Although the applications of this method in turbulence and other dynamical systems often involve vector valued data, we discuss POD using a scalar function
v=v(x,t), x in , t in [0,T]. (2.191)
We define the spatial norm, spatial inner product, and the temporal expectation value by
v2=
v2dx, v,u =
vudx, E{v} = 1 T
T
0
vdt, (2.192) respectively. In POD, we wish to express the data,v, a s
v(x,t)= N
i
fi(t)ui(x), (2.193) where both,fi(t)and the spatial basis functionsui(x), have to be found.
We assume all the functions involved are continuous, and the integral over space and the integral over time commute. To start the process, let us ask ourselves the following: What is the single term f(t)u(x), which is the best approximation forv(x,t)?
The criterion for findinguis the requirement that the expectation value of the square-errorEhas to be a minimum. The square-errorE is defined as
E=
[v(x,t)−f(t)u(x)]2dx, (2.194) and its expectation value, as
E{E} = 1 T
T
0
[v(x,t)−f(t)u(x)]2dxdt. (2.195) We assume our single basis function is normalized. That is,
u =1. (2.196)
The extremization ofE{E}with the constraint u =1 can be done using the modified functional
E{E∗} =E{E} +à[u2−1], (2.197) whereàis a Lagrange multiplier. Explicitly,
E{E∗} = 1 T
T
0
[v(x,t)−f(t)u(x)]2dxdt+à[u2−1]. (2.198)
The first variation of this functional usingf−→f+δfandu−→u+δu, gives
1 T
T
0
2[v−fu](−uδf)dxdt=0, (2.199) 1
T T
0
2[v−fu](−fδu)dxdt+2
uδudx=0. (2.200) Noting thatf is afunction of onlyt, a nduis that of onlyx, we ha ve
f = v,u, (2.201)
1 T
T
0
[vf−f2u]dt−àu=0, (2.202)
where we have usedu =1. Using Eq. (2.201) in(2.202),
E{v(x,t)v(x,t)}u(x)dx= [E{f2} +à]u(x). (2.203) This relation shows that the functionugets mapped into itself through a symmetric integral operatorR(x,x)defined by
R(x,x)=E{v(x,t)v(x,t)}, (2.204)
and the associated eigenvalue problem is
R(x,x)u(x)dx=1
λu(x). (2.205) Equation (2.203) becomes
1
λ=E{f2} +à. (2.206)
Let us compute the first term on the right-hand side.
E{f2} =E{
v(x,t)u(x)dx}2
=E{
v(x,t)u(x)dx
v(x,t)u(x)dx}
=
E{v(x,t)v(x,t)}u(x)dxu(x)dx
=
u2(x) λ dx
=1
λ. (2.207)
Thus, the Lagrange multiplieràturns out to be zero. Also, asE{f2}>0, the eigenvalue is positive.
From the Hilbert-Schmidt theory, our eigenvalue problem with a real symmetric continuous kernel gives a set ofNorthonormal eigen- functionsuiwith eigenvaluesλi, whereNmay be infinite. With this, we extend our approximation to
v(x,t)= N
i
fi(t)ui(x), (2.208)
wherefi= v,uiand the orthonormal eigenfunctionsuisatisfy
ui(x)=λi
R(x,x)ui(x)dx. (2.209) The expectation value of the error becomes
E{E} =E{
[v−
fiui]2dx}, (2.210)
which may be broken into the three integrals:
I1=E{
v(x,t)v(x,t)dx}, I2= −2
E{
v(x,t)fiui(x)dx}, I3=
E{
fi2u2i(x)dx},
where inI3, we have anticipated the orthogonality ofui.
With the expansion of the kernel using Mercer’s theorem stated in Eq. (2.100), we have
E{v(x,t)v(x,t)} =R(x,x)=ui(x)ui(x) λi
, I1=
R(x,x)dx= 1 λi
, I2= −2
E{fi2} = −2 1 λi
, I3=
E{fi2} = 1 λi
. (2.211)
These integrals add up to zero, and our choice of basis functions makes the expectation value of the error zero, provided the complete sequence of eigenfunctions are used for the expansion.
In applications, an approximate representation of the data using a finite number of eigenfunctions is more practical. Also, in order to improve numerical accuracy the spatial mean value of the data is removed before finding the low dimensional representation through the proper orthogonal decomposition.
SUGGESTED READING
Abramowitz, M. and Stegun, I. (1965).Handbookof Mathematical Functions (National Bureau of Standards), Dover.
Barber, J. R. (2002).Elasticity, 2nd ed., Kluwer.
Brebbia, C. A. (1978).The Boundary Element Method for Engineers, Pentech Press, London.
Chatterjee, A. (2000). An introduction to the proper orthogonal decomposi- tion,Current Science, Vol. 78, No. 7, pp. 808–817.
Courant, R. and Hilbert, D. (1953).Methods of Mathematical Physics, Vol. 2, Interscience.
Hartmann, F. (1989).Introduction to Boundary Elements, Springer-Verlag.
Hildebrand, F. B. (1965).Methods of Applied Mathematics, 2nd ed., Prentice- Hall.
Holmes, P., Lumley, J. L., and Berkooz, G. (1996). Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge University Press.
Pozrikides, C. (2002). A Practical Guide to Boundary Element Methods, Chapman & Hall/CRC.
Rahman, M. (2007).Integral Equations and Their Applications, WIT Press.
Sneddon, I. N. (1966).Mixed Boundary-Value Problems in Potential Theory, North Holland.
Tricomi, F. G. (1957).Integral Equations, Interscience.
EXERCISES
2.1 Using differentiation, convert the integral equation u(x)=1
2 1
0 |x−ξ|u(ξ)dξ+x2 2
into a differential equation. Obtain the needed boundary condi- tions, and solve the differential equation.
2.2 Convert
u(x)=λ 1
−1
e−|x−ξ|u(ξ)dξ
into a differential equation. Obtain the required number of boundary conditions.
2.3 Solve the integral equation u(x)=ex+λ
1
−1
e(x−ξ)u(ξ)dξ, and discuss the conditions onλfor aunique solution.
2.4 Solve
u(x)=x+ 1
0
(1−xξ)u(ξ)dξ.
2.5 Solve
u(x)=sinx+ π
0
cos(x−ξ)u(ξ)dξ.
2.6 Obtain the eigenvalues and eigenfunctions of the equation u(x)=λ
2π
0
cos(x−ξ)u(ξ)dξ. 2.7 Find the eigenvalues and eigenfunctions of
u(x)=λ 1
0
(1−xξ)u(ξ)dξ.
2.8 From the differential equation and the boundary conditions obtained for Exercise2.2, find the values ofλfor the existence of non-trivial solutions.
2.9 Show that the equation u(x)=λ
1
0
|x−ξ|u(ξ)dξ
has an infinite number of eigenvalues and eigenfunctions.
2.10 Obtain the values ofλandafor the equation u(x)=λ
1
0 (x−ξ)u(ξ)dξ+a+x2
to have (a) a unique solution, (b) a nonunique solution, and (c) no solution.
2.11 Obtain the resolvent kernel for u(x)=λ
2π
0
ein(x−ξ)u(ξ)dξ+f(x).
2.12 Find the resolvent kernel for u(x)=λ
π
0
cos(x−ξ)u(ξ)dξ+f(x). 2.13 Show that for a Fredholm equation
u(x)=λ b
a
k(x,ξ)u(ξ)dξ+f(x),
the resolvent kernelg(x,ξ)satisfies g(x,ξ)=k(x,ξ)+λ
b
a
k(x,η)g(η,ξ)dη.
Thus, the resolvent kernel satisfies the integral equation for u when the forcing functionfis replaced bykwithξandλkept as parameters.
2.14 Demonstrate the preceding result whenk=1−2xξin the domain 0<x,ξ <1.
2.15 For the Volterraequation, u(x)=
x
0 (1−xξ)u(ξ)dξ+1, starting withu(0)=1, obtain iteratively,u(2). 2.16 For the integral equation of the first kind
2π
0
sin(x+ξ)u(ξ)dξ=sinx, discuss the consequence of assuming
u(x)=
n=0
ancosnx+
n=1
bnsinnx, where the constants,anandbn, are unknown.
2.17 With the quadratic forms, J1[u] =
b
a
b
a
k(x,ξ)u(x)u(ξ)dξdx, J2[u] = b
a
u2dx,
whereuis asmooth function in(a,b), show that the functionsu, which extremize
J[u] =J1[u]
J2[u],
are the eigenfunctions of[k(x,ξ)+k(ξ,x)]/2. Also show that the extremaofJcorrespond to the reciprocal of the eigenvalues of k(x,ξ).
2.18 In the preceding problem, assuming k is symmetric, compute J[v]if
v(x)=
n
anun(x), whereunare normalized eigenfunctions.
2.19 The generalized Abel equation, x
0
u(ξ)dξ
(x−ξ)α =f(x),
is solved by multiplying both sides by(η−x)α−1and integrating with respect to x from 0 to η. Implement this procedure, and discuss the allowable range for the index,α.
2.20 Solve the singular equation x
0
u(ξ)dξ
(x2−ξ2)=f(x), using a change of the independent variable.
2.21 If an elastic, 3D half space (z>0) is subjected to an axi-symmetric z-displacement,w(r), wherer=
x2+y2, we need to solve the integral equation for the distributed pressure on the horizontal surface,z=0,
r
0
g(ρ)dρ r2−ρ2= à
1−νw(r), 0<r<a,
whereàis the shear modulus,νis the Poisson’s ratio, andais the contact radius. Obtain the pressure distributiong(r)(see Barber (2002)).
2.22 In the preceding problem, if the contact displacement is due to arigid sphere of radiusRpressing against the elastic half space, we assume
w(r)=d− r2 2R,
wheredis the maximum indentation. Obtain the pressure distri- bution, the value of the contact radiusa, and the total vertical force.
2.23 Consider the integral equation u(x)=
1
0
k(x,ξ)u(ξ)dξ+x2, where
k(x,ξ)=
x(ξ−1), x< ξ, ξ(x−1), x> ξ.
Obtain an exact solution to this equation. Using the approximate kernelAx(1−x), findAusing the method of least square error.
Obtain an approximate solution foru. Compare the values of the exact and approximate solutions atx=0.5.
2.24 Consider the differential equation
u+u=x, u(0)=0, u(1)=0.
Find an exact solution. Obtain a finite difference solution by dividing the domain into four equal intervals.
Convert the differential equation into an integral equation using the Green’s function for the operator L=d2/dx2. Obtain a numerical solution of the integral equation by dividing the domain into four intervals and using the trapezoidal rule and the collocation method. Compare the results with the exact solution.
2.25 Consider the nonlinear integral equation u(x)−λ
1
0
u2(ξ)dξ=1.
Show that for λ <1/4 this equation has two solutions and for λ >1/4 there are no real solutions. Also show that one of these solutions is singular atλ=0 and two solutions coalesce atλ=1/4.
Sketch the solutions as functions ofλ. (Based on Tricomi(1957).)
F O U R I E R T R A N S F O R M S
The method of Fourier transforms is a powerful technique for solving linear, partial differential equations arising in engineering and physics when the domain is infinite or semi-infinite. This is an extension of the Fourier series, which is applicable to periodic functions defined on an interval−a<x<a. First, let us review the Fourier series and extend it to infinite domains in a heuristic form.