Brownian Motion and Harmonic Functions

Transcription

1 Harini Chandramouli Kiya Holmes Brandon Reeves Nora Stack Michigan State University

2 Laplace s Equation Points interior to the region describe population densities, temperatures, chemical concentrations, etc.

3 Laplace s Equation Points interior to the region describe population densities, temperatures, chemical concentrations, etc. The boundary values are given by u 0 (x,y)

4 Laplace s Equation Points interior to the region describe population densities, temperatures, chemical concentrations, etc. The boundary values are given by u 0 (x,y) After a sufficient amount of time, the temperature at points interior are time-independent

5 Laplace s Equation Points interior to the region describe population densities, temperatures, chemical concentrations, etc. The boundary values are given by u 0 (x,y) After a sufficient amount of time, the temperature at points interior are time-independent u xx + u yy = 0 (Laplace s Equation)

6 Laplace s Equation Points interior to the region describe population densities, temperatures, chemical concentrations, etc. The boundary values are given by u 0 (x,y) After a sufficient amount of time, the temperature at points interior are time-independent u xx + u yy = 0 (Laplace s Equation) In nice regions, the solution is well-known

7 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 )

8 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 )

9 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 )

10 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 )

11 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 )

12 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 )

13 An Alternative Method Kakutani (1944) showed that the solution, u(x 0, y 0 ), to Laplace s equation can be approximated by considering Brownian motion from the point (x 0, y 0 ) We can use random walks to simulate Brownian motion Specifically, the random walks on circles (RWoC) and spheres (RWoS) We simulated Brownian motion in various regions and studied the probability density functions (PDFs) of the point of first encounter in these regions

14 Walk on Circles Pick point (x 0, y 0 ) in the region

15 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center

16 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center Randomly pick a point (x 1, y 1 ) on the circle

17 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center Randomly pick a point (x 1, y 1 ) on the circle (x 1, y 1 ) is either on the boundary or is somewhere else in the region

18 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center Randomly pick a point (x 1, y 1 ) on the circle (x 1, y 1 ) is either on the boundary or is somewhere else in the region If (x 1, y 1 ) if on the boundary then walk on circles ends

19 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center Randomly pick a point (x 1, y 1 ) on the circle (x 1, y 1 ) is either on the boundary or is somewhere else in the region If (x 1, y 1 ) if on the boundary then walk on circles ends If not, create circle with (x 1, y 1 ) as center

20 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center Randomly pick a point (x 1, y 1 ) on the circle (x 1, y 1 ) is either on the boundary or is somewhere else in the region If (x 1, y 1 ) if on the boundary then walk on circles ends If not, create circle with (x 1, y 1 ) as center Randomly pick a point (x 2, y 2 ) on the circle

21 Walk on Circles Pick point (x 0, y 0 ) in the region Create circle with (x 0, y 0 ) as the center Randomly pick a point (x 1, y 1 ) on the circle (x 1, y 1 ) is either on the boundary or is somewhere else in the region If (x 1, y 1 ) if on the boundary then walk on circles ends If not, create circle with (x 1, y 1 ) as center Randomly pick a point (x 2, y 2 ) on the circle Continue until process until you hit boundary of area

22 Walk on Circles in Regions Made programs to simulate walk on circles for:

23 Walk on Circles in Regions Made programs to simulate walk on circles for: Line

24 Walk on Circles in Regions Made programs to simulate walk on circles for: Line Circle (Analytic solution known) Upper Half-Plane (Analytic solution is known)

25 Walk on Circles in Regions Made programs to simulate walk on circles for: Line Circle (Analytic solution known) Upper Half-Plane (Analytic solution is known) Parabola Quarter-Plane

26 Walk on Circles in Regions Made programs to simulate walk on circles for: Line Circle (Analytic solution known) Upper Half-Plane (Analytic solution is known) Parabola Quarter-Plane Square Triangle

27 Walk on Circles in Regions Made programs to simulate walk on circles for: Line Circle (Analytic solution known) Upper Half-Plane (Analytic solution is known) Parabola Quarter-Plane Square Triangle Upper Half-Space Sphere

28 Simulation: Half Plane Beginning at (x 0, y 0 ), with y 0 > 0 we simulate Brownian motion on the upper half plane

29 Simulation: Half Plane Beginning at (x 0, y 0 ), with y 0 > 0 we simulate Brownian motion on the upper half plane

30 Simulation: Half Plane Beginning at (x 0, y 0 ), with y 0 > 0 we simulate Brownian motion on the upper half plane

31 Simulation: Half Plane How did our simulation perform?

32 More General Regions Solution on the half-plane is known:

33 More General Regions Solution on the half-plane is known: u(x 0, y 0 ) = = f y0 (x 0 τ)u 0 (τ)dτ 1 π y 0 (x 0 τ) 2 + y 0 2 u 0(τ)dτ

34 More General Regions Solution on the half-plane is known: u(x 0, y 0 ) = = f y0 (x 0 τ)u 0 (τ)dτ 1 π y 0 (x 0 τ) 2 + y 0 2 u 0(τ)dτ Where f y0 (x 0 τ) = 1 π y 0 (x 0 τ) 2 +y 0 2

35 More General Regions Solution on the half-plane is known: u(x 0, y 0 ) = = f y0 (x 0 τ)u 0 (τ)dτ 1 π y 0 (x 0 τ) 2 + y 0 2 u 0(τ)dτ y 0 (x 0 τ) 2 +y 0 2 Where f y0 (x 0 τ) = 1 π Hence, our PDF is: f (x) = 1 y 0 π (x 0 x) 2 +y 2 0 What about more general regions in the plane?

36 More General Regions Solution on the half-plane is known: u(x 0, y 0 ) = = f y0 (x 0 τ)u 0 (τ)dτ 1 π y 0 (x 0 τ) 2 + y 0 2 u 0(τ)dτ Where f y0 (x 0 τ) = 1 y 0 π (x 0 τ) 2 +y 2 0 Hence, our PDF is: f (x) = 1 y 0 π (x 0 x) 2 +y 2 0 What about more general regions in the plane? Conformal Mappings

37 More General Regions Solution on the half-plane is known: u(x 0, y 0 ) = = f y0 (x 0 τ)u 0 (τ)dτ 1 π y 0 (x 0 τ) 2 + y 0 2 u 0(τ)dτ Where f y0 (x 0 τ) = 1 y 0 π (x 0 τ) 2 +y 2 0 Hence, our PDF is: f (x) = 1 y 0 π (x 0 x) 2 +y 2 0 What about more general regions in the plane? Conformal Mappings Map one region bijectively into another region

38 More General Regions Solution on the half-plane is known: u(x 0, y 0 ) = = f y0 (x 0 τ)u 0 (τ)dτ 1 π y 0 (x 0 τ) 2 + y 0 2 u 0(τ)dτ Where f y0 (x 0 τ) = 1 y 0 π (x 0 τ) 2 +y 2 0 Hence, our PDF is: f (x) = 1 y 0 π (x 0 x) 2 +y 2 0 What about more general regions in the plane? Conformal Mappings Map one region bijectively into another region Riemann Mapping Theorem

39 Using Conformal Mappings PDF x-axis 1 4x 0 y 0 τ π (x 02 y 02 τ 2 ) 2 + (2x 0 y 0 ) 2 y-axis 1 4x 0 y 0 τ π (x 02 y 02 + τ 2 ) 2 + (2x 0 y 0 ) 2 PDF x-axis y 0 f (x) = 1 π (x 0 x) 2 + y 2 0

40 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points?

41 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive

42 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive Our process requires computing many boundary values

43 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive Our process requires computing many boundary values Can we limit the number of times we compute the boundary values while still maintaining accurate approximations?

44 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive Our process requires computing many boundary values Can we limit the number of times we compute the boundary values while still maintaining accurate approximations?

45 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive Our process requires computing many boundary values Can we limit the number of times we compute the boundary values while still maintaining accurate approximations?

46 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive Our process requires computing many boundary values Can we limit the number of times we compute the boundary values while still maintaining accurate approximations?

47 Real World Applications What if we have no expression for our boundary values, but we can compute these values instead at particular points? These points might be expensive Our process requires computing many boundary values Can we limit the number of times we compute the boundary values while still maintaining accurate approximations?

48 Getting the Sum to Equal the Integral Recall: u(x, y) = D(τ)u 0 (τ)dτ where D is the probability density function

49 Getting the Sum to Equal the Integral Recall: u(x, y) = D(τ)u 0 (τ)dτ where D is the probability density function Assumption: u 0 is a polynomial

50 Getting the Sum to Equal the Integral Recall: u(x, y) = D(τ)u 0 (τ)dτ where D is the probability density function Assumption: u 0 is a polynomial Then we can find some numbers D i such that D(τ)u 0 (τ)dτ = 10 i=1 D i u 0 (x i ) where u 0 is up to a 9th degree polynomial

51 Getting the Sum to Equal the Integral Recall: u(x, y) = D(τ)u 0 (τ)dτ where D is the probability density function Assumption: u 0 is a polynomial Then we can find some numbers D i such that D(τ)u 0 (τ)dτ = 10 i=1 D i u 0 (x i ) where u 0 is up to a 9th degree polynomial But we can do better, u 0 can be up to a 2(10) 1 degree polynomial

52 Real World Applications Given D(x) (probability density function) we can construct an inner product on polynomials as (p, q) = p(x)q(x)d(x)dx

53 Real World Applications Given D(x) (probability density function) we can construct an inner product on polynomials as (p, q) = p(x)q(x)d(x)dx Given this inner product, we can find (by Gram-Schmidt Process) the nth degree polynomial, p n (x), orthogonal to all polynomials of lesser degree

54 Real World Applications Given D(x) (probability density function) we can construct an inner product on polynomials as (p, q) = p(x)q(x)d(x)dx Given this inner product, we can find (by Gram-Schmidt Process) the nth degree polynomial, p n (x), orthogonal to all polynomials of lesser degree Pick x i as the roots of p n (x)

55 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1?

56 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1? Let u 0 (τ) be our boundary condition, and deg(u 0 (τ)) = 19 u 0 (τ) = α(τ)p 10 (τ) + r(τ) where deg(r) < deg(p 10 ) = 10 Let D(τ) represent the distribution function u 0 (τ)d(τ)dτ

57 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1? Let u 0 (τ) be our boundary condition, and deg(u 0 (τ)) = 19 u 0 (τ) = α(τ)p 10 (τ) + r(τ) where deg(r) < deg(p 10 ) = 10 Let D(τ) represent the distribution function u 0 (τ)d(τ)dτ = α(τ)p 10 (τ)d(τ)dτ + r(τ)d(τ)dτ

58 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1? Let u 0 (τ) be our boundary condition, and deg(u 0 (τ)) = 19 u 0 (τ) = α(τ)p 10 (τ) + r(τ) where deg(r) < deg(p 10 ) = 10 Let D(τ) represent the distribution function u 0 (τ)d(τ)dτ = α(τ)p 10 (τ)d(τ)dτ + r(τ)d(τ)dτ = r(τ)d(τ)dτ

59 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1? Let u 0 (τ) be our boundary condition, and deg(u 0 (τ)) = 19 u 0 (τ) = α(τ)p 10 (τ) + r(τ) where deg(r) < deg(p 10 ) = 10 Let D(τ) represent the distribution function u 0 (τ)d(τ)dτ = α(τ)p 10 (τ)d(τ)dτ + r(τ)d(τ)dτ = r(τ)d(τ)dτ 10 = r(x i )D i i=1

60 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1? Let u 0 (τ) be our boundary condition, and deg(u 0 (τ)) = 19 u 0 (τ) = α(τ)p 10 (τ) + r(τ) where deg(r) < deg(p 10 ) = 10 Let D(τ) represent the distribution function u 0 (τ)d(τ)dτ = α(τ)p 10 (τ)d(τ)dτ + r(τ)d(τ)dτ = r(τ)d(τ)dτ 10 = r(x i )D i = i= α(x i )p 10 (x i )D i + r(x i )D i i=1 i=1

61 Checking the Efficiency How do we obtain exact answers up to degree 2(10)-1? Let u 0 (τ) be our boundary condition, and deg(u 0 (τ)) = 19 u 0 (τ) = α(τ)p 10 (τ) + r(τ) where deg(r) < deg(p 10 ) = 10 Let D(τ) represent the distribution function u 0 (τ)d(τ)dτ = α(τ)p 10 (τ)d(τ)dτ + r(τ)d(τ)dτ = r(τ)d(τ)dτ 10 = r(x i )D i = = i= α(x i )p 10 (x i )D i + r(x i )D i i=1 10 u 0 (x i )D i i=1 i=1

62 What Have We Done? So if u 0 is a nice (smooth) function, then u(x, y) = 10 i=1 This will be a good approximation D(τ)u 0 (τ)dτ u 0 (x i )D i

63 Summary Brownian Motion and Laplace s Equation Walk on Circles and Spheres Simulating Walk on Circles and Spheres in different regions Probability Density Functions and Conformal Mapping Techniques Less Expensive Real World Applications

64 Acknowledgements NSA Grant: H Dr. Igor Nazarov Dr. Nick Boros

65 Thanks for Listening! Questions?