Basic Numerical Techniques - Jacobs University Bremen · Basic idea of Monte Carlo I Generate inputs (i.e. parameters) randomly I Solve forward problem many times (M realizations)

Basic Numerical TechniquesComputational Electromagnetics

Basic Numerical Techniques

Outline

Introduction

Numerical Integration

Random Number Generation

Monte Carlo Methods

Conclusion


Some useful tools in CEM


I Computing elements of coupling matrix in method of moments

I Near-field to far-field transformations (radiated fields)

I Computing charge and current in FD and FEM methods


I Non-uniform random variables

Monte Carlo Analysis

I Statistical analysis of arbitrary systems


Outline

Introduction



Monte Carlo Methods

Conclusion



Goal is to compute

I =

∫ b

af (x)dx ≈

N∑i=1

ci f (xi ). (1)

I This kind of approximation is called numerical quadrature.

I Several methods exist for choosing the xi and ci .


Uniform Sampling

I Sample interval from a to b uniformly

I Form subintervals

I Approximate f (x) with nth order polynomial

I Closed formulas (use endpoints) open formulas (don’t)

Midpoint Rule

f(x)

x2 xN. . .

∆x

xx1 ba

i = 1, . . . ,N, (2)

xi = (i − 1

2)∆x + a, ∆x =

b − a

N(3)

ci = ∆x . (4)


Higher-order Rules

Trapezoidal Rule

f(x)

. . .

∆x

xxN = ba = x1

xi = (i − 1)∆x + a, (5)

∆x =b − a

N − 1(6)

ci = 0.5, 1, 1, . . . , 1, 0.5∆x . (7)

Simpson’s Rule

xi = (i − 1)∆x + a, (8)

∆x =b − a

N − 1(9)

ci = 1, 4, 2, 4, 2, . . . , 2, 4, 1∆x

3. (10)


Higher-order Rules

Trapezoidal Rule

f(x)

. . .

∆x

xxN = ba = x1

xi = (i − 1)∆x + a, (5)

∆x =b − a

N − 1(6)

ci = 0.5, 1, 1, . . . , 1, 0.5∆x . (7)

Simpson’s Rule

xi = (i − 1)∆x + a, (8)

∆x =b − a

N − 1(9)

ci = 1, 4, 2, 4, 2, . . . , 2, 4, 1∆x

3. (10)


Newton-Cotes Rules

I Generalization of other rules

I Subdivide total interval: equal-length sub-intervals

I Approximate contribution of each subinterval with∫ b′

a′f (x)dx ≈ ∆x

n∑i=0

c(n)i f (xi ). (11)


Newton-Cotes Rules

∆x

f(x)

xb′ = x

na′ = x0

f (x) ≈ g(x) =n∑

i=0

f (xi )n∏

j=0,j 6=i

x − xj

xi − xj︸︷︷︸Pi (x)

.

(12)

Product term Pi (x) is an interpolating polynomial

Pi (x) =

1, x = xi

0, x = xk , k 6= i .(13)


Newton-Cotes Rules

∆x

f(x)

xb′ = x

na′ = x0

f (x) ≈ g(x) =n∑

i=0

f (xi )n∏

j=0,j 6=i

x − xj

xi − xj︸︷︷︸Pi (x)

.

(12)

Product term Pi (x) is an interpolating polynomial

Pi (x) =

1, x = xi

0, x = xk , k 6= i .(13)


Newton-Cotes Rules

∆x

f(x)

xb′ = x

na′ = x0

Uniform sampling: xi = i∆x + a′, ∆x = (b′ − a′)/n

Pi (x) =∏

j=0,j 6=i

x − j∆x − a′

i∆x − j∆x=

∏j=0,j 6=i

s − j

i − j= Li (s), (14)

where s = (x − a′)/∆x .


Example Interpolating Polynomials

Li (s) =∏

j=0,j 6=i

s − j

i − j

0 0.5 1 1.5 2−0.2

0

0.2

0.4

0.6

0.8

1

s

Li(s

)

L0

L1

L2

0 0.5 1 1.5 2 2.5 3−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

1.2

s

Li(s

)

L0

L1

L2

L3


Newton-Cotes: Integration

Since g(x) is now an nth order polynomial, we can integrate itdirectly: ∫ b′

a′g(x)dx =

∫ b′

a′

n∑i=0

f (xi )Li (s)dx (15)

=n∑

i=0

f (xi )

∫ b′

a′Li (s)dx (16)

= ∆xn∑

i=0

f (xi )

∫ n

0Li (s)ds︸︷︷︸c

(n)i

. (17)

From before

∫ b′


n∑i=0

c(n)i f (xi ). (18)


Example: Newton-Cotes (n = 2)

Applying c(n)i =

∫ n0 Li (s)ds, Li (s) =

∏j=0,j 6=i

s−ji−j

c(2)0 =

∫ 2

0

s − 1

−1

s − 2

−2ds =

1

3(19)

c(2)1 =

∫ 2

0

s

1

s − 2

−1ds =

4

3(20)

c(2)2 =

∫ 2

0

s

2

s − 1

1ds =

1

3(21)

leading to ∫ b′


3[f (x0) + 4f (x1) + f (x2)]. (22)

This is just Simpson’s rule on each sub-interval.


Gaussian Quadrature

∫ b

aw(x)f (x)dx ≈

∑i

wi f (xi ), (23)

I Non-uniform sampling

I w(x) is a weighting function

I xi are chosen to be the zeros of an orthogonal polynomialdefined on (a, b)

I Advantages:I Can accommodate cases where a and/or b are infiniteI Exact if function is polynomial of degree 2n − 1 or less

(Maximum accuracy for fixed number of points)

I Disadvantages:I Complexity of non-uniform samplingI More difficult to compute xi


Multi-dimensional Integration

I Natural extension of 1D rules

I Given the rule ∫ b

af (x)dx ≈

N∑i=1

ci f (xi ), (24)

I Can extend to 2D using∫ bx

ax

∫ by

ay

f (x , y)dy dx ≈Nx∑i=1

Ny∑j=1

cx ,icy ,j︸︷︷︸cij

f (xi , yj). (25)


Outline

Introduction



Monte Carlo Methods

Conclusion



I MATLAB/C: Can produce uniform RV on [0,1]

I What if we need other non-uniform RVs?I Two basic methods:

I Inverse methodI Rejection method


Inverse Method

I CDF of RV X is Fx(x) = Pr X ≤ xI Inverse method requires F−1

x (x)

I Let U = Fx(X )

Fu(u) = Pr U ≤ u (26)

= Pr Fx(X ) ≤ u (27)

= PrX ≤ F−1

x (u)

(28)

= Fx [F−1x (u)] (29)

= u. (30)

I Observation: X can be anything, but U always uniform!

I Method: Generate U ∼ U[0, 1], X = F−1x (U)


Inverse Method

I CDF of RV X is Fx(x) = Pr X ≤ xI Inverse method requires F−1

x (x)

I Let U = Fx(X )

Fu(u) = Pr U ≤ u (26)

= Pr Fx(X ) ≤ u (27)

= PrX ≤ F−1

x (u)

(28)

= Fx [F−1x (u)] (29)

= u. (30)

I Observation: X can be anything, but U always uniform!

I Method: Generate U ∼ U[0, 1], X = F−1x (U)


Inverse Method: Example

Rayleigh Distribution

f (x) =x

σ2e−

x2

2σ2 (31)

Fx(x) = 1− e−x2

2σ2 = u (32)

x = F−1x (u) =

√−2σ2 ln(1− u) (33)

We can generate uniform random variables U ∼ U[0, 1] and pluginto the equation above to get variables X that are Rayleighdistributed.


Rejection Method

What happens if we can’t get inverse CDF? Can use rejectionmethod.

PDF of Random Variable

xa b

y

fx(x)

ymax

Compute

U1 ∼ U[0, 1] (34)

U2 ∼ U[0, 1] (35)

and

X1 = a + U1(b − a) (36)

Y1 = ymaxU2 (37)

I Y1 ≤ fx(X1), declareX = X1.

I If not, reject and try again.


Rejection Method: How it works

I Find PDF of X1 under the assumption that X1 is not rejected

I Let R be RV: R = 0 (not rejected) and R = 1 (rejected)

I Conditional PDF on rejection indicator

f (x1|r) =f (r |x1)f (x1)

f (r)(38)

f (x1|r = 0) =Pr R = 0|X1 = x1 f (x1)

Pr R = 0(39)

=Pr Y1 ≤ fx(X1)|X1 = x1 f (x1)

Pr Y1 ≤ fx(X1)(40)

=Pr Y1 ≤ fx(x1) f (x1)

Pr Y1 ≤ fx(X1). (41)


Rejection Method: How it works (cont’d)

f (x1|r = 0) =Pr Y1 ≤ fx(x1) f (x1)

Pr Y1 ≤ fx(X1)

xa b

y

fx(x)

ymax

Pr Y1 ≤ fx(x1) =Height to fx(x1)

Total height=

fx(x1)

ymax(42)

Pr Y1 ≤ fx(X1) =PDF area

Uniform box area=

1

(b − a)ymax(43)

f (x1) = Uniform PDF on [a,b] =1

b − a. (44)

Find that f (x1|r = 0) = fx(x1).


Outline

Introduction



Monte Carlo Methods

Conclusion


Monte-Carlo Analysis

I Analysis of complex devices and systems

I Non-linear, non-invertible input/output relationship

I Deriving distributions of outputs: difficult or impossible

Basic idea of Monte Carlo

I Generate inputs (i.e. parameters) randomly

I Solve forward problem many times (M realizations)

I Compute empirical distributions of outputs

Uses of Monte Carlo

I Finding device yield

I Evaluating algorithmic sensitivity

I Deriving solution confidence

I Validating analytical solutions










Uses of Monte Carlo














Uses of Monte Carlo






Monte-Carlo Analysis: Example

Quarter-Wave Transformer Design

ZS

Zin

ℓ

Z1

ZL

Zin = Z1ZL + jZ1 tanβ`

Z1 + jZL tanβ`. (45)

Ideal Parameters

I Impedance:Z1 =

√ZSZL = 70.71Ω

I Length: β` = π/2

Actual Parameters: Manufacturing tolerance

Z1 ∼ N (µZ , σ2Z )

β` ∼ N (µβ`, σ2β`).

µZ = 70.71

µβ` = 90

σ = Tµ

Analyze M = 105 Monte-Carlo realizations, T=10% or 20%.


Monte-Carlo Analysis: Example resultCDF of 105 realizations

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

X = |Zin

|

Pr(

X <

abscis

sa)

10%20%

Observations

I If require 40Ω < |Zin| < 60Ω

I T=20%: yield is around 40%

I T=10%: yield is around 70%


Outline

Introduction



Monte Carlo Methods

Conclusion


Conclusion

Techniques Learned

I Numerically integrating arbitrary functions

I Generating non-uniform RVs

I Analysis with Monte Carlo

Next Time

I Begin with actual CEM methods

I Finite-difference (FD) techniques


Documents

Basic Numerical Techniques - Jacobs University Bremen · Basic idea of Monte Carlo I Generate inputs (i.e. parameters) randomly I Solve forward problem many times (M realizations)