Stochastic Measures

Z

ΩN

Z(ξ)Φl(ξ)w(ξ)dξ₁· · ·dξ_N. (4.10) Here,γ_l is the norm of thejoint polynomial basis function and w(ξ) is the joint PDF. Due to orthogonality, the norm is obtained as

hΦl(ξ),Φk(ξ)i= ^YN

p=1

γ_lpδ_lp,kp =δ_l,kγ_l. (4.11) To perform the N-dimensional integration in (4.10) is challenging and results in an expo-nential grows of computation costs. A brief overview of approaches to perform this task is given in Section 4.2.4 and two approaches dedicated to reduce the computational burden of this quadrature are proposed in Sections 4.5.3 and 4.6. The effect of exponential growth of computational costs with the number of stochastic variables is known as the curse of dimensionality [178]. This is a general effect occurring in various methods of uncertainty quantification [207].

4.2.3 Stochastic Measures

The representation of a stochastic variable in terms of expansion coefficients (4.5) is favorable when deriving various stochastic measures. For details on how the formulas for the evaluation of the stochastic measures are derived, the reader is referred to [175, Chapter 5] and [208].

Here, only the results needed for further application are introduced.

The most common measure is the mean which is also the first stochastic moment. In terms of PCE the mean is equal to the first expansion coefficient

µ_Z =z0. (4.12)

Furthermore, the variance (the square of standard deviation σ_Z) can be obtained as σ_Z² =^XP

l=1

γ_lz_l². (4.13)

The standard deviation is found in a straight forward manner

σZ =^qσ²_Z =

v u u t

D

X

l=1

γlzl2. (4.14)

4.2 Fundamentals of Polynomial Chaos Expansion (PCE) Other statistic measures like the covariance and higher-order moments are readily available from the expansion coefficients as well [175,178].

Another measure which can directly be obtained from the PCE coefficients are Sobol’

indices⁶ which are conditional variances and represent a global sensitivity [209]. These measures are often interpreted as the relative contribution of a single (or group of) stochastic variables to the overall uncertainty [210,211]. The Sobol’ indices can be computed directly from the PCE coefficients [208,212]. The Sobol’ index S_i for the i^th stochastic variable is computed as

S_i =

P

l∈K_i

γlzl2

σ_Z² , (4.15)

where, K_i is a subset of the chosen multi-index, where the indices corresponding to the i^th variable are non-zero.

Moreover, the actual sensitivity as the derivative of the output variable with respect to an input variable can also be computed [178,213]. This formulation is particularly meaningful in the univariate case and is found to be

∂Z(ξ)

∂ξ =^XP

l=1

z_l ∂Ψl(ξ)

∂ξ

_ξ=0

. (4.16)

The sum skips the zeroth index because the derivative of a constant is zero by definition.

For specific polynomials, this form may be written in a more compact form: e.g. for the case of a Gaussian distributed variable and Hermite Polynomials one can use [1, Eq. 18.9.25]

and rewrite (4.16) as

∂Z(ξ)

∂ξ = 2^XP

l=1

zllHl−1(0). (4.17)

For an uniformly distributed variable andLegendre Polynomialsone can apply [1, Eq. 18.9.12]

and (4.16) becomes

∂Z(ξ)

∂ξ =−

P

X

l=1

z_l(l+ 1)P_l+1(0). (4.18)

6Sometimes calledSobol’ coefficients.

4.2.4 Obtaining the Coecients

The PCE coefficients are found by evaluation of (4.10), namely projecting the function of stochastic variables onto the polynomial space. In very few cases, it may be possible to analytically evaluate the required integration. In general, no analytic solution is available.

There are two approaches to handle this problem numerically: quadrature of the integral or casting the problem into a least squares problem.

The idea behind the least squares approach is to set up a matrix equation expressing the error between the actual function and the PCE representation and to minimize this expression [214–217].

Quadrature rules aim to solve the expression (4.10) numerically. Integrals containing orthogonal polynomials can be evaluated very conveniently using Gaussian quadrature [115].

Furthermore, the quadrature is exact for polynomial functions up to a degree of 2P+ 1 [175, Theorem 3.11]. This procedure requires the evaluation of P + 1 dedicated samples in the single variable case and N^P dedicated samples in the multivariate case. Other integration schemes (likestochastic testing) reduce the number of required samples but with an increased error compared to Gaussian quadrature. Gaussian quadrature, stochastic testing, and a novel quadrature approach are discussed in more detail in Section 4.5.

4.2.5 State-of-the-Art in PCE with Application to CEM

In CEM, PCE has first been applied to model circuits containing stochastic element values by applying SGM to MNA [218]. Later, generalizations, accelerations, and collocation approaches have been proposed [215, 219–221]. Furthermore, PCE has been applied to transmission-line theory [222–229], to transmission-lines in an MNA framework [230], non-linearly terminated lines [231], and non-linear elements in an MNA framework [232–235].

Furthermore, PCE has been applied on a more abstract level, to consider stochastic macromodels [215,216,225,236]. Moreover, PCE has been implemented together with general full-wave solvers, such as time-domain [237] and frequency domainFEM [238–240], 1-D [231]

and 3-D FDTD [211,241], as well as the relatedFIT [242]. Also, static BEM [243,244], 2-D MOM [245, 246], and IEsolvers [247] have been extended in order to take into account stochastic boundary conditions using PCE. In [241], the Fourier transform of a stochastic time signal in terms of a PCE representation is addressed briefly. Beyond methods exclusively considering electromagnetic problems, PCE has been applied to multi-physics systems as well [248].

From an application perspective, PCE has been used for uncertainty quantification in various fields. In the application field of SI/PI, it has been used for on-chip power grids and connections [15,221,225], interconnects [219,222,223,247,249], simple via [250–252]

4.3 Stochastic Galerkin Matching (SGM) and TSV model [253]. Likewise, PCE has been applied in the regime of microwave components [233,254] and planar optics [246,255].

In the context of CEM, different approaches have been proposed to accelerate the simulation of a stochastic problems modeled with PCE. All these techniques try to cure, or at least weaken, the curse of dimensionality. Typically, these approaches are applicable in various simulations involving PCE and are not limited to CEM. The three most widely applied acceleration techniques areModel Order Reduction (MOR) based approaches [221], decoupled PCE [205,206], and stochastic testing [177]. Techniques based on MOR apply PCE in a Galerkin matching or collocation based way, which increases the size of the deterministic formulated problem. In the next step, an MOR technique is applied to reduce the size of the augmented system. Examples can be found in [221,236,239,240,256].

Decoupled PCE [205, 206] makes use of the properties of the Kronecker product, that occurs when formulating a multi-index with lexicographic indexing. By approximation of the operator involved in this formulation, the resulting augmented matrix is cast into a block-diagonal form allowing for parallelization. In [205, 206], decoupled PCE has been proposed for Gaussian distributed variables, but it can also be adapted for other kinds of distributions [257]. Further works that address decoupled PCE can be found in [232, 235, 254]. Stochastic testing [177] refers to an algorithm that can be seen as an approximation of Gaussian quadrature rules. It reduces the number of required sampling nodes from N^P to ^(N+D)!_N!D! when using graded lexicographic indexing. It has been applied and investigated in [233,235,258,259].

4.3 Stochastic Galerkin Matching (SGM)

4.3.1 Fundamentals of SGM

When applying PCE it is often differentiated betweenintrusiveandnon-intrusiveapproaches.

Intrusive approaches require changes in the solution formalism of the associated deterministic problem, whereas non-intrusive approaches are usually sampling based and treat the solution formalism of the associated deterministic problem as ablack-box. In this context, intrusive PCE approaches can be seen as spectral approaches, whereas, non-intrusive approaches are sampling based approaches. In both cases, the result is present in the form of PCE coefficients. The foundation for most intrusive approaches is Galerkin matching. SGM is a widely used approach in UQ [215,220,225,231,234,260].

To introduce the general features of SGM, a stochastic impedanceZ(ξ) is considered which depends on the stochastic variable ξ. A stochastic impedance results in stochastic voltages

and currents in a circuit, therefore, they are considered to be dependent on the same stochastic variable. This way, Ohm’s law is written in a stochastic form

V (ξ) =Z(ξ)I(ξ). (4.19)

Using (4.4), the stochastic voltage, impedance, and current are expanded into the same polynomial basis with the respective coefficientsvi, zi, and ii.

V (ξ) = ^XP

The idea of Galerkin matching is to project both sides of the equation to the same basis using the scalar product. Projection on the basis polynomial of degree P yields

*

Here, e_m,n,l are the so-called linearization coefficients. This equation provides an explicit expression for the expansion coefficients of the voltage as a function of the expansion coeffi-cients of the impedance and the current. In (4.21), the general assumption of a maximum order of approximation of P for all three stochastic quantities is used. This assumption is generally made in SGM, mainly for practical reasons. However, the linearization coefficient is not necessarily zero for l, n≤P and m > P, this means that the order of the product of two expanded variables is larger that the polynomial degrees of the factors [261]. In general, the degree is doubled. By considering only the first P + 1 coefficients a truncation error is introduced. In practice, the error is negligible if P is selected high enough in the first place [261]. In the following, P is assumed to be sufficiently large and the truncation

4.3 Stochastic Galerkin Matching (SGM) error is, therefore, neglected.

Linearization coefficients do not depend on the expansion coefficients but only on the chosen polynomials. Hence, they can be precomputed. For all kinds of polynomials used in PCE, analytic expressions of the linearization coefficients are known, see. Table 4.3. Furthermore, recursive relations exists for the linearization coefficients of the used polynomials [262].

A more convenient representation of (4.22) can be found when using a matrix notation. By writing the expansion coefficients of the voltage and current in a vector, the PCE equivalent of (4.19) can be written as



The matrix ˆZ is generally referred to as an augmented matrix . This matrix can be seen as a representation ofZ(ξ), as it does only depend on the expansion coefficients of Z(ξ). In the case that the expansion coefficients are available, the augmented matrix can be written right away. Among others, the representation with augmented matrices is used to solve stochastic differential equations [175, Chapter 7] and to describe stochastic transmission lines [229, 260]. In general, stochastic scalars are substituted by deterministic matrices.

Usually, operations defined on the stochastic scalar can be represented analogously on the basis of expansion coefficients using the augmented matrix representation. In the following subsections, SGM for multiple variables is introduced and some general properties and observations on linearization coefficients and augmented matrices are stated and discussed.

4.3.2 Analogy to the Fourier Transform

PCE is a so-called spectral method [175]. Such methods differ from others by the use of integral transformations to project onto a more convenient basis. The most famous representative known in electromagnetics is the Fourier transform, where a time dependent periodic signal is represented in a spectral form. Similar to how the Fourier transform eliminates the time dependency from the signal, PCE eliminates stochastic variables. When comparing the Fourier transform and PCE, some analogies are observed. For example, the mean of a periodic signal – which would usually be called the DC part – is represented by the zeroth Fourier coefficient. Likewise, the zeroth expansion coefficient in PCE represents the stochastic mean. A more complete and detailed comparison between PCE and the Fourier transform can be found in [220].

4.3.3 SGM for Multiple Stochastic Variables

Analogous to the single variable case, SGM works in the multivariate case in a similar way.

Consider a generic impedance depending on a set of N stochastic variables. Using the a joint polynomial basis with a corresponding multi-index, (4.22) can be written in a similar way.

The linearization coefficients in the multivariate case follow from the definition of the joint polynomial basis

Similarly to (4.23), (4.24) can be written in a matrix form



The general procedure is similar to the one in the univariate case but the resulting augmented matrix is of size (D+ 1)×(D+ 1). Again, the matrix representation is obtained right away from the expansion coefficients, the linearization coefficients needed to construct the matrix only depend on the chosen basis and can be precomputed.

4.3 Stochastic Galerkin Matching (SGM) 4.3.4 Properties of Linearization Coecients

In the context of SGMlinearization coefficients are of extraordinary importance, as they occur when dealing with augmented matrices and products in general. Before discussing how to implement mathematical operations on the basis of expansion coefficients or augmented matrices, some properties of linearization coefficients shall be stated. For the linearization coefficients of all used polynomials in PCE analytical formulas exist, see Table 4.3. Using the scalar product notation, linearization coefficients are defined as

e_m,n,l = hΨn(ξ) Ψm(ξ),Ψl(ξ)i γm

. (4.27)

From this definition and considering Ψ0(0) = 1 for all polynomials in Table 4.3, some algebraic operations lead to the following identities for index rotations

e_m,n,l=e_m,l,n, (4.28)

en,m,l= γ_m

γ_nem,n,l, (4.29)

e_l,n,m= γ_m γl

e_m,n,l, (4.30)

and when one index is set to zero

e_0,n,l=δ_n,lγ_l, (4.31)

e_m,0,l =δ_m,l, (4.32)

e_m,n,0 =δ_n,m. (4.33)

This properties have some direct implications regarding the matrix E_l used to generate the augmented matrix in (4.23). The zero order matrix becomes the identity matrix.

E₀ =I (4.34)

Table 4.3: Analytical expressions of linearization coefficients Distribution Basis polynomial Reference

Gaussian Hermite [1, Eq. 18.17.49]

gamma Laguerre [263]

beta Jacobi [264] (special cases) uniform Legendre [265, Ch. 5], [266]

Furthermore, the elements of the first row and column of an augmented matrix can be written explicitly as

Zˆ

0,n =^XP

l=0

e_0,n,lz_l =z_nγ_n, (4.35)

Zˆ

m,0 =^XP

l=0

e_m,0,lz_l=z_m. (4.36)

(4.37) Hence, the first column of an augmented matrix contains the expansion coefficients.

As shown in [265, Chapter 5], the factor e_m,n,l appears when expressing the product of two polynomial basis functions in the polynomial basis which motivates the expression linearization coefficient

Ψn(ξ) Ψl(ξ) = ^n+lX

m=|n−l|

e_m,n,lΨm(ξ). (4.38)

In other words, linearization coefficients are only non-zero in a certain band. If the sum of two indices is less than the third, the linearization coefficients are zero

e_m,n,l= 0 if m+n < l orm+l < n orl+n < m. (4.39) Further properties, which are not general but only apply to certain polynomial basis functions are listed in the Appendix D.1.

4.3.5 Mathematical Operations with PCE Coecients

In PCE, stochastic variables are represented by their expansion coefficients. Using the principles of SGM, mathematical operations involving stochastic variables can be expressed in terms of expansion coefficients. In this section, the sum and the product of two stochastic variables represented in terms of expansion coefficients is shown. In the next section, these operations on the basis of augmented matrices is discussed.

4.3 Stochastic Galerkin Matching (SGM)

The Sum of Two Stochastic Functions

Consider the sum of two functions Z⁽³⁾(ξ) = Z⁽¹⁾(ξ) +Z⁽²⁾(ξ) to be dependent on the same set of stochastic variables ξ. The mean of the result Z⁽³⁾(ξ) can be obtained as the sum of the means of Z⁽¹⁾(ξ) and Z⁽²⁾(ξ), but to obtain the PDF of Z⁽³⁾(ξ) convolution operations are involved [174]. Using PCE, we are interested in the expansion coefficients of Z⁽³⁾(ξ) as they are sufficient to restore the PDF and all the other desired stochastic information. Assume the expansion coefficients ofZ⁽¹⁾(ξ) andZ⁽²⁾(ξ) are known and given as z_l⁽¹⁾ and z_l⁽²⁾. Then, the expansion coefficients z_l⁽³⁾ of Z⁽³⁾(ξ) may be obtained using

The sum of two functions depending on the same set of stochastic variables is represented by the element wise sum of the expansion coefficients of the same order.

The Product of Two Stochastic Functions

Analogously to the sum, the product Z⁽³⁾(ξ) =Z⁽¹⁾(ξ)Z⁽²⁾(ξ) of two functions depending on the same set of stochastic variables is investigated. Again, convolution like operations would be required to evaluate the PDF of Z⁽³⁾(ξ) from the PDFs of Z⁽¹⁾(ξ) and Z⁽²⁾(ξ).

On the basis of expansion coefficients, a formula for the product of two stochastic functions is obtained by applying Galerkin matching

z⁽³⁾_m = 1

Please note the similarity between (4.41) and (4.25). In [261], (4.41) is further extended to the product of three variables.

The expansion coefficients of a product of two functions that are dependent on the same set stochastic variables can be evaluated using (4.41). To compute all coefficients, D+ 1 double sums have to be computed, resulting in the summation of (D+ 1)³ =D³+ 3D²+ 1 products. To reduce the complexity an optimized scheme is proposed: First, consider the multiplication in the univariate case

z⁽³⁾_m =^XP

l=0 P

X

n=0

z⁽¹⁾_l z⁽²⁾_n e_m,n,l. (4.42)

The number of summations can be reduced when considering that many linearization coefficients are in fact zero. Using (4.39), the equation can be rewritten as

z⁽³⁾_m =^XP

l=0

min(P,m+l)

X

n=|m−l|

z⁽¹⁾_l z⁽²⁾_n em,n,l. (4.43) The evaluation of this formula requires ¹₂P³+³₂P²+2P+1 multiplications, see Appendix D.2.

Therefore, application of (4.43) requires about half the operations (4.42) requires.

The approach is extended to the multivariate case by considering (4.25). Due to the indexing scheme, it is not possible to simply reduce the boundaries of the sum like in the univariate case. When implementing the multiplication, the reduction of operations can be obtained by storing the multivariate linearization coefficients in a sparse tensor data type storing only nonzero entries.

By defining the sum and product of stochastic functions depending on the same set of stochastic variables various other operations can be constructed [261]. Sequential applica-tions of sums and products allow the evaluation of polynomials in general. Application of theTaylor series allows to evaluate non-polynomial functions using only sums and products.

However, this is mainly of relevance for theoretical aspects, as the numerical effort increases rapidly. In practice, sampling based approaches using Gaussian quadrature formulas are preferred, see [177,231,261], and Section 4.5.

4.3 Stochastic Galerkin Matching (SGM)

4.3.6 Properties of Augmented Matrices

As mathematical functions can be defined on the basis of expansion coefficients, they can be defined likewise on the basis of augmented matrices. Before discussing how mathematical operations are translated to operations on augmented matrices, recapitulate the following properties already outlined in previous subsections:

1. If the expansion coefficients are available, the augmented matrix can be constructed with the effort of a matrix sum with D+ 1 (orP + 1) summands.

2. The augmented matrix contains the expansion coefficients in a plain form in the first column.

The idea when defining mathematical operations using augmented matrices is to construct the matrices from the expansion coefficients, perform the operations and retrieve the expansion coefficients from the resulting matrix.

The Sum of Two Stochastic Functions

Again, we first consider the sum of two stochastic functions Z⁽³⁾(ξ) =Z⁽¹⁾(ξ) +Z⁽²⁾(ξ) depending on the same set of stochastic variables ξ. The augmented matrices Zˆ⁽¹⁾ andZˆ⁽²⁾ are constructed like in (4.26) and summed up, the resulting matrix should contain the expansion coefficients corresponding to the sum in the first column

z_l⁽³⁾ =Zˆ⁽¹⁾+Zˆ⁽²⁾

l,0 =z_l⁽¹⁾+z⁽²⁾_l (4.44) Hence, the sum of two augmented matrices corresponds to the sum of the coefficients and therefore represents the sum of the stochastic functions.

The Product of Two Stochastic Functions

For the product one might expect that the matrix product of two augmented matrices represents theproduct of the functions depending on stochastic variables and in fact the product is mapped quite well. Let us take a closer look at the underlying equations and consider the productZ⁽³⁾(ξ) =Z⁽¹⁾(ξ)Z⁽²⁾(ξ) of two functions depending on the same set of stochastic variables. By augmenting Z⁽¹⁾(ξ) and Z⁽²⁾(ξ) to Zˆ⁽¹⁾ and Zˆ⁽²⁾, respectively, the augmented matrix Zˆ⁽³⁾ corresponding to Z⁽³⁾(ξ) may be written as

Zˆ⁽³⁾ =Zˆ⁽¹⁾Zˆ⁽²⁾. (4.45)

This operation occurs frequently in literature [8,10,224–226]. Results presented in literature involve up to 12 multiplications and show results which are in reasonable agreement with reference simulations.

As the formula (4.45) represents a multiplication of stochastic scalars, the matrices should commute. The order of products should make no difference. If the matrices commute, the entries of the resulting matrix must be equal for both possible orders

D

This can only be generally true, if the products of the linearization coefficients are equal e_m,k,ie_k,n,j =e_m,k,je_k,n,i. (4.48) But this expression is not generally true, counter examples can easily be constructed. For example in the case of one stochastic variable and the choice of n= 1, m= 2, k = 0,i= 1, and j = 2 the expressions yield

e2,0,1e0,1,2 = 0 6=e2,0,2e0,1,1 =γ₂. (4.49) Hence, augmented matrices do not commute in general

Zˆ⁽¹⁾Zˆ⁽²⁾ 6=Zˆ⁽²⁾Zˆ⁽¹⁾. (4.50)

Nevertheless, practical results have shown that the approach provides reasonable results.

To explain this, one needs to take a look at the first column of the resulting matrix. In this case, the linearization coefficients in (4.48) become

em,k,iek,0,j =em,k,jek,0,i, e_m,k,iδ_k,j =e_m,k,jδ_k,i,

e_m,k,i =e_m,k,i.

(4.51)

Hence, the first column – and therefore the retrieved expansion coefficients – are invariant to the order. In other words, even though the permutation property is not preserved in the matrices, it is preserved when extracting the expansion coefficients. After verifying the commutation property for the first column of augmented matrices, let us take a look at the

4.3 Stochastic Galerkin Matching (SGM)

resulting expansion coefficients

z_m⁽³⁾ =Zˆ⁽³⁾

m,0 = ^XD

k=0

Zˆ⁽¹⁾

mk

Zˆ⁽²⁾

k0

= ^XD

k=0 D

X

l=0 D

X

n=0

Zˆ_l⁽¹⁾Zˆ_n⁽²⁾e_m,k,le_k,0,n

=^XD

l=0 D

X

n=0

Zˆ_l⁽¹⁾Zˆ_n⁽²⁾e_m,n,l.

(4.52)

This formula is exactly the same as the one derived for the expansion coefficients (4.41).

Hence, there is no difference in determining the expansion coefficients using a double sum or performing the matrix product. Even though the matrix multiplication can require more operations (depending on the implementation) and requires more memory, sine the

Hence, there is no difference in determining the expansion coefficients using a double sum or performing the matrix product. Even though the matrix multiplication can require more operations (depending on the implementation) and requires more memory, sine the

Im Dokument Extension of the Contour Integral Method for Stochastic Modeling of Waveguiding Structures (Seite 134-0)