Lab 2 Iterative methods and eigenvalue problems. Introduction. Iterative solution of the soap film problem. Beräkningsvetenskap II/NV2, HT (6)

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1 Beräkningsvetenskap II/NV2, HT (6) Institutionen för informationsteknologi Teknisk databehandling Besöksadress: MIC hus 2, Polacksbacken Lägerhyddsvägen 2 Postadress: Box Uppsala Telefon: (växel) Telefax: Hemsida: Department of Information Technology Scientific Computing Visiting address: MIC bldg 2, Polacksbacken Lägerhyddsvägen 2 Postal address: Box 337 SE Uppsala SWEDEN Telephone: (switch) Telefax: Web page: Lab 2 Iterative methods and eigenvalue problems Ideally, you should work together in groups, 2 3 students/group. At the end of the lab, submit your reflections and/or questions. Start by copying the files from bervet2/ht08/lab2 Introduction In this lab we study the dynamics of a trampoline and show how the linear system that arises in the soap film application can be solved efficiently using iterative methods. Iterative solution of the soap film problem In Lab 1, we solved the soap film problem using FEM. Here, we let the computational domain be Ω = [0 1] [0 1] (the unit square), and solve the problem using finite differences. The PDE is given by Ù(Ü Ý) = 0 (Ü Ý) ¾ Ω (1) Ù(Ü Ý) = Ö(Ü Ý) (Ü Ý) ¾ Ω Let Ü = 1 and Æ Ü+1 Ý = 1 Æ Ý+1. Then the discrete problem is Ù+1 2Ù +Ù 1 2 Ü + Ù +1 2Ù +Ù 1 2 Ý Ù = Ö(Ü Ý ) = 0 1 Æ Ü 1 Æ Ý Ü = 0 1 or Ý = 0 1 By ordering the unknown values Ù for example row by row, we can write down a linear system of equations Ù =. So far, we have solved all linear systems by computing the ÄÍ-factorization of and then computing Ù through forward and backward substitution. Now we will explore the effects of this in terms of memory requirements.

2 2 (6) Compute the matrix through A=soapfilm(Nx,Ny). Suitable values for Æ Ü and Æ Ý at this point are around Run the code mylu to study the sparsity patterns of, Ä, and Í. (In the figures, nz is the number of nonzero elements.) How many nonzero diagonals does have? How many nonzero diagonals do Ä and Í have together? Try to come up with a formula for the number of nonzero diagonals in Ä and Í involving Æ Ü and/or Æ Ý. Now we are going to solve the soap film problem with the boundary function given by Ö(0 Ý) = Ö(1 Ý) = cos(2ý) + 05 and Ö(Ü 0) = Ö(Ü 1) = cos(2ü) + 05 Because of the boundary function being non-smooth we will need a fine grid in order to get an accurate solution. Solve the problem using Æ Ü = Æ Ý = 100 through [A,b,x,y,u]=soapfilm(100,100); What is the size of the matrix and how many nonzero diagonals would Ä and Í have in this case? In many cases it is better to solve sparse linear systems using iterative methods. Typically, an iterative method uses less memory than ÄÍ-factorization, and it is often faster. The simplest of the iterative methods is Jacobi s method. Let be a diagonal matrix with the same values as on the diagonal. Ù = Ù + ( )Ù = Ù = 1 ( )Ù + 1

3 3 (6) From this, we can formulate the iterative method in the following way: Choose an arbitrary initial guess Ù 0 for the solution. Let Å = 1 ( ) and = 1. For = 0 1 repeat Ù+1 = ÅÙ + = Ù +1 Ù until tol A more sophisticated iterative method is GMRES, in which the residual Ö = Ù is minimized in spanö 0 Ö 0 Ö 0 in each iteration. Look at the algorithm for Jacobi s method. Why does the iterative method use less memory than ÄÍ-factorization? Solve the soap film problem with Æ Ü = Æ Ý = 100 using Jacobi s method ([u,it]=jacobi(a,b,tol)) and GMRES ([u,it]=gmres(a,b,tol)). Vary the tolerance and write down the numbers of iterations. Suitable tolerance values can be for example between 0.1 and Plot the number of iterations against the log of the tolerance (semilogx(tol,it)) and compare the results. The mathematics of a trampoline We are going to study eigenmodes of a trampoline. Any displacement state of a trampoline at a given time is a linear combination of the eigenmodes. We state the trampoline equation, Û ØØ (Ü Ý Ø) + 2 Û(Ü Ý Ø) = 0 (Ü Ý) ¾ Ω Ø ¾ [0 Ì ] Ù(Ü Ý Ø) = 0 (Ü Ý) ¾ Ω Ø ¾ [0 Ì ] Ù(Ü Ý Ø) = 0 (Ü Ý) ¾ Ω Ø ¾ [0 Ì ] Ù(Ü Ý 0) = 0 (Ü Ý) ¾ Ω Ù Ø (Ü Ý 0) = Ú 0 (Ü Ý) (Ü Ý) ¾ Ω where is the density and is the flexural rigidity of the trampoline. Further Ù(Ü Ý Ø) is the vertical displacement as a function of space and time, Ù(Ü Ý Ø) = 0 at the boundary Ω means that the vertical displacement is zero, i.e. the trampoline is attached to the frame. The other boundary condition Ù(Ü Ý Ø) = 0 models hinges.

4 4 (6) If we can find the eigenmodes of the differential operator 2 we can easily study the dynamics of the trampoline. We let Û denote eigenmode with corresponding eigenvalue i.e., 2 Û (Ü Ý) = Û (Ü Ý) (2) Any function can be written as a linear combination of eigenmodes (Hilbert- Schmidt Theorem). Here we get, Û(Ü Ý Ø) = ½ =1 sin ¼ ½ Ø Û (Ü Ý) (3) where the coefficients are determined from the second initial condition. It is clear that it is crucial to compute the eigenfunctions Û (Ü Ý) and the eigenvalues. If we let Ú and be the eigenvaules and eigenfunctions of the Laplace operator, Ú = Ú (4) we note that, 2 Ú = ( Ú ) = ( Ú ) = 2 Ú, i.e. Û = Ú and = 2. Now we will study a code that computes the eigenfunctions and eigenvalues to the discrete Laplace operator. Again we use finite differences for the discretization. Given these eigenfunctions and eigenvalues we automatically get the eigenfunctions and eigenvalues to the trampoline as Û = Ú and = 2. Data for the trampoline We now specify data for a trampoline, see Figure 1. We let Ω = [0 426] [0 213] (Ñ), = 940 (Ñ 3 ), and = À 3 (12(1 2 )), where À = 001 (half thickness in meters), = (Young s modulus in Pascal), = 05 (ideal elasticity). Note that the matrix we study here is just a scaled version of the one used in the iterative part of the Lab. The scaling is due to different sizes of the computational domain. Given this data it remains to compute the eigenfunctions and eigenvalues to equation (4).

5 5 (6) Figur 1: See Run the code trampoline.m with correct dimensions of the domain Ω. Study different eigenmodes of the trampoline. How does the frequency depend on the size of the eigenvalue? Can we see this in the solution (3)? If two kids are jumping at the same time, where should they stand and how should they jump in order to avoid canceling out each others jumps and thereby decrease the amplitude? The m-file trampoline.m uses an eigenvalue solver called eigs, which is a built in function in Matlab. Now you are going to write your own iterative eigenvalue solver using the power method. The power method and inverse iteration The power method is the simplest algorithm for computing the largest eigenvalue of a matrix. Assume that the Ò Ò matrix has the eigenvalues, = 1 Ò and that Ò. Also assume that the eigenvectors Ü, = 1 Ò are linearly independent. This means that any vector Þ can be written as Þ = È Ò =1 Ü, and Þ = È Ò =1 Ü = È Ò =1 Ü. Repeated multiplication with leads to Þ = Ò =1 Ü = 1 ¼ 1 Ü Ü Ò Ò 1 Ü Ò ½

6 6 (6) This means that Þ will approach the eigenvector associated with the largest eigenvalue 1 as ½. In practise we also need to normalize in each iteration to avoid very large or small entries in the eigenvector (remember an eigenvector is only defined up to a constant). Since we in this application are interested in the smallest eigenvalue rather than the largest, we will apply the power method on the inverse of. Remember that if we assume that is non-singular. Then Ü = Ü Ü = 1 Ü 1 Ü = 1 Ü That is, the eigenvectors stay the same, but the eigenvalues are inverted. Hence, the eigenvector corresponding to the largest eigenvalue of 1 is same as the one corresponding to the smallest eigenvalue of. Applying the power method to the inverse in order to compute the smallest eigenvalue is called inverse iteration. Choose an arbitrary initial guess Þ 0 for the eigenvector. For = 0 1 repeat Þ Ý = Þ 2 Solve Þ +1 = Ý 1 = Ý À Þ +1 until 1 tol The vectors Ý converge to Ü Ò and is an approximation of Ò. Note that a linear system of equations with the matrix must be solved in each iteration. Write a function [lambda,x]=myinvit(a) that applies inverse iteration to the matrix (that you get as output data from trampoline.m) and returns the smallest eigenvalue Ò and the corresponding eigenvector Ü Ò. Compare the results with trampoline.m. You can also check your results against the code invit.m.