Abstract

Mathematical models for phenomena in the physical sciences are typically parameter-dependent, and the estimation of parameters that optimally model the trends suggested by experimental observation depends on how model-observation discrepancies are quantified. Commonly used parameter estimation techniques based on least-squares minimization of the model-observation discrepancies assume that the discrepancies are quantified with the L²-norm applied to a discrepancy function. While techniques based on such an assumption work well for many applications, other applications are better suited for least-squared minimization approaches that are based on other norm or inner-product induced topologies. Motivated by an application in the material sciences, the new alternative least-squares approach is defined and an insightful analytical comparison with a baseline least-squares approach is provided.

Keywords

Laplace Transform, Viscoelastic Composite, Norm Space, Inner Product Space, Least Squares Minimization, Optimal Parameter Estimation

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1. Introduction

In this paper, we assume that X is the space of all continuous functions $f:\left[0,\infty \right)\to ℝ$ having a Laplace transform $F:H\to ℂ$ with $H:=\left\{s\in ℂ:\Re \left(s\right)>0\right\}$.

Parameters $p\in P\subseteq {ℝ}^n$ associates with a time-domain model $m\left(p,\cdot \right):\left[0,\infty \right)\to ℝ$ are considered optimal insofar as they yield a minimal model-observation discrepancy $\epsilon :\left[0,\infty \right)\to ℝ$ defined by $\epsilon \left(t\right):=m\left(p,t\right)-r\left(t\right)$, where function $r:\left[0,\infty \right)\to ℝ$ is obtained as a regression to a set of time-dependent observations. The model-observation discrepancy $\epsilon$ is assumed to be function-valued, so the phrase “minimal discrepancy” only has meaning when $\epsilon$ is understood to be a member of some norm-induced topology $\left(X,\Vert \cdot \Vert \right)$. Having specified the norm-induced topology to which $\epsilon$ belongs, the optimal parameters are then computed as an optimal solution $p^{*}$ to the least squares problem (LSP)

${\min }_{p\in P}{\Vert \epsilon \left(p,\cdot \right)\Vert }^2$ (1)

Two norms on X are considered in formulating the LSP (1).

The first norm, the baseline norm, is denoted by ${\Vert \cdot \Vert }_{T,\gamma }$, while the second norm, the alternative norm, is denoted by ${\Vert \cdot \Vert }_{S,s}$. (The norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ on X are defined in Section 2.) The use of the baseline norm ${\Vert \cdot \Vert }_{T,\gamma }$ in (1) yields a variant of a commonly used LSP for computing optimal model parameters, while the alternative norm ${\Vert \cdot \Vert }_{S,s}$ is motivated by the elegant closed-form expressions for certain models $m\left(p,\cdot \right)$ undertaking the Laplace transform. This is particularly true for certain creep models associated with viscoelastic materials [1] - [7].

While the use of the alternative norm ${\Vert \cdot \Vert }_{S,s}$ in LSP (1) has been successfully applied for computing optimal parameter estimates in [5], a theoretical foundation and justification for the use of the alternative form ${\Vert \cdot \Vert }_{S,s}$ in LSP (1) is in need of further development. Refining the developments began in [8] [9], this paper addresses the above need in Section 2, where 1) two inner products ${\langle \cdot ,\cdot \rangle }_{T,\gamma }:X\times X\to ℂ$ and ${\langle \cdot ,\cdot \rangle }_{S,s}:X\times X\to ℂ$ are defined over X and verified with respect to the inner product properties; 2) the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ are induced from the respective inner products ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ and ${\langle \cdot ,\cdot \rangle }_{S,s}$ ; 3) from the inner product properties, a bounding relationship is established between the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ ; and 4) insight is obtained from the bounding relationship into how the parameter solutions $p\in P$ to LSP (1) $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{T,\gamma }$ relate to the parameter solutions $p\in P$ to LSP (1) $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{S,s}$. The first three contributions represent a substantial refinement and streamlining of the developments in [8] [9], thus paving the way for the fourth contribution which, furthermore, builds on the developments in [8] [9].

The remainder of the paper is organized as follows. From the developments in Section 2, a more simple and improved implementation of a previous application [5] becomes evident, and this is presented in Section 3. Computational setup and results are presented and discussed briefly in this same section. Lastly Section 4 concludes this paper and provides comments on future work.

2. Definition and Analysis of the Norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$

The two norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ are induced, respectively, by the following two inner products ${\langle \cdot ,\cdot \rangle }_{T,\gamma }:X\times X\to ℂ$ and ${\langle \cdot ,\cdot \rangle }_{S,s}:X\times X\to ℂ$ defined in the following manner for each pair $f,g\in X$ and parameters $\gamma >0$ and $s\in H$ :

${\langle f,g\rangle }_{T,\gamma }:={\displaystyle {\int }_0^{\infty }f\left(t\right)\overline{g\left(t\right)}{\text{e}}^{-\gamma t}\text{d}t}$ (2)

${\langle f,g\rangle }_{S,s}:=\left({\displaystyle {\int }_0^{\infty }f\left(t\right){\text{e}}^{-st}\text{d}t}\right)\stackrel{¯}{\left({\displaystyle {\int }_0^{\infty }g\left(t\right){\text{e}}^{-st}\text{d}t}\right)}$ (3)

It is now shown that (2) and (3) are, in fact, inner products.

Proposition 2.1. The mappings ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ given by (2) and ${\langle \cdot ,\cdot \rangle }_{S,s}$ given by (3) are defined for all $f,g\in X$ and are furthermore inner products over X.

Proof: Because X contains the continuous functions $f:\left[0,\infty \right)\to ℝ$ having a Laplace transform, the inner product ${\langle \cdot ,\cdot \rangle }_{S,s}$ is defined for all $f,g\in X$. Also, the function fg defined by multiplying $f\in X$ and $g\in X$ is continuous and of exponential order [10] (follows from the same properties of f and g), and so the Laplace transform $\mathcal{L}\left\{fg\right\}$ exists, and (2) is simply the Laplace transform $\mathcal{L}\left\{fg\right\}$ evaluated at $s=\gamma$. Thus, the inner product ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ is also defined for all $f,g\in X$.

Recall that, for any vector space V, an inner product $\langle \cdot ,\cdot \rangle :V\times V\to ℂ$ satisfies the following rules for each $u,v,w\in V$ and $\lambda \in ℂ$ (e.g., see [11]):

I1: $\langle u,v\rangle =\overline{\langle v,u\rangle }$

I2: $\langle \lambda u,v\rangle =\lambda \langle u,v\rangle$

I3: $\langle u,v+w\rangle =\langle u,v\rangle +\langle u,w\rangle$

I4: $\langle u,v\rangle \ge 0$, and $\langle u,u\rangle =0\Rightarrow u=0$

Property I1 follows readily for ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ by noting that f and g are real-valued functions ${\text{e}}^{-\gamma t}$ is real-values, and so the integrand is real-valued. For ${\langle \cdot ,\cdot \rangle }_{S,s}$, Property I1 follows from (3) by computing

$\begin{array}{c}{\langle f,g\rangle }_{S,s}=\left({\displaystyle {\int }_0^{\infty }f\left(t\right){\text{e}}^{-st}\text{d}t}\right)\stackrel{¯}{\left({\displaystyle {\int }_0^{\infty }g\left(t\right){\text{e}}^{-st}\text{d}t}\right)}\\ =F\left(s\right)\overline{G\left(s\right)}\\ =G\left(s\right)\overline{F\left(s\right)}\\ ={\overline{\langle g,f\rangle }}_{S,s}\end{array}$

where F(s) and G(s) denote the Laplace transform of f and g, respectively.

Properties I2 and I3 follow easily for both ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ and ${\langle \cdot ,\cdot \rangle }_{S,s}$ from elementary properties of integrals.

Property I4 applies to ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ because: 1) for each $f\in X$, the integrand of ${\langle f,f\rangle }_{T,\gamma }$ is always nonnegative; and 2) if $f\cancel{\equiv }0$, then by the continuity of f over $\left[0,\infty \right)$, there exist $t_0\in \left[0,\infty \right)$, $ϵ>0$, and $\delta >0$ over which $f\left(T\right)\ge \delta$ for all $T\in \left[t_0-\frac{ϵ}{2},t_0+\frac{ϵ}{2}\right]$. Thus, for each $\gamma >0$, we have ${\langle f,f\rangle }_{T,\gamma }\ge ϵ{\delta }^2{\text{e}}^{-\gamma \left(t_0+ϵ\right)}>0$ if $f\cancel{\equiv }0$. From this, the implication ${\langle f,f\rangle }_{T,\gamma }=0\Rightarrow f\equiv 0$ follows.

To show that property I4 applies to ${\langle \cdot ,\cdot \rangle }_{S,s}$, first note that ${\langle f,f\rangle }_{S,s}\ge 0$ for all $f\in X$ follows from the definition (3), and so it remains to show that ${\langle f,f\rangle }_{S,s}=0\Rightarrow f=0$. This latter claim holds under application of Lerch’s theorem (see, e.g., [11] [12]) to the setting where f is continuous. Namely, if ${\langle f,f\rangle }_{S,s}=0$ (so that $F\left(s\right)\equiv 0$ ), then ${\displaystyle {\int }_0^{a}f\left(t\right)\text{d}t}=0$ for all $a>0$. The assumed continuity of f on $\left[0,\infty \right)$ and the Fundamental Theorem of Calculus imply that $f\equiv 0$. Thus, I4 holds for ${\langle \cdot ,\cdot \rangle }_{S,s}$. Hence, it has been shown that ${\langle \cdot ,\cdot \rangle }_{T,\gamma }$ and ${\langle \cdot ,\cdot \rangle }_{S,s}$ are both inner products over X.

One possible relationship between two different norms ${\Vert \cdot \Vert }_a$ and ${\Vert \cdot \Vert }_b$ called equivalence is now explored. The equivalence of two norms ${\Vert \cdot \Vert }_a$ and ${\Vert \cdot \Vert }_b$ is characterized by the existence of $0<\mathcal{l}\le u<\infty$ such that

$\mathcal{l}{\Vert f\Vert }_a\le {\Vert f\Vert }_b\le u{\Vert f\Vert }_a$ for all $f\in X$ (4)

(See, e.g., [11].) Using the inner-product structures defined on X, the Cauchy-Schwartz inequality can be used to show a bounding relationship of the form ${\Vert f\Vert }_{S,s}\le u{\Vert f\Vert }_{T,\gamma }$ for all $f\in X$, $s\in H$, and $\gamma <R\left(s\right)$ via the computation

${\Vert f\Vert }_{S,s}^2={\left|{\displaystyle {\int }_0^{\infty }f\left(t\right){\text{e}}^{-st}\text{d}t}\right|}^2={\left|{\langle f\left(t\right),{\text{e}}^{-\left(s-\gamma \right)t}\rangle }_{T,\gamma }\right|}^2\le {\langle f,f\rangle }_{T,\gamma }{\langle {\text{e}}^{-\left(s-\gamma \right)t},{\text{e}}^{-\left(s-\gamma \right)t}\rangle }_{T,\gamma }$ (5)

$=\left({\displaystyle {\int }_0^{\infty }{\left|f\left(t\right)\right|}^2{\text{e}}^{-\gamma t}\text{d}t}\right)\left({\displaystyle {\int }_0^{\infty }{\text{e}}^{-\left(s+\bar{s}-\gamma \right)}\text{d}t}\right)\Rightarrow {\Vert f\Vert }_{S,s}^2\le u{\Vert f\Vert }_{T,\gamma }^2$ (6)

where $u={\displaystyle {\int }_0^{\infty }{\text{e}}^{-\left(s+\bar{s}-\gamma \right)}\text{d}t}=\frac{1}{s+\bar{s}-\gamma }$.

Whereas the upper bound coefficient u is established in (6), the lower bound coefficient $\mathcal{l}>0$ necessary to establish the equivalence (4) for each fixed $s\in H$ and $0<\gamma <R\left(s\right)$ is shown not to exist through two counterexamples:

Counterexample 1: Let f be of the form $f\left(t\right)={\text{e}}^{-\omega t},\omega >0$. Then ${\Vert f\Vert }_{T,\gamma }=\sqrt{\frac{1}{2\omega +\gamma }}$ and ${\Vert f\Vert }_{S,s}=\frac{1}{\left|\omega +s\right|}$. So $\mathcal{l}\le \frac{{\Vert f\Vert }_{S,s}}{{\Vert f\Vert }_{T,\gamma }}=\sqrt{\frac{2\omega +\gamma }{{\left|\omega +s\right|}^2}}$. Both ${\lim }_{\omega \to \infty }{\Vert f\Vert }_{T,\gamma }=0$ and ${\lim }_{\omega \to \infty }{\Vert f\Vert }_{S,s}=0$. Furthermore, since ${\lim }_{\omega \to \infty }\sqrt{\frac{2\omega +\gamma }{{\left|\omega +s\right|}^2}}=0$, there is no $\mathcal{l}>0$ serving as a lower bound coefficient.

Counterexample 2: Let f be of the form $f\left(t\right)=\sin \left(\omega t\right)$, $\omega >0$. Then ${\Vert f\Vert }_{T,\gamma }=\sqrt{\frac{1}{2}\left(\frac{1}{\gamma }-\frac{\gamma }{{\gamma }^2+4{\omega }^2}\right)}$ and ${\Vert f\Vert }_{S,s}=\frac{\omega }{\left|s^2+{\omega }^2\right|}$. Now ${\lim }_{\omega \to \infty }{\Vert f\Vert }_{T,\gamma }=\sqrt{\frac{1}{2\gamma }}>0$ and ${\lim }_{\omega \to \infty }{\Vert f\Vert }_{S,s}=0$. Thus, ${\lim }_{\omega \to \infty }\frac{{\Vert f\Vert }_{S,s}}{{\Vert f\Vert }_{T,\gamma }}=0$, and so there is no lower bound $\mathcal{l}>0$ on $\frac{{\Vert f\Vert }_{S,s}}{{\Vert f\Vert }_{T,\gamma }}$.

The lack of a lower bound coefficient $\mathcal{l}>0$ is also depicted in Figure 1 and Figure 2 for the same two counterexamples. Thus, it is established that due to the lack of the lower bound coefficient $\mathcal{l}>0$, the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ over X are not equivalent.

The bounding relationship (6) between the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ is also described via inclusion relationships between sublevel sets. The sublevel set $L_{\Vert \cdot \Vert }\left(f,P,\delta \right)$ is defined by

$L_{\Vert \cdot \Vert }\left(f,P,\delta \right):=\left\{p\in P:\Vert f\left(p,\cdot \right)\Vert \le \delta \right\}$

for each f, P, and $\delta >0$. By the existence of the bounding coefficient $u,0<u<\infty$, in (6), we have the inclusion

$L_{\Vert \cdot \Vert }\left(f,P,\frac{1}{u}\delta \right)\subseteq L_{{\Vert \cdot \Vert }_{S,s}}\left(f,P,\delta \right)$ (7)

The sublevel set inclusion (7) provides a sense in which the norm ${\Vert \cdot \Vert }_{S,s}$ penalizes model-observation discrepancy more leniently than the norm ${\Vert \cdot \Vert }_{T,\gamma }$. This leniency is observed, for example, in the plot of Figure 2 where the increasing frequency of $f\left(t\right)=\sin \left(\omega t\right)$ due to $\omega \to \infty$ leads to ${\Vert f\Vert }_{S,s}\to 0$

while ${\Vert \cdot \Vert }_{T,\gamma }f\to \sqrt{\frac{1}{2\gamma }}>0$.

For application purposes, the preference between the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ in formulating the LSP (1) depends on 1) the desired degree of leniency in penalizing imperfect model-observation fit due to the use of parameter $p\in P$ ; and 2) the ease and accuracy of evaluating the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$. Next, in Section 3, the material science application of solving LSP (1) motivating the contributions of this paper is revisited where the use of each of the two norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ is evaluated in terms of the above two preference criteria.

3. Application for Modeling Time Dependent Properties of Viscoelastic Materials

A time-dependent model $m\left(p,\cdot \right)$ for modeling creep of viscoelastic materials under an applied stress load is given by

$m\left(p,t\right):=\frac{\sigma }{E}\left[1+\lambda {\displaystyle {\sum }_{n=0}^{\infty }\frac{{\left(-\beta \right)}^n{t}^{\left(1-\alpha \right)\left(n+1\right)}}{\Gamma \left[\left(1-\alpha \right)\left(n+1\right)+1\right]}}\right]$ (8)

where the stress level σ and Young’s modulus E are determined experimentally, and the material-specific kernel parameters $\left(\alpha ,\beta ,\lambda \right)$ satisfy

$\left(\alpha ,\beta ,\lambda \right)\in \left\{\left(\alpha ,\beta ,\lambda \right):0<\alpha <1,\beta \in ℝ,\lambda \in ℝ\right\}$

(See [1] [3] [5] for details.) The parameter $\alpha$ can be found from the first term of the infinite series expansion in (8) [3]. Thus, only the model parameters $\beta$ and $\lambda$ need to be determined as an optimal solution $p=\left(\beta ,\lambda \right)$ to problem (1) with $P=\left\{p:p=\left(\beta ,\lambda \right),\beta \in ℝ,\lambda \in ℝ\right\}$.

The regression function $r:\left[0,\infty \right)\to ℝ$ is fit to observations based on experiments performed for three types of composites with nanofillers [5]:

1) Pure polyamide (PA).

2) Polyamide with ultra-dispersed diamonds (PA + UDD).

3) Polyamide with carbon nanotube fillers (PA + CNT).

For each material, the tests with the corresponding three loading levels ${\sigma }_{0.3}$, ${\sigma }_{0.4}$, and ${\sigma }_{0.5}$ are performed, where the subscript of $\sigma$ indicates that the stress applied to the materials is 30%, 40%, and 50%, respectively, of the ultimate stress, which was taken equivalent to the yielding stress of each of the tested materials. Using these experimental data, the regression functions $r\left(t\right)$ used for each data set take the form

$r\left(t\right)=c_0+c_1{\text{e}}^{-0.1t}+c_2{\text{e}}^{-0.5t}+c_1{\text{e}}^{-0.02t}$ (9)

where the coefficients $c_i,i=0,1,2,3$ are estimated for each data set using standard linear regression techniques. The resulting regression functions and the material-specific vales for ${\sigma }_{0.3}$, ${\sigma }_{0.4}$, and ${\sigma }_{0.5}$ are given in Table 1.

For each computation, the norm ${\Vert \cdot \Vert }_{T,\gamma }$ parameter $\gamma =0.005$ and the norm ${\Vert \cdot \Vert }_{S,s}$ parameter $s=0.01\left(1+i\right)$ are used; furthermore, the experimentally determined parameters α, E, and $\sigma ={\sigma }_i,i=0.3,0.4,0.5$ associated with $m\left(p,t\right)$ are provided in Table 2.

The optimal parameters $p^{*}=\left({\beta }^{*},{\lambda }^{*}\right)$ are computed as optimal solutions to LSP (1) using the baseline norm $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{T,\gamma }$ and the alternative norm $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{S,s}$. These computations are performed with Maple^TM [13]. The computed parameter estimates are presented in Table 3 and the resulting wellness-of-fit between the parameterized models and experimental observations are illustrated in Figure 3.

As observed earlier [1] [3], the model $m\left(p,t\right)$ has an elegant simplification under its Laplace transformation

$M\left(p,s\right):=\mathcal{L}\left\{m\left(p,t\right)\right\}=\frac{\sigma }{E}\frac{1}{s}\left[1+\frac{\lambda }{s^{1-\alpha }+\beta }\right]$ (10)

Furthermore, each function r with the form (9) has a closed-form Laplace transform denoted by $R\left(s\right)$. Thus, for each s satisfying $\Re \left(s\right)>0$, problem (1) takes the following elegant form when $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{S,s}$ :

${\min }_{p\in P}{\Vert M\left(p,s\right)-R\left(s\right)\Vert }_2^2$ (11)

Solving the LSP (11) is computationally more accurate and less expensive than solving the corresponding LSP (1) with $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{T,\gamma }$. This is consistent with the

motivation and observation seen in earlier works [1] [3] [14] associated with the use of Laplace transform-based approaches to estimating the optimal model parameters.

4. Conclusions

This paper contributes a mathematical foundation for the comparison between time domain least squares parameter estimation problems formulated using the norm ${\Vert \cdot \Vert }_{T,\gamma }$ and Laplace domain least squares parameter estimation problems introduced in [1] [3], applied in [5] [8], and formulated using the alternative norm ${\Vert \cdot \Vert }_{S,s}$ as defined in Section 2. A relationship between the norms ${\Vert \cdot \Vert }_{T,\gamma }$ and ${\Vert \cdot \Vert }_{S,s}$ is analyzed in terms of norm equivalence, and in exploring this equivalence, the existence of the necessary upper bound coefficient $u,0<u<\infty$ was shown to exist in Section 2 using the two inner product structures (2) and (3) defined on X. However, the non-existence of the corresponding lower bound coefficient $\mathcal{l},0<\mathcal{l}<u$, is demonstrated through two counterexamples. From the bounding relationship (6), inclusion relationships (7) of sublevel sets follow that provides a sense in which the norm ${\Vert \cdot \Vert }_{S,s}$ penalizes certain types of model-observation deviation more leniently than the norm ${\Vert \cdot \Vert }_{T,\gamma }$.

The plots of Figure 3 suggest that the solutions $p^{*}=\left({\beta }^{*},{\lambda }^{*}\right)$ to LSP (1) with $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{S,s}$ yield improved model-observation fit over the corresponding solutions with $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{T,\gamma }$. In addition to the computational advantages associated with solving (11), the improvement is also attributed to the relatively lenient (in a sense derived from the inclusion relationships (7)) penalization of certain types of model-observation by ${\Vert \cdot \Vert }_{S,s}$ as compared with ${\Vert \cdot \Vert }_{T,\gamma }$. If the types of model-observation deviations that are penalized leniently are subjectively negligible to the model user, then the computation of the optimal solutions $\left({\beta }^{*},{\lambda }^{*}\right)$ to LSP (1) with $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{S,s}$ is more flexible, and this results in subjectively improved model-observation fit as compared with the fit obtained with the use of the norm $\Vert \cdot \Vert ={\Vert \cdot \Vert }_{T,\gamma }$.

Acknowledgements

The authors thank Mr. Brian Dandurand of Argon National Laboratory, Chicago, IL. For valuable insights, discussions, and computations provide.

The List of the Variables Used in This Paper

p: parameters in time-domain model

$m\left(p,\cdot \right)$ : model equation

$\epsilon \left(t\right)$ : strain

$r\left(t\right)$ : regression function

$\left(X,\Vert \cdot \Vert \right)$ : norm induced topology

X: space of all condition functions of real variables

F: Laplace transformation

${\Vert \cdot \Vert }_{T,\gamma }$ : baseline norm in real domain

${\Vert \cdot \Vert }_{S,s}$ : alternative norm in Laplace complex domain

V: vector space

u, v, w: vectors

λ: constant

s: complex variable

t: real variable

f, g: real valued functions

F(s), G(s): Laplace transforms of f andg functions

L: lower bound coefficient

ω: real parameter > 0

γ: complex valued parameter

δ: small real number

σ: stress level

E: Young’s modulus

α, β, λ: material specific kernel parameters

Γ: Gamma function

c_i: regression function coefficients

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

References