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This paper constructs and studies a nonlinear multivariate regression-tensor model for substantiation of necessary/sufficient conditions of optimization of technological calculation of multifactor physical and chemical process of hardening of complex composite media for metal coatings. An adaptive a posteriori procedure for parametric formation of the target quality functional of integrative physical and mechanical properties of the designed metal coating has been proposed. The results of the research may serve as elements of a mathematical language when creating automated design of precision nanotechnologies for surface hardening of complex composite metal coatings on the basis of complex tribological and anticorrosive tests.

Researchers pay a lot of attention to increasing the strength characteristics of the technological process of hardening metal coatings (see, for example, 1) Munz W.-D., Lewis D.B., Hovsepian P.E. et al. Industrial scale manufacturing of superlattice hard PVD Coatings//Surface Engineering. -2001. -V. 17. -pp. 15-17; 2) Mitterer C., Holler F., Ustel F. et al. Application of hard coatings in aluminum die casting//Surface & Coating Technology. -2000. -V. 125. -pp. 233-239). Nonlinear integrative physical and chemical (PC) processes lie at the root of the methods of hardening the working surfaces of modern power machines, which actualizes the issues related to formalization/development of their mathematical models. In this context, regression models [

In the article, you will find the development of the tasks set in the conclusions of [

Let R be a field of real numbers, R^{n} be a n-dimensional vector space over R with Euclidean norm ‖ ⋅ ‖ R n , col ( w 1 , ⋯ , w n ) ∈ R n be a column vector with elements w 1 , ⋯ , w n ∈ R and let M n , m ( R ) be the space of all n × m -matrices with elements from R. Moreover, let us assume that T m k is the space of all covariant tensors of k-th valency, i.e. real polylinear forms f k , m : R 1 m × ⋯ × R k m → R with norm ‖ f k , m ‖ T m k : = ( ∑ t ... j ... 2 ) 1 / 2 , where { t ... j ... } is the “coordinate matrix” of the tensor f k , m with respect to the canonical basis [

Let v ∈ R m be the vector of varying PC predictors [

w ( ω + v ) = col ( ∑ j = 0 , ⋯ , k f 1 j , m ( v , ⋯ , v ) , ⋯ , ∑ j = 0 , ⋯ , k f n j , m ( v , ⋯ , v ) ) + ε ( ω , v ) . (1)

Here f i j , m ∈ T m j , ε ( ω , ⋅ ) : R m → R n is a nonparameterizable class vector-function

‖ ε ( ω , v ) ‖ R n = o ( ( v 1 2 + ⋯ + v m 2 ) k / 2 ) , (2)

v = col ( v 1 , ⋯ , v m ) , f i 0 , m is the 0-rank tensor, representing the tribological index w i , i = 1 , n ¯ of the PM quality of the investigated PC process in its reference mode, given by the vector ω ∈ R m .

Note 1. The precision of nonlinear simulation of the PC process in the class of regression-tensor systems (1) (and adaptation of their parameters) is correct because of the continuous dependence ( [

The problem of the multidimensional nonlinear regression-tensor modeling of multifactor physical and chemical process of hardening of metal coatings, optimal with respect to some target “tribological criterion”, was set and investigated in detail in work [

(I) for a fixed vector-predictor ω ∈ R m and its open neighborhood V ⊂ R m , analytical conditions are defined, under which the vector function w ( ⋅ ) : V → R n of PM property indices satisfies the multivariate regression-tensor system (1);

(II) a direct algorithm is constructed for identifying tensor coordinates f i j , m , i = 1 , n ¯ , j = 0 , 2 ¯ in a 2-valent regression-tensor model (1) based on a numerical solution of a two-criteria LSM problem of optimal a posteriori PM modeling written as:

{ min ( ∑ l = 1 , ⋯ , q ( ‖ w ( l ) − col ( ∑ j = 0 , ⋯ , k f 1 j , m ( v ( l ) , ⋯ , v ( l ) ) , ⋯ , ∑ j = 0 , ⋯ , k f n j , m ( v ( l ) , ⋯ , v ( l ) ) ) ‖ R n ) 2 ) 1 / 2 , min ( ∑ i = 1 , ⋯ , n ∑ j = 0 , ⋯ , k ‖ f i j , m ‖ T m j 2 ) 1 / 2 , (3)

where w ( l ) ∈ R n , v ( l ) ∈ R m , l = 1 , q ¯ are, respectively, the vectors of experimental factor-predictors of the PC process, i.e. w ( l ) is the a posteriori response to the target variation v ( l ) relative to the coordinates of the reference vector ω under the condition ‖ v ( l ) ‖ R m < 1 (this inequality is methodologically dictated by condition (2)), q is the number of tribological experiments conducted (determined by representativeness of model (1)), carried out with the dynamics of PC processes [

(III) for the 2-valent regression-tensor model (1) with the given predictor ω ∈ R m and nominal condition ε ( ω , ⋅ ) ≡ 0 , the analytical solution of the optimization problem as a non-linear “v-optimization” of the varied (relative to the vector ω ) factor-predictors of the prognostic PM characteristics of the designed composite metal coatings was obtained:

max v ∈ R m F ( v ) : = r T w ( ω + v ) = r 1 w 1 ( ω + v ) + ⋯ + r n w n ( ω + v ) , (4)

where the vector function v ↦ w ( ω + v ) = col ( w 1 ( ω + v ) , ⋯ , w n ( ω + v ) ) has a coordinate representation according to the LSM-identified model (1)-(3), r i > 0 are weight factors reflecting the priority of PM indices; we can also investigate Problem (III) for some r j < 0 , which corresponds to the position when w j should be minimized in PM indices.

The significance of the nonlinear multifactor regression-tensor analysis is not only in the exact theorems already obtained by this method [

Problem statement (according to analytical conclusions of [

(i) to determine necessary and sufficient conditions of solvability of the optimization problem (4) for a 3-valent ( k = 3 ) functional regression-tensor system (1);

(ii) to construct an algorithm for correction of sufficient conditions of extremum of stationary point of Problem (i) based on the r-parametric adjustment of the r ↦ r T w ( ω + v ) PM functional

v ↦ F ( v ) = r T w ( ω + v ) . (5)

Consider Problem (i) on optimization of the PM characteristics of metal coatings at k = 3 ; note that the solution of the accompanying Problem (II) of the parametric identification for k = 3 is an non-complicated modification of Assertion 3 of [

In such a mathematical formulation, the nonlinear multivariate prognostic equation (1) can be given in the following vector-matrix-tensor form:

w ( ω + v ) = c + A v + col ( v T B 1 v + f 1 3 , m ( v , v , v ) , ⋯ , v T B n v + f n 3 , m ( v , v , v ) ) + ε ( ω , v ) (6)

where c ∈ R n , A ∈ M n , m ( R ) , B i ∈ M m , m ( R ) , i = 1 , n ¯ . Without loss of generality, we believe that each matrix B i has an upper triangular structure; this substantially simplifies the numerical implementation of the ANC-algorithm (3). Additionally, we note that the vector function ε ( ω , ⋅ ) : R m → R n satisfies (according to (2)) the qualitative estimate of ‖ ε ( ω , v ) ‖ R n = o ( ( v 1 2 + ⋯ + v m 2 ) 3 / 2 ) .

According to (1), at k = 3 PM functional of the total tribological indices (5) are twice continuously differentiable, which guarantees the equality of the mixed derivatives

∂ 2 F ( v 1 , ⋯ , v m ) / ∂ v g ∂ v p = ∂ 2 F ( v 1 , ⋯ , v m ) / ∂ v p ∂ v g ∀ g , p = 1 , m ¯ . (7)

Therefore, in the solution of optimization Problem (4) for 3-valent model (6) the main result, according to Theorem 3 ( [

B i * : = ( B i + B i T ) ∈ M m , m ( R ) , i = 1 , n ¯ , (8)

where each B i is a matrix of the system (6) (the matrix of the tensor f i 2 , m in such a statement when it is not considered to be symmetric in the system (1)). Moreover, let us consider a vector function

v ↦ Φ ( v ) : = ( r 1 B 1 * + ⋯ + r n B n * ) − 1 ( A T + [ ∇ v f 1 3 , m ( v , v , v ) , ⋯ , ∇ v f n 3 , m ( v , v , v ) ] ) r , (9)

where ∇ v f i 3 , m ( v , v , v ) is the gradient of the functional v ↦ f i 3 , m ( v , v , v ) .

Assertion 1. The stationary points v * ∈ R m of Problem (i)are the essence of the solutions of equation

v * + Φ ( v * ) = 0 . (10)

A sufficient condition F ( v * ) = max { F ( v ) : v ∈ R m } is that v * ,as a stationary point of the functional (5),must be of elliptic type. In other words,the point v * for the Hessian G ( v , r ) of the functional (5)must satisfy the inequalities

det [ b i j ] p < 0 , p = 1 , m ¯ , (11)

where [ b i j ] p ∈ M p , p ( R ) , p = 1 , m ¯ are the principal submatrices of the Hessian, det is determinant

G ( v * , r ) = r 1 ( B 1 * + [ ∂ 2 f 1 3 , m ( v , v , v ) / ∂ v g ∂ v p | v * ] ) + ⋯ + r n ( B n * + [ ∂ 2 f n 3 , m ( v , v , v ) / ∂ v g ∂ v p | v * ] ) ∈ M m , m ( R ) ,

which is equivalent: characteristic numbers λ p ( v * , r ) of the matrix G ( v * , r ) satisfy the

λ p ( v * , r ) < 0 , p = 1 , m ¯ . (12)

Corollary 1. In case of k = 2 the Hessian of the functional (5)and conditions (11), (12)are invariant to the position of the stationary point v * ,and the Hessian equals

G ( r ) = r 1 B 1 * + ⋯ + r n B n * , (13)

which leads to a linear dependence of the numbers λ p ( r ) , p = 1 , m ¯ on the normalization of the vector r.

If rank G ( r ) = m ,the solution of Equation (10)is unique and has the form of

v * = − G − 1 ( r ) A T r , (14)

which makes the position of the point v * invariant to the normalization of the vector r.

According to vector functions ∇ v f i 3 , m ( v , v , v ) , the equation (10) is geometrically defined by the intersection of m quadrics ( [

One of the factors affecting the stationary point v * geometry of Assertion 1 is the digital adaptive parametric adjustment of r ↦ G ( v * , r ) , which leads to elliptic conditions (11) or (12). This is the subject of the next section.

Consider statement (ii): For a stationary point of the optimization problem (i), construct a numerical procedure for correction of weight factors r ∈ R n , based on fulfillment of spectral conditions (12), i.e. providing elliptic nature of the stationary point v * of Statement 1. This formulation is relevant for optimization of v * -parameters of the PM process when some target PM indices have to be minimized (i.e. r j < 0 ).

Note 2. Despite the algebraic equivalence of conditions (11)~(12), the use of expansion of determinants (11) in construction of adaptive correction r ↦ G ( v * , r ) is almost inevitably doomed to failure (even by means of computer algebra) due to a large number of terms expressed through multivariate regression coefficients.

The solvability conditions for a problem similar to (ii) can be obtained only in exceptional cases. Therefore, below we shall discuss an approach to this problem based on the ideas of the theory of localization and perturbations of eigenvalues [

Let some initial vector r 0 ∈ R n of weight factors from Statement (ii) be given. For example, the heuristic choice of the vector r 0 can be made based on the equality of its coordinates r 0 i , i = 1 , n ¯ to the values of some functions Ψ i : R → R (with a clear physical context) that depend on the values of functionals J i ( v ) : = w i ( ω + v ) , i = 1 , n ¯ from auxiliary problems of optimal prediction of PM quality by individual target tribological indices w i . In particular, for the 2-valent regression model (1), this position, according to Corollary 2 of [

Assertion 2. If the maximal valency of tensors k is two,then the vector of initial weight factors r 0 = col ( r 01 , ⋯ , r 0 n ) with coordinates

r 0 i = Ψ i ( z i ) , z i = max { J i ( v ) : v ∈ R m } , i = 1 , n ¯

has an analytic representation

r 0 = col ( Ψ 1 ( c 1 − e 1 T A B 1 * − 1 A T e 1 / 2 ) , ⋯ , Ψ n ( c n − e n T A B n * − 1 A T e n / 2 ) ) ,

where { e i } i = 1 , n ¯ is the canonical basis in R n .

Let us denote by v 0 ∈ R m some stationary point of the functional (5) in the case when the r-priority of the probing points is r 0 . Correspondingly, we denote by G 0 ∈ M m , m ( R ) the Hessian of the given functional calculated for the pair ( r 0 , v 0 ) and let

G i : = B i * + [ ∂ 2 f i 3 , m ( v , v , v ) / ∂ v g ∂ v p | v 0 ] , i = 1 , n ¯ .

Then for the admissible linear variation Δ r of vector r 0 = col ( r 01 , ⋯ , r 0 n ) coordinates, given (due to comments to formula (4)) by the region of this variation W ⊂ R n written as

Δ r : = col ( Δ r 1 , ⋯ , Δ r n ) ∈ W ,

r i = r 0 i + Δ r i > 0 , i = 1 , n ¯ ,

the Δ r -parametric family of linear variations of the Hessian G ( v 0 , r 0 + Δ r ) is defined by a matrix m × m -multiverse written as:

G 0 + ∑ i = 1 , ⋯ , n Δ r i G i , Δ r ∈ W . (15)

By virtue of (7), the matrices of the family (15) are symmetric.

For the matrices of the manifold (15), the eigenvalues can be characterized as a series of optimization problems by means of the Courant-Fischer Theorem [

Taking into account the introduced constructions, the adaptive adjustment of the PC process tribological quality functional F ( v ) = r T w ( ω + v ) , which ensures that inequality (12) is fulfilled when varying the vector r ∈ R n at the stationary point, contains the following Assertion 3 below. In essence, this assertion is a non-complicated modification (in the version of the strong derivative d G ( v 0 , r ) / d r | r 0 ) of Theorem 6.3.12 [

Assertion 3. Let r = r 0 + Δ r , { ( λ p ( r 0 ) , x p ) , p = 1 , m ¯ } ⊂ R × R m be the set of eigenpairs of the Hessian G 0 ,i.e. λ p ( r 0 ) x p = G 0 x p , p = 1 , m ¯ ,and let,given the realization of the manifold (15),the numbers

g p i = x p T G i x p / x p T x p , p = 1 , m ¯ , i = 1 , n ¯

are set.

Then the eigenvalues λ p ( v 0 , r 0 + Δ r ) , p = 1 , m ¯ of the Hessian G ( v 0 , r 0 + Δ r ) have the form

λ 1 ( v 0 , r 0 + Δ r ) = λ 1 ( r 0 ) + ∑ i = 1 , ⋯ , n g 1 i Δ r i + o ( ‖ Δ r ‖ R n ) , ⋮ λ m ( v 0 , r 0 + Δ r ) = λ m ( r 0 ) + ∑ i = 1 , ⋯ , n g m i Δ r i + o ( ‖ Δ r ‖ R n ) . (16)

System (16) gives an estimate of the sensitivity of the Hessian G ( v 0 , r 0 + Δ r ) spectrum to linear variations Δ r i , i = 1 , n ¯ of the weight factors. For nonlinear variations we can refer to the recurrence formulas from p. (b) ( [

Corollary 2. Let k = 2 , n = m , Λ ( r 0 ) : = col ( λ 1 ( r 0 ) , ⋯ , λ m ( r 0 ) ) be a vector of characteristic numbers of the matrix ( r 01 B 1 * + ⋯ + r 0 m B m * ) and { x p } p = 1 , m ¯ be their corresponding eigenvectors. Moreover,let Λ * : = col ( λ 1 * , ⋯ , λ m * ) be a vector of characteristic numbers that are “benchmark/reference” by criterion (12),and B : = [ b p i ] be a m × m -matrix with elements

b p i = x p T B i * x p / x p T x p .

Then for r 0 + Δ r ,where the variation vector has the representation Δ r = B − 1 ( Λ * − Λ ( r 0 ) ) ,the eigenvalues of the Hessian G ( r 0 + Δ r ) will be o ( ‖ Δ r ‖ R n ) close to the benchmark { λ p * } p = 1 , m ¯ .

Note 3. Since Corollary 2 is valid for small ‖ Δ r ‖ R m , the question remains whether the iterative computational process will converge to

r j = ( r j − 1 + Δ r j − 1 ) ∈ R m , j = 1 , 2 , ⋯ ,

constructed from the calculation Δ r j − 1 = B − 1 ( Λ * − Λ ( r j − 1 ) ) , if the initial divergence ‖ Λ * − Λ ( r 0 ) ‖ R m is significant enough. Moreover, according to the structure of the target functional (5), at each iteration step j for vector r j ∈ R m coordinates it is necessary (within the physical statement of Problem (4)) to check the coordinate conditions r i j > 0 , i = 1 , n ¯ .

Note 4. For adaptive systems, the evaluation of input signals (in our case ‖ v ( l ) ‖ R m < 1 in (3)) is essential (which is why adaptive techniques with learning are used). In this context, it is important to obtain sufficient conditions for the adaptive system to have robust bounded solutions [

In the context of Note 3, let us provide the result of calculating an upper bound estimate for perturbation ‖ Δ r ‖ R m . To this end, assume that ‖ ⋅ ‖ M is the matrix norm in M m , m ( R ) , consistent with the norm in Euclidean space ‖ ⋅ ‖ R m , and ‖ E ‖ M = 1 , E ∈ M m , m ( R ) is the unit matrix. For example, the Frobenius norm

‖ D ‖ F : = ( m − 1 ∑ d i j 2 ) 1 / 2 , D = [ d i j ] ∈ M m , m ( R ) ,

or the spectral (induced) matrix norm

‖ D ‖ S : = sup { ‖ D x ‖ R m : x ∈ R m , ‖ x ‖ R m = 1 } = max 1 ≤ i ≤ m λ i 1 / 2 ( D T D )

can serve as such.

Returning to Corollary 2, we have B Δ r = Λ * − Λ ( r 0 ) , det B ≠ 0 . Now suppose that the vector of characteristic numbers Λ * − Λ ( r 0 ) turns into a perturbed vector Λ * − Λ ( r 0 ) + δ (in particular, due to the members o ( ‖ Δ r ‖ ) R m of system (16)), and the matrix B turns into B + D . Then the vector Δ r will get (due to a modification of Corollary 2) some increment θ , passing to the value Δ r + θ , which satisfies the equation ( B + D ) ( Δ r + θ ) = Λ * − Λ ( r 0 ) + δ .

It is obvious that δ ∈ R m , D ∈ M m , m ( R ) models the perturbations of the vector Λ * − Λ ( r 0 ) , and the inaccuracy of the parametric estimation of the matrix B (if ‖ D ‖ M ‖ B − 1 ‖ M < 1 , then ‖ D ‖ M < ‖ B ‖ M ; see ( [

Corollary 3. Let s ( B ) : = ‖ B ‖ M ‖ B − 1 ‖ M ,the conditional number of matrix B,where ‖ ⋅ ‖ M is the norm ‖ ⋅ ‖ F or ‖ ⋅ ‖ S ,be added to the assumptions of Corollary 2. Then the following estimate is valid for θ , Δ r

‖ θ ‖ R m / ‖ Δ r ‖ R m ≤ s ( B ) ( 1 − s ( B ) ‖ D ‖ M / ‖ B ‖ M ) − 1 × ( ‖ δ ‖ R m / ‖ Λ * − Λ ( r 0 ) ‖ R m + ‖ D ‖ M / ‖ B ‖ M ) .

If ‖ ⋅ ‖ M = ‖ ⋅ ‖ S and λ 1 , λ m are,respectively,the smallest and the largest eigenvalues of the matrix B T B ,then in the last inequality we can assume that s ( B ) = ( λ m / λ 1 ) 1 / 2 .

Note 5. The construction of the conditional number s ( B ) = ( λ m / λ 1 ) 1 / 2 obtained using the spectral norm ‖ ⋅ ‖ S is transparent due to equality s ( B ) = ‖ B ‖ S ‖ B − 1 ‖ S .

Alternative approaches [

The aim of the article was, in development of the results [

Assertion 3 essentially asked: what can we say about the eigenvalues of the matrix G 0 + ∑ i = 1 , ⋯ , n Δ r i G i , if each variation of Δ r i is a small parameter? Thus, we were only interested in the purely formal aspect of the mathematical modeling problem under study, when we do not consider the question of what the real value of the increment Δ r i must be for the term “small parameter” to be really relevant. In this case, the result of Assertion 3 is based on the fact that the eigenvalues (12) smoothly r depend on the Hessian G ( v , r ) elements during the current parametric r-correction of the target functional (5). However, it should be noted that some information is lost when we deal only with the characteristic polynomial, because there are many different matrices with a given characteristic polynomial. It is therefore not surprising that the stronger results on modeling the Hessian spectrum G ( v , r ) , in particular Assertion 3 and Corollary 2, take into account the structure of G ( v , r ) . The latter ones admit technical simplifications by means of specialized computer algebra proceeding from the geometrical assumption that any Hessian matrix is orthogonally similar to a real diagonal matrix.

Numerical methods for finding eigenvalues and eigenvectors represent one of the most important parts of matrix theory. The analysis of the vector Λ * − Λ ( r 0 ) and matrix B from Corollary 2 has not touched on any aspect of this topic above, but Corollary 3 gives an upper estimate for the perturbation Δ r via relative perturbations Λ * − Λ ( r 0 ) , B and the conditional number s ( B ) . The s ( B ) is involved in the estimation in all cases, whether the perturbations occur in Λ * − Λ ( r 0 ) , only in B, or in Λ * − Λ ( r 0 ) and B at the same time.

Finally, we denote another approach (essentially cybernetic) in adaptive correction r ↦ r T w ( ω + v ) , related to the use of sufficient robust stability conditions for the 2-valent model of the matrix G ( r ) , which also leads to conditions (12). In this context, it is required that with interval tolerances on the vector r coordinates one can construct a Lyapunov function V ( x ) = x p T P x p , where P ∈ M m , m ( R ) is the symmetric positively-defined matrix for which the Lyapunov equation G ( r ) P + P G ( r ) = − Q has a solution given a symmetric positively-defined m × m -matrix Q. The transition to adaptive robust quadratic stability [

The research was carried out with funding from the Ministry of Education and Science of the Russian Federation (project: 121041300056-7).

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

Anatolievich, R.V., Viktorovich, A.S., Vasilyevich, D.A. and Alecseevich, G.I. (2021) On the Regression-Tensor Analysis of the Hardening Process of Metal Coatings. Journal of Applied Mathematics and Physics, 9, 1639-1651. https://doi.org/10.4236/jamp.2021.97110