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We chose a ‘rectangular plate with a circular hole under plane strain constraint’ and unidirectional monotonous loading for numerical tests of the different error sources and their indicators, Fig. 1a.

Figure 1a:. Benchmark system: Stretched square plate with a hole in plane strain, h = 200, r = 10

Three material parameters describe linear isotropic elastic and perfect plastic deformations, while mixed linear and nonlinear isotropic hardening with saturation need additionally three parameters, Table 2.

Figure 1b:. Upper left quarter, using symmetry conditions, with point numbers of computed data

Figure 2a:. v. Mises stresses at λ = 4.5 for hardening material

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1.2

Show that an a priori error bound for the result of the bisection method for finding √a with 0 < a < 1 is

|x−√a| ≤ (1−a)/2N,

where N is the number of iterations completed.

1.3

In Algorithm 1.1 we have at each stage

|x−√a| ≤ x1−x02.

Show that if

(x1−x0)2<4aε2,

then the relative error in x as an approximation to √a satisfies

|x−√a√a|<ε.

Hence modify the algorithm so that the iterations stop when the relative error is less than ε.

The aim of this paragraph is to evaluate the mean square performance of some of the proposed architectures at steady state. In particular, we are interested in the derivation of the EMSE of the nonlinear adaptive filter [2]. For simplicity we limit our analysis to the case of the Wiener and Hammerstein systems. The steady-state analysis is performed in a separate manner, by considering first the case of the linear filter alone and then the nonlinear spline function alone. Note that, since we are interested only in the steady state, the fixed subsystem in each substep can be considered adapted. The analysis is conducted by using a model of the same type of the used one, as shown in Fig. 3.5 for the particular case of the WSAF. Similar schemes are valid for the other kinds of architectures.

Figure 3.5. Model used for statistical performance evaluation, in the particular case of the WSAF.

In the following we denote with ε[n] the a priori error of the whole system, with εw[n] the a priori error when only the linear filter is adapted while the spline control points are fixed and similarly with εq[n] the a priori error when only the control points are adapted while the linear filter is fixed. In order to make tractable the mathematical derivation, some assumption must be introduced:

A1.

The noise sequence v[n] is independent and identically distributed with variance σv2 and zero mean.

A2.

The noise sequence v[n] is independent of xn, sn, ε[n], εw[n] and εq[n].

A3.

The input sequence x[n] is zero mean with variance σx2 and at steady state it is independent of wn.

A4.

At steady state, the error sequence e[n] and the a priori error sequence ε[n] are independent of φi′(un), φi′2(un), ‖xn‖, ‖sn‖, ‖CTun‖ and ‖CTUi,nwn‖.

The EMSE ζ is defined at steady state as

(3.30)ζ≡limn→∞E{ε2[n]}.

Similar definitions hold for the EMSE ζw due to the linear filter component wn only and the EMSE ζq due to the spline nonlinearity qn only.

The MSE analysis has been performed by the energy conservation method [2]. Following the derivation presented in [41] and [44] we obtain an interesting property.

The values of ζw and ζq are dependent on the particular SAF architecture considered. Specifically, for the Wiener case, we have [44]

(3.32)ζw=μw[n]σv2E{φi′2(un)‖xn‖2}2Δx2E{φi′(un)c3qi,n}−μw[n]E{φi′2(un)‖xn‖2},

where c3 is the third row of the C matrix and

(3.33)ζq=μq[n]σv2E{‖CTun‖2}2−μq[n]E{‖CTun‖2}.

Other useful properties on the steady-state performances of the WSAF can be found in [44]. For the Hammerstein case, we have [41]

(3.34)ζw=μw[n]σv2E{‖sn‖2}2−μw[n]E{‖sn‖2}

and

(3.35)ζq=μq[n]σv2E{‖CTUi,nwn‖2}2−μq[n]E{‖CTUi,nwn‖2}.

Both in metal and polymer forming, the viscoplastic behavior of the material can be modeled by a flow formulation, the viscosity being a function of the second invariant of the strain rate tensor ε(v):

(1)ηεv=K2|εv|m−1.

On a time depending domain Ω, inertia effects and volume forces being negligible, the velocity v and pressure p fields are solution of the following problem:

(2)∇⋅2ηεvεv−∇p=0∇⋅v=0inΩon∂Ω∩∂Ov−vO.n>0andτ=0orv−vO.n=0,andτ=αv−vOv−vO,

The error estimation procedures can be grouped into two distinct types: a priori and a posteriori estimates. A priori error estimates are based upon the characteristics of the solution and provide qualitative information about the asymptotic rate of convergence. The classic a priori error estimate is given by:

[88]‖u−uh‖m≤chp,p=k+1−m>0

where k is the degree of the interpolation polynomial, h is the characteristic length of an element, p is the rate of convergence, and c is a constant. Eqn (88) gives not only the rate at which the FE solution approaches the exact solution, but also indicates how one can improve the solution. We can decrease the mesh size (called h-convergence) or increase the order of the interpolation polynomial (called p-convergence). Incorporating both h— and p-convergences produces the h − p convergence technique. For illustration, consider the differential equation:

[89]d2udx2+2=0;0<x<1

with:

[90]u(0)=u(1)=0

The exact solution of the problem is:

[91]u(x)=x(1−x)

and the FE solutions are given by, for different number of elements, N:

ForN=2:uh(x)={hx,0≤x≤hh2(2−xh),h≤x≤2h}

[93]ForN=3:uh(x)={2hx,0≤x≤h2h2,h≤x≤2h2h2(3−xh),2h≤x≤3h}

[94]ForN=4:uh(x)={3hx,0≤x≤h2h2+hx,h≤x≤2h6h2−hx,2h≤x≤3h3h2(4−xh),3h≤x≤4h}

The residual type of error estimation can be understood by considering the FE solution of the following boundary value problem, stated in operator form:

[95]Au+f=0

When the FE finite element solution u_h, is used, eqn (95) leads to the residual R:

[96]Auh+f=R≠0

Eqn (96) shows that the residual effectively acts as a separate forcing function in the differential equation. This leads to the concept that the FE solution is the exact solution to a perturbed problem with a supplementary pseudo-force system. Subtracting eqn (96) from [95] produces:

[97]Ac+R=0

where e = u − u_h is the error. Hence the estimation of the error in the problem amounts to finding the solution of this problem. The residual R is composed of residuals in the interior of the domain as well as residuals on the boundary. The domain residuals are determined by substituting the FE solution into the differential equation. The boundary residuals, however, are more complicated since the derivatives of u_h, may not be continuous along the boundary.

For the postprocessing type of estimator, the error in the FE stress solution (obtained by differentiating the FE displacement solution) is defined as:

[98]eσ=σ−σh

Then the error in the energy norm is computed as:

[99]‖e‖=u−uh‖=(∫VeσTD−1eσdV)1/2

where D is the elasticity matrix. An estimator to e_σ can be produced by replacing σ in eqn (98) by the recovered solution σ^*:

[100]eσ≈eσ*=σ*−σh

where σ* is interpolated from the nodal values using the same basis functions N as those used for u_h.

Extending the error magnitude bound over P≤M outputs y(k),y(k−1),…,y(k−P+1) and using selective partial updates to limit update complexity leads to the set-membership partial-update APA. At each coefficient update the set-membership partial-update APA solves the following constrained optimization problem:

(4.74a)minIM(k)minwM(k+1)‖wM(k+1)−wM(k)︸δwM(k+1)‖22

subject to:

(4.74b)ep(k)=γ(k)

where γ(k)=[γ1(k),…,γP(k)]T is an error constraint vector with |γi(k)|≤γ. Equation (4.74b) can be rewritten as:

For fixed IM(k), (4.74) is equivalent to solving the following underdetermined minimum-norm least-squares problem for δwM(k+1):

(4.76)XM(k)δwM(k+1)=e(k)−γ(k),‖δwM(k+1)‖22 is minimum

with the unique solution given by:

(4.77)δwM(k+1)=XM†(k)(e(k)−γ(k))=XMT(k)(XM(k)XMT(k))−1(e(k)−γ(k))

where XM(k)XMT(k) is assumed to be invertible. Introducing a regularization parameter to avoid rank-deficient matrix inversion we finally obtain the set-membership partial-update APA recursion for fixed IM(k):

(4.78)wM(k+1)=wM(k)+μ(k)XMT(k)(ϵI+XM(k)XMT(k))−1(e(k)−γ(k))

where:

(4.79)μ(k)={1if |e(k)|>γ0otherwise

There are infinitely many possibilities for the error constraint vector γ(k). Two particular choices for γ(k) were proposed in (Werner and Diniz, 2001). The first one sets γ(k)=0 which leads to:

(4.80)w(k+1)=w(k)+μ(k)IM(k)XT(k)(ϵI+X(k)IM(k)XT(k))−1e(k)

This recursion, which we call the set-membership partial-update APA-1, uses the selective-partial-update APA whenever |e(k)|>γ and applies no update otherwise. The other possibility for γ(k) is defined by:

(4.81)γ(k)=[γe(k)|e(k)|e2(k)⋮eP(k)]

where e(k)=[e(k),e2(k),…,eP(k)]T with |ei(k)|≤γ for i=2,…,P. This results in:

which we shall refer to as the set-membership partial-update APA-2.

The optimum coefficient selection matrix IM(k) solving (4.74) is obtained from:

which is similar to the minimization problem for the selective-partial-update APA in (4.70). Thus, a simplified approximate coefficient selection matrix for the set-membership partial-update APA is given by (4.64). Note that the full implementation of (4.83) would be computationally expensive and, therefore, we do not advocate its use.

Let us assume a linear adaptive filter with output signal y(k)=wT(k)x(k). Then we have:

The a posteriori error gives the error signal at the adaptive filter output when the current adaptive filter coefficients are replaced with the updated coefficients. Using the principle of minimum disturbance, a selective-partial-update adaptive filtering algorithm is derived from the solution of the following constrained optimization problem:

(2.83a)minIM(k)minwM(k+1)‖wM(k+1)−wM(k))‖22

subject to:

(2.83b)ep(k)−(1−μ‖xM(k)‖22ϵ+‖xM(k)‖22)e(k)=0

where ϵ is a small positive regularization constant which also serves the purpose of avoiding division by zero, wM(k) is an M×1 subvector of w(k) comprised of entries given by the set {wj(k):ij(k)=1}, and xM(k) is an M×1 subvector of x(k) defined similarly to wM(k). Here the ij(k) are the diagonal elements of the coefficient selection matrix IM(k). Note that wT(k)IM(k)w(k)=‖wM(k)‖22 and xT(k)IM(k)x(k)=‖xM(k)‖22. Thus wM(k) is the subset of adaptive filter coefficients selected for update according to IM(k). For example, if N=3 and M=2, IM(k) and wM(k) are related as follows:

IM(k)=diag(1,1,0),wM(k)=[w1(k),w2(k)]TIM(k)=diag(1,0,1),wM(k)=[w1(k),w3(k)]TIM(k)=diag(0,1,1),wM(k)=[w2(k),w3(k)]T.

Using this relationship between IM(k) and wM(k), we have ‖wM(k+1)−wM(k))‖22=‖IM(k)(w(k+1)−w(k))‖22 in (2.83). Since only the coefficients identified by IM(k) are updated, an implicit constraint of any partial-update technique, including (2.83), is that:

(2.84)w(k+1)−w(k)=IM(k)(w(k+1)−w(k))

We will have occasion to resort to this partial-update ‘constraint’ in the sequel.

The constrained optimization problem in (2.83) may be solved either by using the method of Lagrange multipliers as we shall see next or by formulating it as an underdetermined minimum-norm estimation problem. We will use the latter approach when we discuss the ℓ1 and ℓ∞-norm constrained optimization problems later in the section. Suppose that IM(k) is fixed for the time being. Then we have:

(2.85a)minwM(k+1)‖wM(k+1)−wM(k))‖22

subject to:

(2.85b)ep(k)−(1−μ‖xM(k)‖22ϵ+‖xM(k)‖22)e(k)=0.

The cost function to be minimized is:

(2.86)J(wM(k+1),λ)=‖wM(k+1)−wM(k))‖22+λ(ep(k)−(1−μ‖xM(k)‖22ϵ+‖xM(k)‖22)e(k))

where λ is a Lagrange multiplier. Setting:

(2.87)∂J(wM(k+1),λ)∂wM(k+1)=0and∂J(wM(k+1),λ)∂λ=0

yields:

From (2.88a) we have:

since only the adaptive filter coefficients selected by IM(k) are updated [see (2.84)]. Substituting (2.89) into (2.88b) yields:

(2.90)λ=2μe(k)1ϵ+‖xM(k)‖22

Using this result in (2.88a) gives the desired adaptation equation for fixed IM(k):

(2.91)wM(k+1)=wM(k)+μϵ+‖xM(k)‖22e(k)xM(k)

What remains to be done is to find IM(k) that results in the minimum-norm coefficient update. This is done by solving:

(2.92)minIM(k)‖μϵ+‖xM(k)‖22e(k)xM(k)‖22

which is equivalent to:

(2.93)maxIM(k)‖xM(k)‖22+ϵ2‖xM(k)‖22

If any M entries of the regressor vector were zero, then the solution would be:

(2.94)minIM(k)‖xM(k)‖22

which would select the adaptive filter coefficients corresponding to the M zero input samples. For such an input (2.83) would yield the trivial solution wM(k+1)=wM(k). However, this solution is not what we wanted. We get zero update and therefore the adaptation process freezes while we could have selected some non-zero subset of the regressor vector yielding a non-zero update. This example highlights an important consideration when applying the principle of minimum disturbance, viz. the requirement to avoid the trivial solution by ruling out zero regressor subsets from consideration. Letting ϵ→0 does the trick by replacing (2.93) with:

(2.95)maxIM(k)‖xM(k)‖22

The solution to this selection criterion is identical to the coefficient selection matrix of the M-max LMS algorithm:

(2.96)IM(k)=[i1(k)0⋯00i2(k)⋱⋮⋮⋱⋱00⋯0iN(k)],ij(k)={1if |x(k−j+1)|∈max1≤l≤N(|x(k−l+1)|,M)0otherwise

If the step-size parameter μ in the optimization constraint obeys the inequality:

(2.97)|1−μ‖xM(k)‖22ϵ+‖xM(k)‖22|<1

then the a posteriori error magnitude is always smaller than the a priori error magnitude for non-zero e(k). This condition is readily met if 0<μ<2 with μ=1 yielding ep(k)≈0.

Combining (2.91) and (2.96) we obtain the solution to (2.83) as:

(2.98)w(k+1)=w(k)+μϵ+‖IM(k)x(k)‖22e(k)IM(k)x(k)

This algorithm is known as the selective-partial-update NLMS algorithm. When M=N (i.e. all coefficients are updated) it reduces to the conventional NLMS algorithm. The M-max NLMS algorithm is defined by:

(2.99)w(k+1)=w(k)+μϵ+‖x(k)‖22e(k)IM(k)x(k)

The selective-partial-update NLMS algorithm differs from the M-max NLMS algorithm in the way the step-size parameter μ is normalized. The following discussion will show that the selective-partial-update NLMS algorithm is a natural extension of Newton’s method.

The update recursion of the proposed DS-APA is written as follows:

(1.16)wk+1=wk+μATkδI+AkATk−1ekifγminσn2≤ek2≤γmaxσn2wkotherwise

In order to obtain steady-state MSE, Eq. (1.7) is taken. Substituting for d(k) and y(k) in e(k) we get e(k) = w_o^Tx(k) + n(k) − w^Tx(k).The error consists of two terms, namely, a priori error and a posteriori error vectors as follows:

(1.17)eak=Akw~k

(1.18)epk=Akw~k+1

where w~k=woT−wTk is the weights error vector. In order to resolve the nonlinearity present in Eq. (1.16), the model introduced in Section 1.2 is used in Eq. (1.16) as

(1.19)wn+1=wk+μPupdateATkδI+AkATk−1ek

In this section we consider the case of the interpolation of a given continuous function f:[a,b]→R. For any x∈R, the interpolation error is defined as

(3.29)Enf(x):=f(x)−Πnf(x).

The principal aim of the polynomial interpolation process is the uniform convergence of Πnf to f as a function of the polynomial degree n.

The following a priori error estimate can be proved.