Adaptive Quadrature for Nystrm Yuchen Su 12/6/17 Quadrature - - PowerPoint PPT Presentation

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Adaptive Quadrature for Nystrm Yuchen Su 12/6/17 Quadrature - - PowerPoint PPT Presentation

Aug 18, 2023 125 likes •793 views

Adaptive Quadrature for Nystrm Yuchen Su 12/6/17 Quadrature Quadrature Quadrature Quadrature Quadrature Quadrature Composite Newton-Cotes Quadrature Composite Gaussian Error Estimation When should we be satisfied with our Quadrature?

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Adaptive Quadrature for Nyström

Yuchen Su 12/6/17

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Quadrature

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Quadrature

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Quadrature

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Quadrature

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Quadrature

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Quadrature

Composite Newton-Cotes

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Quadrature

Composite Gaussian

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Error Estimation

When should we be satisfied with our Quadrature?

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Error Estimation

When should we be satisfied with our Quadrature? Consider error:

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Automatic Quadrature

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Automatic Quadrature

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Automatic Quadrature

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Automatic Quadrature

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Automatic Quadrature

Algorithm:

  • 1. Set an error tolerance
  • 2. Compute
  • 3. If
  • Increase
  • Loop to Step 2
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Automatic Quadrature

This seems unintelligent

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Automatic Quadrature

This seems unintelligent Can we do better?

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Adaptive Quadrature

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Adaptive Quadrature

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Adaptive Quadrature

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Adaptive Quadrature

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Adaptive Quadrature

Algorithm:

  • 1. Set an error tolerance , usually 1
  • 2. Compute
  • 3. If
  • Let
  • Compute
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Amazing Quadrature

Can we make this jump directly?

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Error Estimation

Main approaches: Error ~ 1. 2. 3. 4.

Gonnet – A Review of Error Estimation in Adaptive Quadrature, ACM Computing Surveys (2012)

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Nyström Method

Consider the Fredholm equation of the second kind Where the integral operator is

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Nyström Method

Consider the Fredholm equation of the second kind Where the integral operator is If we approximate by some quadrature rule We can solve the system at the quadrature points

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Nyström Method

Once we have the density at the quadrature points Use Nyström interpolation to get density for all

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Nyström Method

Once we have the density at the quadrature points Use Nyström interpolation to get density for all For example if , we may interpolate by: where ,

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Nyström Error Estimate

Corollary 10.11, Kress – LIE (2nd ed.)

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Nyström Error Estimate

Corollary 10.11, Kress – LIE (2nd ed.)

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Example 1

Consider the following second kind integral equation Where We can compute the solution as

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Ex 1 Density Error: Gauss-Legendre

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Ex 1 Density Approx: Gauss-Legendre

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Ex 1 Density Error: Trapezoid Rule

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Ex 1 Density Approx: Trapezoid Rule

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Ex 1 Density Approx: Trapezoid Rule

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Ex 1 Density Approx: Trapezoid Rule

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Ex 1: Trapezoid Rule Differences

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Example 2

Consider the following second kind integral equation Where We can compute the solution as

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Ex 1 Density Error: Gauss-Legendre

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Ex 1: Gauss-Legendre Differences

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Ex 1 Density Error: Trapezoid Rule

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Ex 1: Trapezoid Rule Differences

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Nyström Local Estimates

Not so good =( Can we do better?

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Nyström Local Estimates

Consider the constant

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Nyström Local Estimates

Consider the constant Remove the norms (abuse notation)

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Nyström Local Estimates

Consider the constant Remove the norms (abuse notation) Let

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Nyström Local Estimates

Our equation becomes It turns out

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Nyström Local Estimates

Our equation becomes It turns out Shuffling around From collective compactness, we have

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Nyström Local Estimates

Stare at this real hard and consider

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Nyström Local Estimates

Stare at this real hard and consider 2 things happen

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Nyström Local Estimates

Stare at this real hard and consider 2 things happen

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Nyström Local Estimates

Stare at this real hard and consider 2 things happen

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Nyström Local Estimates

Maybe take a look at

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Nyström Local Estimates

Let’s assume we have infinite computational power and compute directly Look at

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Ex 1: Trapezoid, n = 11

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Ex 1: Recall Trapezoid Rule Error

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Just to check it works

n=26 n=51

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How can we estimate this?

This is expensive! Maybe ?

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Ex 1: Estimator

n = 11 vs n = 21

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Ex 1: Estimator Comparison

Real Estimator

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Ex 2: Trapezoid, n = 11

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Ex 1: Trapezoid Rule Differences

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Ex 2: Estimator

n = 11 vs n = 21

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Ex 2: Estimator comparison

Real Estimator

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Some Thoughts

(done verbally) =)