Resonant MEMS, Eigenvalues, and Numerical Pondering

Transcription

1 Resonant MEMS, Eigenvalues, and Numerical Pondering David Bindel UC Berkeley, CS Division Resonant MEMS, Eigenvalues,and Numerical Pondering p.1/27

2 Introduction I always pass on good advice. It is the only thing to do with it. It is never of any use to oneself. Oscar Wilde Resonant MEMS and some applications Eigenvalue computations: basic notions Some comments (rants) on numerical computing Resonant MEMS, Eigenvalues,and Numerical Pondering p.2/27

3 What are MEMS? Micro-Electro-Mechanical Systems Frequently made from etched Si Involve multiple energy domains (electrical, mechanical, thermal,...) Lots of applications Inertial sensors (accelerometers, gyroscopes) Microfluidic devices (ink-jet print heads, DNA chips) Optics (displays, switches) RF devices (cell phones, RFID tags) Resonant MEMS, Eigenvalues,and Numerical Pondering p.3/27

4 Resonant MEMS A common microsystem design technique: Periodically excite a mechanical structure Measure the position or velocity somewhere Deduce something from the response Examples: Sense capacitance (accelerometers, pressure sensors) Sense frequency shift (chemical and pressure sensors) Filter mechanically (cell phones) Resonant MEMS, Eigenvalues,and Numerical Pondering p.4/27

5 Micromechanical filters and RF Radio signal Mechanical filter Filtered signal Capacitive drive Capacitive sense Mechanical high-frequency (high MHz-GHz) filter Saves power and cost over electronic filters Already used (piezo-actuated quartz SAW filters) Resonant MEMS, Eigenvalues,and Numerical Pondering p.5/27

6 Shear ring resonator 0.9 hw = e Value = 2.66E+07 Hz Ring is driven in a shearing motion Can couple ring to other resonators Resonant MEMS, Eigenvalues,and Numerical Pondering p.6/27

7 Checkerboard resonator 9.27 MHz MHz MHz MHz. Resonant MEMS, Eigenvalues,and Numerical Pondering p.7/27

8 What do we want from simulation? The purpose of computing is insight, not numbers. The purpose of computing numbers is not yet in sight. R.W. Hamming (1961 and 1997) We would mostly like modes, frequencies, and frequency response information. Natural frequencies and mode shapes Frequency response plots Dependence of frequencies on parameters Sensitivity to fabrication errors Sources and effects of damping Resonant MEMS, Eigenvalues,and Numerical Pondering p.8/27

9 Flavors of eigenvalue problem Different problems need different methods: Is the problem Standard? Generalized? Ax = λx Kx = λm x Quadratic? (λ 2 M + λb + K)x = 0 Nonlinear? f(λ)x = 0 How big is the problem? Is it symmetric (Hermitian?) Do we want all or only a few eigenvalues? Do we want corresponding eigenvectors? Do we have the matrices explicitly? Resonant MEMS, Eigenvalues,and Numerical Pondering p.9/27

10 Power iteration Example: 0.1 A = [ ] [ and x 0 = ] 1 1 Compute x j = A j x 0 / A j x 0. Resonant MEMS, Eigenvalues,and Numerical Pondering p.10/27

11 Power iteration Pick random x and compute x j = A j x/ A j x. If A = QΛQ then A j x 0 = QΛ j Q x 0 Let ˆx = Q x and suppose λ 1 > λ 2... λ n. ˆx 1 A j x 0 = QΛ j ˆx = λ j 1 Q (λ 1 /λ 2 ) j ˆx 2. (λj1ˆx 1)q 1 (λ n /λ 2 ) j ˆx n Resonant MEMS, Eigenvalues,and Numerical Pondering p.11/27

12 Power iteration notes Markov chain steps are power method steps Power method is rarely used as is. But it s the idea behind more powerful algorithms. We don t need the matrix explicitly for the power method. Only have to multiply by the matrix. Useful if the matrix is a Jacobian (finite difference) Also useful if the matrix is hidden in another program Resonant MEMS, Eigenvalues,and Numerical Pondering p.12/27

13 Accelerating power iteration Spectral transformations change the eigenvalues of the matrix, but not the eigenvectors: A 1 = QΛ 1 Q A σi = Q(Λ σi)q (A σi) 1 = Q(Λ σi) 1 Q If σ is a good estimate to λ i, then (λ i σ) 1 gets very big, and power iteration on (A σi) 1 converges quickly. Resonant MEMS, Eigenvalues,and Numerical Pondering p.13/27

14 Big problems and small problems For dense problems (n a few hundred, all eigenvalues): Fastest solver for symmetric problems uses inverse iteration together with bisection. Several other methods use the same idea. LAPACK: ( For big problems where we only want a few eigenvalues: Run something like power iteration (possibly with spectral transformation) Look for approximate eigenvectors in the space spanned by the iterates. Templates for Algebraic Eigenvalue Problems ( Resonant MEMS, Eigenvalues,and Numerical Pondering p.14/27

15 How else could we do it? The characteristic polynomial of A is p(λ) = det(a λi). Could we solve the equation p(λ) = 0? Resonant MEMS, Eigenvalues,and Numerical Pondering p.15/27

16 Wilkinson s example Suppose p(λ) = 21 i=1 (x i). Compute the coefficients and try to recover the roots. 0.6 Error in computed roots of Wilkinson s polynomial Absolute error Computed root Resonant MEMS, Eigenvalues,and Numerical Pondering p.16/27

17 What went wrong? I rounded the coefficients to double precision (16 decimal digits) My error would be little better even if I committed no additional rounding errors. The roots are very sensitive functions of the coefficients. But the eigenvalues of a symmetric matrix are not so sensitive to the matrix coefficients! Resonant MEMS, Eigenvalues,and Numerical Pondering p.17/27

18 Problem sensitivity Roots function Polynomials near p(λ) Associated roots How can I quantify the sensitivity of the problem? Resonant MEMS, Eigenvalues,and Numerical Pondering p.18/27

19 Problem sensitivity The condition number κ measures sensitivity of the roots (λ) to changes in polynomial coefficients (p): δλ λ κ = κ δp p λ p p λ + higher order terms If κ is very large, we call the problem ill-conditioned. The same idea works for other problems. Resonant MEMS, Eigenvalues,and Numerical Pondering p.19/27

20 Forward and backward error Forward error: How wrong did I get the answer? λ computed = λ exact (A) + δλ Backward error: How wrong did I get the problem? λ computed = λ exact (A + δa) We usually seek algorithms with small backward error. Resonant MEMS, Eigenvalues,and Numerical Pondering p.20/27

21 Forward = sensitivity backward We try to design algorithms so that backward error input value Cɛ where C is some constant depending on the dimension and ɛ is the rounding threshold (around for double precision). Then to first order forward error output value κ backward error input value Cκɛ Resonant MEMS, Eigenvalues,and Numerical Pondering p.21/27

22 Sensitivity and eigenvalue problems For the symmetric eigenvalue problem: Eigenvalues are perfectly conditioned ( δλ δa ) Cannot even define vectors continuously near a multiple eigenvalue Unless admissible perturbations are restricted to those compatible with some symmetry group The algorithm usually won t preserve all the symmetries of the physical problem. Resonant MEMS, Eigenvalues,and Numerical Pondering p.22/27

23 Choosing numerical software: accuracy Want intermediates to more digits than data deserves. Better maintain relationships from original problem Compensate for accuracy lost in composing answers Significance arithmetic model is very pessimistic May want control over several error metrics Forward error? Backward error? Residual error? Which norm? Want estimate of problem sensitivity (condition number) Resonant MEMS, Eigenvalues,and Numerical Pondering p.23/27

24 Choosing numerical software: speed Premature optimization is the root of all evil. D. Knuth Speed is much more than operation counting. Human cycles are more valuable than human cycles. Making the fast part faster is futile. Resonant MEMS, Eigenvalues,and Numerical Pondering p.24/27

25 Where to go Lesser artists borrow. Great artists steal. I. Stravinsky Numerical accuracy software should be fast, accurate, and robust. It s easy to build code which is none of these. Good sources: Netlib ( Guide to Available Mathematical Software (gams.nist.gov) Environments like Matlab and Mathematica Resonant MEMS, Eigenvalues,and Numerical Pondering p.25/27

26 Conclusions Several MEMS based on steady-state frequency-response Some ideas in eigenvalue calculations Some numerical comments Thinking about accuracy Building for speed and accuracy Where to look for codes Resonant MEMS, Eigenvalues,and Numerical Pondering p.26/27

27 Some concluding quotes Think. Then discretize. V. Rokhlin Oh, I knew All the answers But I couldn t get the questions right. W.A. Yankovic, I Lost on Jeopardy Given a solution to a problem, it is always possible to find a sense in which it is optimal. W. Kahan paraphrasing R. Karp Resonant MEMS, Eigenvalues,and Numerical Pondering p.27/27