A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex‐valued advection–diffusion–reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second‐order advection–diffusion–reaction discretization is more accurate than the standard second‐order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.     

Point estimation and some applications to iterative methods

We consider one of the crucial problems in solving polynomial equations concerning the construction of such initial conditions which provide a safe convergence of simultaneous zero-finding methods. In the first part we deal with the localization of polynomial zeros using disks in the complex plane. These disks are used for the construction of initial inclusion disks which, under suitable conditions, provide the convergence of the Gargantini-Henrici interval method. They also play a key role in the convergence analysis of the fourth order Ehrlich-Aberth method with Newton's correction for the simultaneous approximation of all zeros of a polynomial. For this method we state the initial condition which enables the safe convergence. The initial condition is computationally verifiable since it depends only on initial approximations, which is of practical importance.     

Foregone with the Wind: Indirect Payoff Information and its Implications for Choice

Examination of the effect of information concerning foregone payoffs on choice behavior reveals a complex pattern. Depending on the environment, this information can facilitate or impair maximization. Our study of nine experimental tasks suggests that the complex pattern can be summarized with the assumption that initially people tend to be highly sensitive, and sometimes too sensitive, to recent foregone payoffs. However, over time, people can learn to adjust their sensitivity depending on the environment they are facing. The implications of this observation to models of human adaptation and to problems of mechanism design are discussed.We thank Nick Feltovich, Ernan Haruvy, Rosemarie Nagel and Rajiv Sarin for helpful comments and suggestions. This research was initiated while the first author was visiting the Technion in Haifa. We very much appreciate the hospitality of the Minerva Center.     

Singularities of the Hele-Shaw flow and shock waves in dispersive media

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.     

Single molecule observations of the adsorption sites of methyl isocyanide on Pt(111) by low-temperature scanning tunneling microscopy

Scanning tunneling microscopy (STM) has been used to directly investigate the local structure of methyl isocyanide (CNCH3) adsorbed on Pt(111). At low coverages, CNCH3 is preferentially adsorbed at on-top sites, in agreement with earlier deductions based on vibrational spectroscopy. When dosed at low coverages at 50 K, the molecules tend to adsorb near other CNCH3 molecules with preferred distances of a and a, where a = 2.78 A is the lattice constant of Pt. Annealing the surface to 120 K, however, results in a more uniform separation of the molecules. At higher coverages, the CNCH3 molecules are observed to occupy both on-top and two-fold bridge sites. On the basis of STM image analysis, CNCH3 forms an ordered layer of (2 x 3) periodicity at 0.33 ML. Additional details on the structures of CNCH3 adsorbed at the on-top and two-fold bridge sites are provided by density functional theory (DFT) calculations. At a coverage that saturates the first layer (0.33 ML), the occupation ratio for the on-top and two-fold bridge bonded CNCH3 is 1:1, which is consistent with the results obtained from the combined use of experimental reflection absorption infrared spectroscopy (RAIRS) data and DFT calculations.     

Numerical methods for A-optimal designs with a sparsity constraint for ill-posed inverse problems

We consider the problem of experimental design for linear ill-posed inverse problems. The minimization of the objective function in the classic A-optimal design is generalized to a Bayes risk minimization with a sparsity constraint. We present efficient algorithms for applications of such designs to large-scale problems. This is done by employing Krylov subspace methods for the solution of a subproblem required to obtain the experiment weights. The performance of the designs and algorithms is illustrated with a one-dimensional magnetotelluric example and an application to two-dimensional super-resolution reconstruction with MRI data.     

Crystal Structure and Characterization of [Ni(RPOEt)<Subscript>2</Subscript>(C<Subscript>2</Subscript>H<Subscript>5</Subscript>OH)<Subscript>2</Subscript>](ClO<Subscript>4</Subscript>)<Subscript>2</Subscript> [RPOEt = bis{(diphenylphosphinyl)methyl}ethyl phosphinate]

Abstract  

The compound [Ni(RPOEt)₂(C₂H₅OH)₂](ClO₄)₂ (1) [RPOEt = bis{(diphenylphosphinyl)methyl}ethyl phosphinate] has been prepared in the form of single crystals and characterized by elemental and thermal analyses, IR spectroscopy and by a single crystal X-ray diffraction study. The complex crystallizes in the triclinic P[`1] P\bar{1} space group with a = 9.8773(5), b = 13.741(1), c = 14.287(1) ?, α = 64.313(9), β = 70.280(5), γ = 89.973(7)o, V = 1620.5(2) ?³, and Z = 1. The nickel(II) atom in 1, situated at the centre of symmetry, is coordinated by four oxygen atoms from two RPOEt ligands and two oxygen atoms from two ethanol molecules in a slightly distorted octahedral environment. The third phosphoryl-oxygen atom from the phosphinate ligand is hydrogen bonded to the oxygen atom of the ethanol molecule coordinated to nickel(II). The complex [Ni(RPOEt)₂(C₂H₅OH)₂]²⁺ cations are linked by the C–H···O hydrogen bonds into one-dimensional chains and by the composite π···π and C–H···π phenyl interactions into a final three-dimensional structure.     

Bis[bis{(diphenylphosphinyl)methyl}ethyl phosphinate](ethanol) copper(II) perchlorate: synthesis, crystal and molecular structure

Bis[bis{(diphenylphosphinyl)methyl}ethyl phoshinate]bis(ethanol) metal(II) perchlorate complexes, where METAL = Co, Ni or Cu, have been prepared by the reaction of metal perchlorates and bis[(diphenylphosphinyl)methyl]ethyl phosphinate, (RPOEt), in absolute ethanol. The crystal structure of the copper complex is triclinic, space group P , a = 13.688(7) Å, b = 14.424(10) Å, c = 9.865(2) Å, = 110.43(4)°, β = 90.13(2)°, γ = 115.54(4)°, V = 1619.8 ų, Z = 1 and refined to R = 0.048 (R_w = 0.057). The structure consists of complex cations surrounded by perchlorate anions. Two RPOEt ligands and two ethanols are coordinated to the copper atom situated at the centre of symmetry 0, 0, 0. The RPOEt ligand is bidentate [Cu---O distances 1.969(4) Å, 2.312(4) Å], but the third oxygen atom from phosphoryl bonds is hydrogen bonded [2.669(6) Å], to the oxygen atom of ethanol molecule coordinated to the copper at 1.988(4) Å. The conformations of the ligand and chelate rings, magnetic data, molar conductance values, infrared and electronic spectra are given.