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Review

Review. Paul Heckbert Computer Science Department Carnegie Mellon University. y’=y. y’=-y, stable but slow solution. y’=-y, unstable solution. Jacobian of ODE. ODE: y’=f(t,y) , where y is n -dimensional Jacobian of f is a square matrix

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Review

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  1. Review Paul Heckbert Computer Science Department Carnegie Mellon University 15-859B - Introduction to Scientific Computing

  2. y’=y 15-859B - Introduction to Scientific Computing

  3. y’=-y, stable but slow solution 15-859B - Introduction to Scientific Computing

  4. y’=-y, unstable solution 15-859B - Introduction to Scientific Computing

  5. Jacobian of ODE • ODE: y’=f(t,y), where y is n-dimensional • Jacobian of f is a square matrix • if ODE homogeneous and linear then J is constant and y’=Jy • but in general J varies with t and y 15-859B - Introduction to Scientific Computing

  6. Stability of ODE depends on Jacobian At a given (t,y) find J(t,y) and its eigenvalues find rmax = maxi { Re[i(J)] } if rmax<0, ODE stable, locally rmax =0, ODE neutrally stable, locally rmax >0, ODE unstable, locally 15-859B - Introduction to Scientific Computing

  7. Stability of Numerical Solution • Stability of numerical solution is related to, but not the same as stability of ODE! • Amplification factor of a numerical solution is the factor by which global error grows or shrinks each iteration. 15-859B - Introduction to Scientific Computing

  8. Stability of Euler’s Method • Amplification factor of Euler’s method is I+hJ • Note that it depends on h and, in general, on t & y. • Stability of Euler’s method is determined by eigenvalues of I+hJ • spectral radius (I+hJ)= maxi | i(I+hJ) | • if (I+hJ)<1 then Euler’s method stable • if all eigenvalues of hJ lie inside unit circle centered at –1, E.M. is stable • scalar case: 0<|hJ|<2 iff stable, so choose h < 2/|J| • What if one eigenvalue of J is much larger than the others? 15-859B - Introduction to Scientific Computing

  9. Implicit Methods • use information from future time tk+1 to take a step from tk • Euler method: yk+1 = yk+f(tk,yk)hk • backward Euler method: yk+1 = yk+f(tk+1,yk+1)hk • example: • y’=Ay f(t,y)=Ay • yk+1 = yk+Ayk+1hk • (I-hkA)yk+1=yk -- solve this system each iteration 15-859B - Introduction to Scientific Computing

  10. Stability of Backward Euler’s Method • Amplification factor of B.E.M. is (I-hJ)-1 • B.E.M. is stable independent of h (unconditionally stable) as long as rmax<0, i.e. as long as ODE is stable • Implicit methods such as this permit bigger steps to be taken (larger h) 15-859B - Introduction to Scientific Computing

  11. Large steps OK with Backward Euler’s method 15-859B - Introduction to Scientific Computing

  12. Popular IVP Solution Methods • Euler’s method, 1st order • backward Euler’s method, 1st order • trapezoid method (a.k.a. 2nd order Adams-Moulton) • 4th order Runge-Kutta • If a method is pth order accurate then its global error is O(hp) 15-859B - Introduction to Scientific Computing

  13. Solving Boundary Value Problems • Shooting Method • transform BVP into IVP and root-finding • Finite Difference Method • transform BVP into system of equations (linear?) • Finite Element Method • choice of basis: linear, quadratic, ... • collocation method • residual is zero at selected points • Galerkin method • residual function is orthogonal to each basis function 15-859B - Introduction to Scientific Computing

  14. 2nd Order PDE Classification • We classify conic curve ax2+bxy+cy2+dx+ey+f=0 as ellipse/parabola/hyperbola according to sign of discriminant b2-4ac. • Similarly we classify 2nd order PDE auxx+buxy+cuyy+dux+euy+fu+g=0: b2-4ac < 0 – elliptic (e.g. equilibrium) b2-4ac = 0 – parabolic (e.g. diffusion) b2-4ac > 0 – hyperbolic (e.g. wave motion) For general PDE’s, class can change from point to point 15-859B - Introduction to Scientific Computing

  15. Example: Wave Equation • utt = c uxx for 0x1, t0 • initial cond.: u(0,x)=sin(x)+x+2, ut(0,x)=4sin(2x) • boundary cond.: u(t,0) = 2, u(t,1)=3 • c=1 • unknown: u(t,x) • simulated using Euler’s method in t • discretize unknown function: 15-859B - Introduction to Scientific Computing

  16. k+1 k k-1 j+1 j j-1 Wave Equation: Numerical Solution u0 = ... u1 = ... for t = 2*dt:dt:endt u2(2:n) = 2*u1(2:n)-u0(2:n) +c*(dt/dx)^2*(u1(3:n+1)-2*u1(2:n)+u1(1:n-1)); u0 = u1; u1 = u2; end 15-859B - Introduction to Scientific Computing

  17. Wave Equation Results 15-859B - Introduction to Scientific Computing

  18. Poor results when dt too big dx=.05 dt=.06 Euler’s method unstable when step too large 15-859B - Introduction to Scientific Computing

  19. PDE Solution Methods • Discretize in space, transform into system of IVP’s • Discretize in space and time, finite difference method. • Discretize in space and time, finite element method. • Latter methods yield sparse systems. • Sometimes the geometry and boundary conditions are simple (e.g. rectangular grid); • Sometimes they’re not (need mesh of triangles). 15-859B - Introduction to Scientific Computing

  20. PDE’s and Sparse Systems • A system of equations is sparse when there are few nonzero coefficients, e.g. O(n) nonzeros in an nxn matrix. • Partial Differential Equations generally yield sparse systems of equations. • Integral Equations generally yield dense (non-sparse) systems of equations. • Sparse systems come from other sources besides PDE’s. 15-859B - Introduction to Scientific Computing

  21. Example: PDE Yielding Sparse System • Laplace’s Equation in 2-D: 2u = uxx +uyy = 0 • domain is unit square [0,1]2 • value of function u(x,y) specified on boundary • solve for u(x,y) in interior 15-859B - Introduction to Scientific Computing

  22. Sparse Matrix Storage • Brute force: store nxn array, O(n2) memory • but most of that is zeros – wasted space (and time)! • Better: use data structure that stores only the nonzeros col 1 2 3 4 5 6 7 8 9 10... val 0 1 0 0 1 -4 1 0 0 1... 16 bit integer indices: 2, 5, 6, 7,10 32 bit floats: 1, 1,-4, 1, 1 • Memory requirements, if kn nonzeros: • brute force: 4n2 bytes, sparse data struc: 6kn bytes 15-859B - Introduction to Scientific Computing

  23. An Iterative Method: Gauss-Seidel • System of equations Ax=b • Solve ith equation for xi: • Pseudocode: until x stops changing for i = 1 to n x[i]  (b[i]-sum{ji}{a[i,j]*x[j]})/a[i,i] • modified x values are fed back in immediately • converges if A is symmetric positive definite 15-859B - Introduction to Scientific Computing

  24. Conjugate Gradient Method • Generally for symmetric positive definite, only. • Convert linear system Ax=b • into optimization problem: minimize xTAx-xTb • a parabolic bowl • Like gradient descent • but search in conjugate directions • not in gradient direction, to avoid zigzag problem • Big help when bowl is elongated (cond(A) large) 15-859B - Introduction to Scientific Computing

  25. Convergence ofConjugate Gradient Method • If K = cond(A) = max(A)/ min(A) • then conjugate gradient method converges linearly with coefficient (sqrt(K)-1)/(sqrt(K)+1) worst case. • often does much better: without roundoff error, if A has m distinct eigenvalues, converges in m iterations! 15-859B - Introduction to Scientific Computing

  26. Example: Poisson’s Equation • Sweep of Gauss-Seidel “relaxes” each grid value to be the average of its four neighbors plus an f offset • Many relaxations required to solve this on a fixed grid. • Multigrid solves it on a hierarchy of grids. 15-859B - Introduction to Scientific Computing

  27. Elements of Multigrid Method • relax on a given grid a few times • coarsen (restrict) a grid • refine (interpolate) a grid 15-859B - Introduction to Scientific Computing

  28. final solution finest grid kk k/2k/2 ... coarsest grid 11 time =relax =interpolate =restrict A Common Multigrid Schedule Full Multigrid V Cycle: 15-859B - Introduction to Scientific Computing

  29. Some Iterative Methods • Gauss-Seidel • converges for all symmetric positive definite A • Conjugate Gradient (CG) Method • convergence rate determined by condition number • note that condition number typically larger for finer grids • Preconditioned Conjugate Gradient • instead of solving Av=f, solve M-1Av=M-1f where M-1 is cheap and M is close to A • often much faster than CG, but conditioner M is problem-dependent • Multigrid • convergence rate is independent of condition number, problem size • but algorithm must be tuned for a given problem; not as general as others note: don’t need matrix A in memory – can compute it on the fly! 15-859B - Introduction to Scientific Computing

  30. Cost Comparison on 2-D Poisson Equation, kk grid, n=k2 unknowns METHOD COST Gaussian Elimination O(k6) = O(n3) Gauss-Seidel O(k4logk) = O(n2logn) Conjugate Gradient O(k3) = O(n1.5) FFT/cyclic reduction O(k2logk) = O(nlogn) multigrid O(k2) = O(n) optimal! 15-859B - Introduction to Scientific Computing

  31. Function Representations • sequence of samples (time domain) • finite difference method • pyramid (hierarchical) • polynomial • sinusoids of various frequency (frequency domain) • Fourier series • piecewise polynomials (finite support) • finite element method, splines • wavelet (hierarchical, finite support) • (time/frequency domain) 15-859B - Introduction to Scientific Computing

  32. What Are Wavelets? In general, a family of representations using: • hierarchical (nested) basis functions • finite (“compact”) support • basis functions often orthogonal • fast transforms, often linear-time 15-859B - Introduction to Scientific Computing

  33. Nested Function Spaces for Haar Basis • Let Vj denote the space of all piecewise-constant functions represented over 2j equal sub-intervals of [0,1] • Vj has basis 15-859B - Introduction to Scientific Computing

  34. Function Representations –Desirable Properties • generality – approximate anything well • discontinuities, nonperiodicity, ... • adaptable to application • audio, pictures, flow field, terrain data, ... • compact – approximate function with few coefficients • facilitates compression, storage, transmission • fast to compute with • differential/integral operators are sparse in this basis • Convert n-sample function to representation in O(nlogn) or O(n) time 15-859B - Introduction to Scientific Computing

  35. Time-Frequency Analysis • For many applications, you want to analyze a function in both time and frequency • Analogous to a musical score • Fourier transforms give you frequency information, smearing time. • Samples of a function give you temporal information, smearing frequency. • Note: substitute “space” for “time” for pictures. 15-859B - Introduction to Scientific Computing

  36. Comparison to Fourier Analysis • Fourier analysis • Basis is global • Sinusoids with frequencies in arithmetic progression • Short-time Fourier Transform (& Gabor filters) • Basis is local • Sinusoid times Gaussian • Fixed-width Gaussian “window” • Wavelet • Basis is local • Frequencies in geometric progression • Basis has constant shape independent of scale 15-859B - Introduction to Scientific Computing

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