Applied/Numerical Analysis Qualifying Exam

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1 Applied/Numerical Analysis Qualifying Exam Cover Sheet Applied Analysis Part Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem so that it becomes trivial. Name

2 Combined Applied Analysis/Numerical Analysis Qualifier Applied Analysis Part Instructions: Do any 3 of the 4 problems in this part of the exam. Show all of your work clearly. Please indicate which of the 4 problems you are skipping. Problem 1. Recall that the DFT and inverse DFT are given by ŷ k = n 1 j=0 y j w jk and y j = 1 n 1 n j=0 ŷkw jk, where w = e 2πi/n. (a) State and prove the Convolution Theorem for the DFT. (b) Let a, x, y be column vectors with entries a 0,..., a n 1, x 0,..., x n 1, y 0,..., y n 1. In addition, let α, ξ and η be n-periodic sequences, the entries for one period, k = 0,..., n 1, being those of a, x, and y, respectively. Consider the circulant matrix a 0 a n 1 a n 2 a 1 a 1 a 0 a n 1 a 2 A = a 2 a 1 a 0 a a n 1 a n 2 a n 3 a 0 Show that the matrix equation Ax = y is equivalent to convolution η = α ξ. (c) Use parts (a) and (b) above to find the eigenvalues of A = Problem 2. Let Lu = (e x u ), u(0) = 0, u (1) = 0. (a) Find the Green s function G(x, y) for Lu = (e x u ) = f, u(0) = 0, u (1) = 0. (b) Why is Kf(x) = 1 G(x, y)f(y)dy compact? (One sentence will do.) 0 (c) Consider the eigenvalue problem Lu = λu, u(0) = 0, u (1) = 0. Show that the (orthonormal) set of eigenfunctions for L form a complete set in L 2 [0, 1]. Problem 3. Let H be a (separable) Hilbert space and let C(H) be the set of compact operators on H. (a) State and prove the Closed Range Theorem. (b) Let H = L 2 [0, 1]. Define the kernel k(x, y) := x 2 y 9 and let Ku(x) = 1 k(x, y)u(y)dy. 0 Show the K is in C(L 2 [0, 1]). (c) Let L = I λk, λ C, with K as defined in part (b) above. Find all λ for which Lu = f can be solved for all f L 2 [0, 1]. For these values of λ, find the resolvent (I λk) 1. 1

3 Problem 4. Recall that a geodesic on a surface provides the path of shortest distance between two points on a surface. Let S be the unit sphere in R 3. In the coordinates shown above, the differential arc length is given by ds = dθ 2 + sin 2 (θ)dϕ 2. If P 0 = (θ 0, 0) and P 1 = (θ 1, 0), 0 < θ 0 < θ 1 < π, show that the geodesic is the arc of the great circle given by θ 0 θ θ 1, ϕ = 0. Hint: describe curves joining the two points by ϕ = u(θ), where u C 2 [θ 0, θ 1 ] and satisfies u(θ 0 ) = u(θ 1 ) = 0. Minimize the arc-length functional. 2

4 Applied/Numerical Analysis Qualifying Exam Cover Sheet Numerical Analysis Part Policy on misprints: The qualifying exam committee tries to proofread exams as carefully as possible. Nevertheless, the exam may contain a few misprints. If you are convinced a problem has been stated incorrectly, indicate your interpretation in writing your answer. In such cases, do not interpret the problem so that it becomes trivial. Name

5 NUMERICAL ANALYSIS PART Problem 1. Let b be a strictly positive constant and consider the problem: find u(x, t) such that u t + b u x = 0, 0 < x < 1, 0 < t u(x, 0) = u 0 (x), 0 < x < 1, u(0, t) = u(1, t), t > 0 where u 0 is a smooth function. Let J and N be positive integers, x i = ih where h = 1/J and t n = nτ where τ = 1/N. Also denote by u n j the approximation of u(x j, t n ). Set u 0 j = u 0 (x j ) and define reccursively u n j u n+1 j Show that for all j = 1,..., J and n 0 provided τb h 1. by the following Lax-Friedrichs scheme = 1 2 (un j+1 + u n j 1) τb 2h (un j+1 u n j 1), j = 1,..., J. min(u 0 i ) u n j max(u 0 i ) i i Problem 2. Below, C i, for i = 1, 2, 3 denote positive constants. consider solutions u H 1 (Ω) to (2.1) A(u, φ) = fφ, for all φ H 1 (Ω). For f L 2 (Ω), we Here Ω is a polyhedral domain in R n and A(, ) is a (non-coercive) bounded bilinear form on H 1 (Ω). It is assumed that A satisfies a Gärding inequality, i.e., there are positive constants K and α satisfying (2.2) α v 2 H 1 (Ω) A(v, v) + K v 2 L 2 (Ω), for all v H1 (Ω). We assume that solutions of (2.1) and those of the adjoint problem: u H 1 (Ω) satisfying (2.3) A(φ, u) = fφ, for all φ H 1 (Ω), exist, are unique and satisfy Ω u H 2 (Ω) C 1 f L 2 (Ω). We finally assume that {V h }, h (0, 1] is collection of conforming finite element subspaces satisfying the standard approximation properties and consider the finite element approximation: u h V h satisfying (2.4) A(u h, θ) = fθ, for all θ V h. Ω 1

6 (a) Suppose that u solves (2.1) and u h V h satisfies (2.4) (we do not assume that u h is unique). Show that u u h L 2 (Ω) C 2 h u u h H 1 (Ω). (b) Use (2.2) and Part (a) to show that there is an h 0 > 0 such that if h h 0, α 2 u u h 2 H 1 (Ω) A(u u h, u u h ). (c) Use Part (b) to show that the solutions of (2.4) are unique when h h 0. This also implies existence. (d) Prove that the unique solution (when h h 0 ) of (2.4) satisfies u u h H 1 (Ω) C 3 inf u v h H v h V 1 (Ω). h Problem 3. For this problem, for M 1, S M is a finite dimensional subspace of H 2 (Ω) with Ω = (0, 1). Also, we are given linear operators, P c : H 2 (Ω) S M and P M : L 2 (Ω) S M. We further assume that there is a constant C 1 not depending on M, u or s and satisfying (I P M )u H s (Ω) C 1 M s 2 u H 2 (Ω), for all u H 2 (Ω), s = {0, 1, 2}. Here H s (Ω) denotes the H s (Ω) semi-norm. We set Ω M = (0, M). For u defined on Ω, we define û(x) for x Ω M by û(x) = u(x/m) and define P M (û) = P M u and Pc (û) = P c u. We finally assume there is a constant C 2 (not depending on M) satisfying and that P c PM = P M. P c û L 2 (Ω M ) C 2 û H 2 (Ω M ), for all û H 2 (Ω M ), (a) Derive a relationship between u H s (Ω) and û H s (Ω M ). (b) Show that there is a constant C 3 not depending on M satisfying (I P M )û H 2 (Ω M ) C û H 2 (Ω M ). (c) Show that there is a constant C 3 not depending on M satisfying (I P c )u L 2 (Ω) C 3 M 2 u H 2 (Ω), for all u H 2 (Ω). 2