Split Sampling: Expectations, Normalisation and Rare Events

In this paper we develop a methodology that we call split sampling methods to estimate high dimensional expectations and rare event probabilities. Split sampling uses an auxiliary variable MCMC simulation and expresses the expectation of interest as an integrated set of rare event probabilities. We derive our estimator from a Rao-Blackwellised estimate of a marginal auxiliary variable distribution. We illustrate our method with two applications. First, we compute a shortest network path rare event probability and compare our method to estimation to a cross entropy approach. Then, we compute a normalisation constant of a high dimensional mixture of Gaussians and compare our estimate to one based on nested sampling. We discuss the relationship between our method and other alternatives such as the product of conditional probability estimator and importance sampling. The methods developed here are available in the R package: SplitSampling

A preconditioned Newton-Krylov method for computing steady-state pulse solutions of mode-locked lasers

Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.Includes bibliographical references (p. 47-48).We solve the periodic boundary value problem for a mode-locked laser cavity using a specially preconditioned matrix-implicit Newton-Krylov solver. Solutions are obtained at least an order of magnitude faster than with dynamic simulation, the standard method. Our method is demonstrated experimentally on a one-dimensional temporal model of an eight femtosecond mode-locked laser operating in the dispersion-managed soliton regime. Our solver is applicable to finding the steady-state solution of any nonlinear optical cavity with moderate self phase modulation, such as those of solid state lasers, and requires only a model for the round-trip action of the cavity. We conclude by proposing avenues of future work to improve the method's convergence and expand its applicability to lasers with higher degrees of cavity nonlinearity. Our approach can be extended to spatio-temporal cavity models, potentially allowing for the first feasible simulation of the full dynamics of Kerr-lens mode locking.by Jonathan R. Birge.S.M

Prior Reduced Fill-In in Solving Equations in Interior Point Algorithms

The efficiency of interior-point algorithms for linear programming is related to the effort required to factorize the matrix used to solve for the search direction at each iteration. When the linear program is in symmetric form (i.e., the constraints are Ax b, x > 0 ), then there are two mathematically equivalent forms of the search direction, involving different matrices. One form necessitates factoring a matrix whose sparsity pattern has the same form as that of (A AT). The other form necessitates factoring a matrix whose sparsity pattern has the same form as that of (ATA). Depending on the structure of the matrix A, one of these two forms may produce significantly less fill-in than the other. Furthermore, by analyzing the fill-in of both forms prior to starting the iterative phase of the algorithm, the form with the least fill-in can be computed and used throughout the algorithm. Finally, this methodology can be applied to linear programs that are not in symmetric form, that contain both equality and inequality constraints

Measurement of spin memory lengths in PdNi and PdFe ferromagnetic alloys

Weakly ferromagnetic alloys are being used by several groups in the study of superconducting/ferromagnetic hybrid systems. Because spin-flip and spin-orbit scattering in such alloys disrupt the penetration of pair correlations into the ferromagnetic material, it is desirable to have a direct measurement of the spin memory length in such alloys. We have measured the spin memory length at 4.2 K in sputtered Pd0.88Ni0.12 and Pd0.987Fe0.013 alloys using methods based on current-perpendicular-to-plane giant magnetoresistance. The alloys are incorporated into hybrid spin valves of various types, and the spin memory length is determined by fits of the Valet-Fert spin-transport equations to data of magnetoresistance vs. alloy thickness. For the case of PdNi alloy, the resulting values of the spin memory length are lsf(PdNi) = 2.8 +/- 0.5 nm and 5.4 +/- 0.6 nm, depending on whether or not the PdNi is exchange biased by an adjacent Permalloy layer. For PdFe, the spin memory length is somewhat longer, lsf(PdFe) = 9.6 +/- 2 nm, consistent with earlier measurements indicating lower spin-orbit scattering in that material. Unfortunately, even the longer spin memory length in PdFe may not be long enough to facilitate observation of spin-triplet superconducting correlations predicted to occur in superconducting/ferromagnetic hybrid systems in the presence of magnetic inhomogeneity.Comment: 7 pages, 8 figure

Quadratic Electro-Optic Effects in Bacteriorhodopsin: Measurement of γ(-ω;0,0,ω) in Dried Gelatin Thin Films

Quadratic electro-optic effects (dc or low frequency Kerr effect) of bacteriorhodopsin dispersed in dried gelatin thin films are examined in the near resonance region at three wavelengths: 633, 647, and 676 nm. The films show relatively large quadratic electro-optic effects compared to other molecular dispersed systems. The purple membrane is fixed within the polymerized gelatin matrix, and we show that the electronic contribution to γ dominates over possible orientational contributions. At 676 nm. the quadratic electro-optic coefficient s1133( - ω;0,0,ω) is 6.7 × 10-20 m2/V2 and the third order nonlinear susceptibility X1133(3)(-ω;0,0,ω) is 7.0 × 10-13 cm4 statCoulomb-2, with both values obtained for a protein concentration of 6.9 × 1018 cm-3. The orientationally averaged second molecular hyperpolarizability 〈γ(-ω;0.0,ω)〉 determined from the quadratic electro-optic coefficients at 676 nm assuming an Onsager ellipsoidal local field factor is (10.8±5.1)×10-32 cm7 statCoulomb-2 [(1.34±0.63) × 10-56 F3 m4C-2]. The 〈γ(- ω;0,0,ω)〉 value increases roughly tenfold when the probe wavelength is decreased to 633 nm. The behavior of γ(-ω;0,0,ω), when fit to a two-state model, predicts that γ(- ω;0,0,ω) is strongly enhanced via type III processes. Thus, the magnitude of γ(-ω;0,0,ω) is dominated by a term (Δμ210×μ210)/(ω10-ω)3, where Δμ10 is the change in dipole moment, μ10 is the transition moment, and ω10 is the transition energy of the lowest-lying allowed 1Bw*+-like π,π* state. We calculate that Δμ10 is 12.8±1.2 D, in good agreement with previous Stark and two-photon experimental values. Time-dependent Hartree-Fock methods based on the MNDO Hamiltonian yield reasonable agreement with experiment, underestimating γ(-ω;0,0,ω) by factors of only 2-4, with the error increasing as the frequency approaches resonance

On the Convergence of L-shaped Algorithms for Two-Stage Stochastic Programming

In this paper, we design, analyze, and implement a variant of the two-loop L-shaped algorithms for solving two-stage stochastic programming problems that arise from important application areas including revenue management and power systems. We consider the setting in which it is intractable to compute exact objective function and (sub)gradient information, and instead, only estimates of objective function and (sub)gradient values are available. Under common assumptions including fixed recourse and bounded (sub)gradients, the algorithm generates a sequence of iterates that converge to a neighborhood of optimality, where the radius of the convergence neighborhood depends on the level of the inexactness of objective function estimates. The number of outer and inner iterations needed to find an approximate optimal iterate is provided. Finally, we show a sample complexity result for the algorithm with a Polyak-type step-size policy that can be extended to analyze other situations. We also present a numerical study that verifies our theoretical results and demonstrates the superior empirical performance of our proposed algorithms over classic solvers.Comment: 39 pages, 2 figure

Redistricting to maximize the preservation of political boundaries

Redefining legislative districts is a task undertaken by the states after each census in order to ensure equitable representation. Many criteria have been proposed as objectives in forming districts but specific definitions of an optimal plan have not been enforced. In attempting to eliminate political concerns from the effort, the Michigan Supreme Court defined criteria based on the preservation of county and municipality borders. A quadratic programming formulation is given for this problem, and a heuristic solution procedure is proposed.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/25116/1/0000549.pd

The value of the stochastic solution in stochastic linear programs with fixed recourse

Stochastic linear programs have been rarely used in practical situations largely because of their complexity. In evaluating these problems without finding the exact solution, a common method has been to find bounds on the expected value of perfect information. In this paper, we consider a different method. We present bounds on the value of the stochastic solution, that is, the potential benefit from solving the stochastic program over solving a deterministic program in which expected values have replaced random parameters. These bounds are calculated by solving smaller programs related to the stochastic recourse problem.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47912/1/10107_2005_Article_BF01585113.pd

Spin Triplet Supercurrent in Co/Ni Multilayer Josephson Junctions with Perpendicular Anisotropy

We have measured spin-triplet supercurrent in Josephson junctions of the form S/F'/F/F'/S, where S is superconducting Nb, F' is a thin Ni layer with in-plane magnetization, and F is a Ni/[Co/Ni]n multilayer with out-of-plane magnetization. The supercurrent in these junctions decays very slowly with F-layer thickness, and is much larger than in similar junctions not containing the two F' layers. Those two features are the characteristic signatures of spin-triplet supercurrent, which is maximized by the orthogonality of the magnetizations in the F and F' layers. Magnetic measurements confirm the out-of-plane anisotropy of the Co/Ni multilayers. These samples have their critical current optimized in the as-prepared state, which will be useful for future applications.Comment: 4 pages, 4 figures, formatted in RevTeX version 4. Submitted to Physical Review B on August 13th, 201