Y$	$N$ dimensional vector $Y = U^{T} {\bar{G}}_{N}$
$z (t_{i})$	$additive noise of relaxation modulus measurement, i = 1, \dots, N$ $, P a$
$z_{N}$	$N$ dimensional vector $of measurement noises z (t_{i})$ $, i = 1, \dots, N$ , $P a$
$α$	time-scaling factor of the basis functions $h_{k} (v)$ $, s$
$λ$	regularization parameter introduced in the optimization task (14), –
$λ_{G C V}$	optimal regularization parameter determined by GCV method, Equation (17)
$Λ_{λ}$	$K \times N$ diagonal matrix defined by (20)
$σ_{i}$	the non-zero singular values of $Φ_{N, K}$ $, i = 1,2, \dots, r$ , Equation (18)
$Σ$	$N \times K$ diagonal matrix of singular values $σ_{1}, \dots, σ_{r}$ of $Φ_{N, K}$ , Equation (18)
$Φ_{N, K}$	$N \times K$ matrix defined by (11) basis for least-squares task (13)
$Φ_{N, K}^{†}$	$the Moore - Penrose pseudoinverse of matrix Φ_{N, K}$
$ϕ_{k} (t)$	$basis functions defined by (9) of the model G_{K} (t)$ $, Equation (8), s^{- 1 / 2}$
$E R R$	integral square error of $H^{M} (v)$ approximation, Equation (60), $P a \cdot s^{1 / 2}$