numerical approaches to singular value asymptotics for

OutlineMotivation

Sturm-Liouville problemNumerical approaches

Conclusion

Numerical approaches to singular value asymptoticsfor specific integral operators

Melina Freitag

Fakultat fur MathematikTechnische Universitat Chemnitz

Professur Inverse Probleme

Minisymposium Inverse Probleme23rd September 2004

Melina Freitag Numerical approaches to singular value asymptotics

Page 2: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

1 Outline

2 Motivation

3 The Sturm-Liouville problem

4 Numerical approaches to the problemFinite difference methods for the Sturm-Liouville problemGalerkin method for integral equations of the first kindSummary of numerical results

5 Conclusion

Page 3: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Problem formulation

Given: B = M ◦ J as Fredholm integral equation

[Bx ](s) =

∫ 1

0k(s, t)x(t)dt, (0 ≤ s ≤ 1)

with

k(s, t) =

{

m(s), (0 ≤ t ≤ s ≤ 1)0, (0 ≤ s ≤ t ≤ 1).

where m(s) has got a zero, for example m(s) = sα or

m(s) = e−1

sα

Problem: finding the singular value asymptotics of B

Page 4: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Problem formulation

Given: B = M ◦ J as Fredholm integral equation

[Bx ](s) =

∫ 1

0k(s, t)x(t)dt, (0 ≤ s ≤ 1)

with

k(s, t) =

{

m(s), (0 ≤ t ≤ s ≤ 1)0, (0 ≤ s ≤ t ≤ 1).

where m(s) has got a zero, for example m(s) = sα or

m(s) = e−1

sα

Problem: finding the singular value asymptotics of B

Page 5: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Problem formulation

well-known: singular values of integral operator J:

σn(J) =2

π(2n − 1)

analytical results for the singular value decomposition (Chang(1952), Reade (1984), minimax principle):

σn(B) = O(n−1)

only lower bounds on the degree of ill-posedness if m(s) hasgot a zero

Page 6: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Problem formulation

well-known: singular values of integral operator J:

σn(J) =2

π(2n − 1)

analytical results for the singular value decomposition (Chang(1952), Reade (1984), minimax principle):

σn(B) = O(n−1)

only lower bounds on the degree of ill-posedness if m(s) hasgot a zero

Page 7: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Formulation as a Sturm-Liouville problem

Formulate

[Bx ](s) =

∫ s

0m(s)x(t)dt, (0 ≤ s ≤ 1),

as a boundary value problem, using the eigenvalue equationB∗Bu = σ2u:

σ2

(

u′(τ)

m2(τ)

)

′

= −u(τ), m(τ) 6= 0, u ∈ C 2[0, 1]

or−(a(τ)u′(τ))′ = λu(τ), u(1) = u′(0) = 0,

where λ =1

σ2and a(τ) =

1

m2(τ).

Page 8: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Finite difference methods for the Sturm-Liouville problemGalerkin method for integral equations of the first kindSummary of numerical results

Finite difference methods for the Sturm-Liouville problem

Boundary value problem:

−(a(τ)u′(τ))′ = λu(τ)

u(1) = 0 and limτ→0

a(τ)u′(τ) = 0.

apply classical finite difference method:

(

ai+1 − ai−1

4h2−

ai

h2

)

ui−1 +2ai

h2ui

+

(

ai−1 − ai+1

4h2−

ai

h2

)

ui+1 = λui

right hand side τ0 := ε.

Page 9: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

−(a(τ)u′(τ))′ = λu(τ)

a(τ)u′(τ) = 0.

(

ai+1 − ai−1

4h2−

ai

h2

)

ui−1 +2ai

h2ui

+

(

ai−1 − ai+1

4h2−

ai

h2

)

ui+1 = λui

Page 10: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

−(a(τ)u′(τ))′ = λu(τ)

a(τ)u′(τ) = 0.

(

ai+1 − ai−1

4h2−

ai

h2

)

ui−1 +2ai

h2ui

+

(

ai−1 − ai+1

4h2−

ai

h2

)

ui+1 = λui

Page 11: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Results for m(s) = sα

101

102

104

106

108

1010

1012

1014

1016

number of eigenvalue in logarithmic scale

eige

nval

ues

in lo

garit

hmic

sca

le

Logarithmic scaling of eigenvalue asymptotics

α=0α=1α=2α=3α=4exact solution for α=0

Figure: Computed eigenvalues of Sturm-Liouville problem −(au ′)′ = λu

for n = 500 and different values for α and exact eigenvalues for α = 0 in

logarithmic scales

λapproxn (A) = (α + 1)2π2n2 + O(n)

Page 12: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Results for m(s) = e− 1

sα

101

102

103

104

105

106

107

108

number of eigenvalue in logarithmic scale

eige

nval

ues

in lo

garit

hmic

sca

le

Logarithmic scaling of eigenvalue asymptotics

α=0α=0.2α=0.5α=1exact solution for α=0

Figure: Computed eigenvalues of Sturm-Liouville problem −(au ′)′ = λu

for n = 100 and different values for α and exact eigenvalues for α = 0 in

logarithmic scales

λapproxn (A) = f (α)n2 + O(n)

Page 13: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Galerkin method for integral equations

[B(x)](s) =

∫ 1

0k(s, t)x(t)dt, 0 ≤ s ≤ 1, 0 ≤ t ≤ 1

singular value expansion for square integrable kernels:

k(s, t) =

∞∑

j=1

σjuj(t)vj(s), 0 ≤ s ≤ 1, 0 ≤ t ≤ 1.

algebraic singular value decomposition of A ∈ Rn,n:

A = UΣV T =n

∑

j=1

sjujvTj ,

‖B‖2HS :=

∞∑

j=1

σ2j < ∞ ‖A‖2

F :=n

∑

i=1

n∑

j=1

a2ij =

n∑

j=1

s2j .

Page 14: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Galerkin method for integral equations

[B(x)](s) =

∫ 1

0k(s, t)x(t)dt, 0 ≤ s ≤ 1, 0 ≤ t ≤ 1

singular value expansion for square integrable kernels:

k(s, t) =

∞∑

j=1

σjuj(t)vj(s), 0 ≤ s ≤ 1, 0 ≤ t ≤ 1.

algebraic singular value decomposition of A ∈ Rn,n:

A = UΣV T =n

∑

j=1

sjujvTj ,

‖B‖2HS :=

∞∑

j=1

σ2j < ∞ ‖A‖2

F :=n

∑

i=1

n∑

j=1

a2ij =

n∑

j=1

s2j .

Page 15: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Algorithm for Galerkin’s method

Choose {Ψj} and {Φj} orthonormal sets of basis functions inIt = (0, 1), Is = (0, 1).

Determine matrix A ∈ Rn,n with

aij = 〈BΦj ,Ψi〉L2(0,1) i , j = 1, . . . , n.

Compute the SVD of this matrix

Av = s(n)u.

Page 16: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

aij = 〈BΦj ,Ψi〉L2(0,1) i , j = 1, . . . , n.

Av = s(n)u.

Page 17: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

aij = 〈BΦj ,Ψi〉L2(0,1) i , j = 1, . . . , n.

Av = s(n)u.

Page 18: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Matrix structure for m(s) = s

05

1015

2025

30

0

5

10

15

20

25

300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Page 19: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Matrix structure for m(s) = s2

05

1015

2025

30

0

5

10

15

20

25

300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Page 20: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Matrix structure for m(s) = s3

05

1015

2025

30

0

5

10

15

20

25

300

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Page 21: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Matrix structure for m(s) = e− 1

s

05

1015

2025

30

0

5

10

15

20

25

300

0.002

0.004

0.006

0.008

0.01

0.012

0.014

Page 22: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Matrix structure for m(s) = e− 1

s2

05

1015

2025

30

0

5

10

15

20

25

300

0.002

0.004

0.006

0.008

0.01

0.012

Page 23: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Approximation properties

Proposition Let

‖B‖2 :=

∫ 1

0

∫ 1

0|k(s, t)|2dtds =

∞∑

i=1

σ2i .

Thens(n)i ≤ s

(n+1)i ≤ σi , i = 1, . . . , n.

Furthermore, the errors of the approximate singular values s(n)i are

bounded by

0 ≤ σi − s(n)i ≤ δn, i = 1, . . . , n,

whereδ2n = ‖B‖2 − ‖A‖2

F .

Furthermore

s(n)i ≤ σi ≤ [(s

(n)i )2 + δ2

n]12 , i = 1, . . . , n.

Page 24: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Results for m(s) = sα

101

102

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

number of singular value in logarithmic scale

sing

ular

val

ues

in lo

garit

hmic

sca

le

Logarithmic scaling of singular value asymptotics

α=0α=1α=2α=3α=4α=5

Figure: Computed singular values of integral equation Bv = σu for n =

100 and different values for α in logarithmic scales

σapproxn (B) ∼

1

(α + 1)πnMelina Freitag Numerical approaches to singular value asymptotics

Page 25: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Discrete singular values for n → ∞

m(s) = s, n = 100

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

sing

ular

val

ues

in lo

garit

hmic

sca

le

Page 26: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

m(s) = s, n = 150

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

sing

ular

val

ues

in lo

garit

hmic

sca

le

Page 27: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

m(s) = s, n = 200

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

sing

ular

val

ues

in lo

garit

hmic

sca

le

Page 28: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

m(s) = s, n = 400

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

sing

ular

val

ues

in lo

garit

hmic

sca

le

Page 29: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

m(s) = s, n = 1000

100

101

102

10−6

10−5

10−4

10−3

10−2

10−1

100

sing

ular

val

ues

in lo

garit

hmic

sca

le

Page 30: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Results for m(s) = e− 1

sα

101

102

10−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

sing

ular

val

ues

in lo

garit

hmic

sca

le

Logarithmic scaling of singular value asymptotics

α=0α=0.2α=0.5α=1α=2

Figure: Computed singular values of integral equation Bv = σu for n =

100 and different values for α in logarithmic scales

σapproxn (B) ∼

1

g(α)nMelina Freitag Numerical approaches to singular value asymptotics

Page 31: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Further numerical approaches and summary

Rayleigh-Ritz method for symmetric kernel B ∗B

Orthonormal/Non-orthonormal basis functions

Generalized singular value problem/Generalized eigenproblem

Comparison yields

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

for the integral operator B = M ◦ J.

Combination of analytical and numerical results:

Cn−1 = σapproxn (B) ≤ σexact

n (B) ≤ Cn−1, n → ∞

Page 32: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Comparison yields

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

n (B) ≤ Cn−1, n → ∞

Page 33: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Comparison yields

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

n (B) ≤ Cn−1, n → ∞

Page 34: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Comparison yields

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

n (B) ≤ Cn−1, n → ∞

Page 35: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Comparison yields

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

n (B) ≤ Cn−1, n → ∞

Page 36: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Conclusions

numerical computation of the SVD of the discretized problemthrough various methods

error estimates and approximation properties

relationship

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

no influence of decay rate of m(s) → 0∫ 10 m(s)ds is important (generalization of the results by Vu

Kim Tuan and Gorenflo (1994)) for m(s) = s−α)

Page 37: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Conclusions

relationship

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

Page 38: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Conclusions

relationship

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

Page 39: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Conclusions

relationship

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

Page 40: Numerical approaches to singular value asymptotics for

OutlineMotivation

Conclusion

Conclusions

relationship

σn(B) =

∫ 1

0m(s)ds · σn(J) =

∫ 1

0m(s)ds ·

2

π(2n − 1),

numerical approaches to singular value asymptotics for

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