Mathematics Research Works
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Item Detection of co-eluted peptides using database search methods(Springer Nature, 2008-07-02) Alves, Gelio; Ogurtsov, Aleksey Y; Kwok, Siwei; Wu, Wells W; Wang, Guanghui; Shen, Rong-Fong; Yu, Yi-KuoCurrent experimental techniques, especially those applying liquid chromatography mass spectrometry, have made high-throughput proteomic studies possible. The increase in throughput however also raises concerns on the accuracy of identification or quantification. Most experimental procedures select in a given MS scan only a few relatively most intense parent ions, each to be fragmented (MS2) separately, and most other minor co-eluted peptides that have similar chromatographic retention times are ignored and their information lost. We have computationally investigated the possibility of enhancing the information retrieval during a given LC/MS experiment by selecting the two or three most intense parent ions for simultaneous fragmentation. A set of spectra is created via superimposing a number of MS2 spectra, each can be identified by all search methods tested with high confidence, to mimick the spectra of co-eluted peptides. The generated convoluted spectra were used to evaluate the capability of several database search methods – SEQUEST, Mascot, X!Tandem, OMSSA, and RAId_DbS – in identifying true peptides from superimposed spectra of co-eluted peptides. We show that using these simulated spectra, all the database search methods will gain eventually in the number of true peptides identified by using the compound spectra of co-eluted peptides. Reviewed by Vlad Petyuk (nominated by Arcady Mushegian), King Jordan and Shamil Sunyaev. For the full reviews, please go to the Reviewers' comments section.Item RECOVERY OF EDGES FROM SPECTRAL DATA WITH NOISE—A NEW PERSPECTIVE(Copyright: Society for Industrial and Applied Mathematics, 2008) ENGELBERG, SHLOMO; TADMOR, EITANWe consider the problem of detecting edges—jump discontinuities in piecewise smooth functions from their N-degree spectral content, which is assumed to be corrupted by noise. There are three scales involved: the “smoothness” scale of order 1/N, the noise scale of order √η, and the O(1) scale of the jump discontinuities. We use concentration factors which are adjusted to the standard deviation of the noise √η ≫ 1/N in order to detect the underlying O(1)-edges, which are separated from the noise scale √η ≪ 1.Item LONG-TIME EXISTENCE OF SMOOTH SOLUTIONS FOR THLong time existence of smooth solutions for the rapidly rotating shallow-water and Euler equationsE(Copyright: Society for Industrial and Applied Mathematics, 2008) CHENG, BIN; TADMOR, EITANWe study the stabilizing effect of rotational forcing in the nonlinear setting of two-dimensional shallow-water and more general models of compressible Euler equations. In [Phys. D, 188 (2004), pp. 262–276] Liu and Tadmor have shown that the pressureless version of these equations admit a global smooth solution for a large set of subcritical initial configurations. In the present work we prove that when rotational force dominates the pressure, it prolongs the lifespan of smooth solutions for t ≲ ln(δ^−1); here δ ≪ 1 is the ratio of the pressure gradient measured by the inverse squared Froude number, relative to the dominant rotational forces measured by the inverse Rossby number. Our study reveals a “nearby” periodic-in-time approximate solution in the small δ regime, upon which hinges the long-time existence of the exact smooth solution. These results are in agreement with the close-to-periodic dynamics observed in the “near-inertial oscillation” (NIO) regime which follows oceanic storms. Indeed, our results indicate the existence of a smooth, “approximate periodic” solution for a time period of days, which is the relevant time period found in NIO observations.Item CENTRAL DISCONTINUOUS GALERKIN METHODS ON OVERLAPPING CELLS WITH A NONOSCILLATORY HIERARCHICAL RECONSTRUCTION(Copyright: Society for Industrial and Applied Mathematics, 2007) LIU, YINGJIE; SHU, CHI-WANG; TADMOR, EITAN; ZHANG, MENGPINGThe central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408–463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241–282] employs a variable control volume, which in turn yields a semidiscrete nonstaggered central scheme. Another approach, which we advocate here, is to view the staggered meshes as a collection of overlapping cells and to realize the computed solution by its overlapping cell averages. This leads to a simple technique to avoid the excessive numerical dissipation for small time steps [Y. Liu, J. Comput. Phys., 209 (2005), pp. 82–104]. At the heart of the proposed approach is the evolution of two pieces of information per cell, instead of one cell average which characterizes all central and upwind Godunov-type finite volume schemes. Overlapping cells lend themselves to the development of a central-type discontinuous Galerkin (DG) method, following the series of works by Cockburn and Shu [J. Comput. Phys., 141 (1998), pp. 199–224] and the references therein. In this paper we develop a central DG technique for hyperbolic conservation laws, where we take advantage of the redundant representation of the solution on overlapping cells. The use of redundant overlapping cells opens new possibilities beyond those of Godunov-type schemes. In particular, the central DG is coupled with a novel reconstruction procedure which removes spurious oscillations in the presence of shocks. This reconstruction is motivated by the moments limiter of Biswas, Devine, and Flaherty [Appl. Numer. Math., 14 (1994), pp. 255–283] but is otherwise different in its hierarchical approach. The new hierarchical reconstruction involves a MUSCL or a second order ENO reconstruction in each stage of a multilayer reconstruction process without characteristic decomposition. It is compact, easy to implement over arbitrary meshes, and retains the overall preprocessed order of accuracy while effectively removing spurious oscillations around shocks.Item Non-oscillatory central schemes for one- and two-dimensional MHD equations. II: high-order semi-discrete schemes.(Copyright: Society for Industrial and Applied Mathematics, 2006) Balbas, Jorge; Tadmor, EitanWe present a new family of high-resolution, nonoscillatory semidiscrete central schemes for the approximate solution of the ideal magnetohydrodynamics (MHD) equations. This is the second part of our work, where we are passing from the fully discrete staggered schemes in [J. Balb´as, E. Tadmor, and C.-C. Wu, J. Comput. Phys., 201 (2004), pp. 261–285] to the semidiscrete formulation advocated in [A. Kurganov and E. Tadmor, J. Comput. Phys., 160 (2000), pp. 241–282]. This semidiscrete formulation retains the simplicity of fully discrete central schemes while enhancing efficiency and adding versatility. The semidiscrete algorithm offers a wider range of options to implement its two key steps: nonoscillatory reconstruction of point values followed by the evolution of the corresponding point valued fluxes. We present the solution of several prototype MHD problems. Solutions of one-dimensional Brio–Wu shock-tube problems and the two-dimensional Kelvin–Helmholtz instability, Orszag–Tang vortex system, and the disruption of a high density cloud by a strong shock are carried out using third- and fourth-order central schemes based on the central WENO reconstructions. These results complement those presented in our earlier work and confirm the remarkable versatility and simplicity of central schemes as black-box, Jacobian-free MHD solvers. Furthermore, our numerical experiments demonstrate that this family of semidiscrete central schemes preserves the ∇ · B = 0-constraint within machine round-off error; happily, no constrained-transport enforcement is needed.Item SPECTRAL VANISHING VISCOSITY METHOD FOR NONLINEAR CONSERVATION LAWS(Copyright: Society for Industrial and Applied Mathematics, 2001) GUO, BEN-YU; MA, HE-PING; TADMOR, EITANWe propose a new spectral viscosity(SV) scheme for the accurate solution of nonlinear conservation laws. It is proved that the SV solution converges to the unique entropysolution under appropriate reasonable conditions. The proposed SV scheme is implemented directlyon high modes of the computed solution. This should be compared with the original nonperiodic SV scheme introduced byMada y, Ould Kaber, and Tadmor in [SIAM J. Numer. Anal., 30 (1993), 321–342], where SV is activated on the derivative of the SV solution. The new proposed SV method could be viewed as a correction of the former, and it offers an improvement which is confirmed byour numerical experiments. A postprocessing method is implemented to greatlyenhance the accuracyof the computed SV solution. The numerical results show the efficiencyof the new method.Item A MULTISCALE IMAGE REPRESENTATION USING HIERARCHICAL (BV,L2) DECOMPOSITIONS(Copyright: Society for Industrial and Applied Mathematics, 2004) TADMOR, EITAN; NEZZAR, SUZANNE; VESE, LUMINITAWe propose a new multiscale image decomposition which offers a hierarchical, adaptive representation for the different features in general images. The starting point is a variational decomposition of an image, f = u0 + v0, where [u0, v0] is the minimizer of a J-functional, J(f, λ0; X, Y ) = infu+v=f u X + λ0 v p Y . Such minimizers are standard tools for image manipulations (e.g., denoising, deblurring, compression); see, for example, [M. Mumford and J. Shah, Proceedings of the IEEE Computer Vision Pattern Recognition Conference, San Francisco, CA, 1985] and [L. Rudin, S. Osher, and E. Fatemi, Phys. D, 60 (1992), pp. 259–268]. Here, u0 should capture “essential features” of f which are to be separated from the spurious components absorbed by v0, and λ0 is a fixed threshold which dictates separation of scales. To proceed, we iterate the refinement step [uj+1, vj+1] = arginf J(vj, λ02j ), leading to the hierarchical decomposition, f = k j=0 uj + vk. We focus our attention on the particular case of (X, Y) = (BV,L2) decomposition. The resulting hierarchical decomposition, f ∼ j uj , is essentially nonlinear. The questions of convergence, energy decomposition, localization, and adaptivity are discussed. The decomposition is constructed by numerical solution of successive Euler–Lagrange equations. Numerical results illustrate applications of the new decomposition to synthetic and real images. Both greyscale and color images are considered.Item CRITICAL THRESHOLDS IN 2D RESTRICTED EULER–POISSON EQUATIONS(Copyright: Society for Industrial and Applied Mathematics, 2003) LIU, HAILIANG; TADMOR, EITANWe provide a complete description of the critical threshold phenomenon for the twodimensional localized Euler–Poisson equations, introduced by the authors in [Comm. Math. Phys., 228 (2002), pp. 435–466]. Here, the questions of global regularity vs. finite-time breakdown for the two-dimensional (2D) restricted Euler–Poisson solutions are classified in terms of precise explicit formulae, describing a remarkable variety of critical threshold surfaces of initial configurations. In particular, it is shown that the 2D critical thresholds depend on the relative sizes of three quantities: the initial density, the initial divergence, and the initial spectral gap, that is, the difference between the two eigenvalues of the 2 × 2 initial velocity gradient.Item CRITICAL THRESHOLDS IN A CONVOLUTION MODEL FOR NONLINEAR CONSERVATION LAWS(Copyright: Society for Industrial and Applied Mathematics, 2001) LIU, HAILIANG; TADMOR, EITANIn this work we consider a convolution model for nonlinear conservation laws.Due to the delicate balance between the nonlinear convection and the nonlocal forcing, this model allows for narrower shock layers than those in the viscous Burgers’ equation and yet exhibits the conditional finite time breakdown as in the damped Burgers’ equation.W e show the critical threshold phenomenon by presenting a lower threshold for the breakdown of the solutions and an upper threshold for the global existence of the smooth solution.The threshold condition depends only on the relative size of the minimum slope of the initial velocity and its maximal variation.W e show the exact blow-up rate when the slope of the initial profile is below the lower threshold.W e further prove the L1 stability of the smooth shock profile, provided the slope of the initial profile is above the critical threshold.Item High order time discretization methods with the strong stability property(Copyright: Society for Industrial and Applied Mathematics, 2001) Gottlieb, Sigal; Shu, Chi-Wang; Tadmor, EitanIn this paper we review and further develop a class of strong stability-preserving (SSP) high-order time discretizations for semidiscrete method of lines approximations of partial differential equations.Previously termed TVD (total variation diminishing) time discretizations, these high-order time discretization methods preserve the strong stability properties of first-order Euler time stepping and have proved very useful, especially in solving hyperbolic partial differential equations.The new developments in this paper include the construction of optimal explicit SSP linear Runge–Kutta methods, their application to the strong stability of coercive approximations, a systematic study of explicit SSP multistep methods for nonlinear problems, and the study of the SSP property of implicit Runge–Kutta and multistep methods.