Minimal Smoothing Research Articles

Experimental comparison of FORC and remanent Preisach diagrams

First-order reversal curve (FORC) and remanence-based Preisach diagrams are alternative ways of determining the Preisach distribution of a sample, which incorporates information about the coercivity spectrum and the distribution of interactions and self-demagnetizing fields. We compare results of the two methods for well-characterized synthetic and natural samples containing single-domain (SD) and pseudo-SD (PSD) magnetite, maghemite, titanomagnetite and titanomaghemite. The greater time requirements of remanence as opposed to in-field measurements limited our Preisach diagrams to a few hundred points, compared to several thousand points for the corresponding FORC diagrams. Only minimal smoothing could be applied in order to limit the regions near the axes of the diagrams in which function values must be extrapolated. In spite of these restrictions, we find excellent agreement between the essential features of the distributions determined by the two methods. The main features, the location and spreading of the distribution peak, are very consistent. However, the low-coercivity part of the Preisach distribution is sometimes poorly resolved or not imaged at all for remanence-only measurements. Features in this region can be diagnostic of PSD and multidomain (MD) grains. The essential agreement between our FORC and Preisach diagrams in the region where they overlap justifies using the much faster FORC routine instead of traditional remanence-based Preisach methods to determine the Preisach distribution of palaeomagnetic samples without strong interactions. We propose a symmetric FORC protocol that would permit separation of the irreversible and reversible parts of the Preisach distribution. The irreversible part is what is determined by remanence-only methods and what is desired for characterization of the remanence behaviour of palaeomagnetic samples. The reversible part is most significant in detecting MD behaviour and screening out samples containing large PSD and MD grains.

Statistical inversion of South Atlantic circulation in an abyssal neutral density layer

This paper introduces a Bayesian inversion approach to estimating steady state ocean circulation and tracer fields. It is based on a quasi-horizontal flow model and a PDE solver for the forward problem of computing solutions to the tracer field advection-diffusion equations. A typical feature of existing ocean circulation inverse methods is a preprocessing stage in which the tracer data are interpolated over a regular grid and the interpolation error is ignored in the subsequent inversion. Our approach only uses interpolated data at those grid points that have neighboring hydrographic stations. By exploiting physically-based models in an integrated fashion, the method provides a statistically unified inversion and tracer field reconstruction with minimal data smoothing. Solving the problem consists of finding information about the circulation and tracer fields in the presence of a number of assumptions (prior information); the resulting posterior probability distribution summarizes what we can know about these fields. We develop a Markov chain Monte Carlo simulation procedure to extract information from the (analytically intractable) posterior distribution of all the parameters in the model; uncertainty about the “solution” is represented by variation in the output of the simulation runs. Our approach is aimed at finding the time-averaged quasi-horizontal flow and tracer fields for an abyssal neutral density layer in the South Atlantic. The collection of oceanographic data during the past century has formed the basis for our understanding of the distribution of properties and the circulation in the ocean. However, a description of the steady flow and associated property fields in the oceanic interior that is dynamically consistent with those collective observations and that accounts for the inherent uncertainty associated with measurements, physical variability and model parameterizations, for example, is still lacking. It may be that the steady-state assumption is a poor approximation to the equations of motion, but this hypothesis deserves close attention as a test of our understanding of the basic physics of the ocean circulation. In this paper we introduce a physically-based statistical inversion approach to estimate the

Distributional Assumptions in Voxel-Based Morphometry

In this paper we address the assumptions about the distribution of errors made by voxel-based morphometry. Voxel-based morphometry (VBM) uses the general linear model to construct parametric statistical tests. In order for these statistics to be valid, a small number of assumptions must hold. A key assumption is that the model's error terms are normally distributed. This is usually ensured through the Central Limit Theorem by smoothing the data. However, there is increasing interest in using minimal smoothing (in order to sensitize the analysis to regional differences at a small spatial scale). The validity of such analyses is investigated. In brief, our results indicate that nonnormality in the error terms can be an issue in VBM. However, in balanced designs, provided the data are smoothed with a 4-mm FWHM kernel, nonnormality is sufficiently attenuated to render the tests valid. Unbalanced designs appear to be less robust to violations of normality: a significant number of false positives arise at a smoothing of 4 and 8 mm when comparing a single subject to a group. This is despite the fact that conventional group comparisons appear to be robust, remaining valid even with no smoothing. The implications of the results for researchers using voxel-based morphometry are discussed.

Distributional Assumptions in Voxel-Based Morphometry

In this paper we address the assumptions about the distribution of errors made by voxel-based morphometry. Voxel-based morphometry (VBM) uses the general linear model to construct parametric statistical tests. In order for these statistics to be valid, a small number of assumptions must hold. A key assumption is that the model's error terms are normally distributed. This is usually ensured through the Central Limit Theorem by smoothing the data. However, there is increasing interest in using minimal smoothing (in order to sensitize the analysis to regional differences at a small spatial scale). The validity of such analyses is investigated. In brief, our results indicate that nonnormality in the error terms can be an issue in VBM. However, in balanced designs, provided the data are smoothed with a 4-mm FWHM kernel, nonnormality is sufficiently attenuated to render the tests valid. Unbalanced designs appear to be less robust to violations of normality: a significant number of false positives arise at a smoothing of 4 and 8 mm when comparing a single subject to a group. This is despite the fact that conventional group comparisons appear to be robust, remaining valid even with no smoothing. The implications of the results for researchers using voxel-based morphometry are discussed.

Block Descent Methods and Hybrid Procedures for Linear Systems

In this paper, we define and study several types of block descent methods for the simultaneous solution of a system of linear equations with several right hand sides. Then, improved block EN methods will be proposed. Finally, block hybrid and minimal residual smoothing procedures will be considered.

Modeling the high‐frequency barotropic response of the ocean to atmospheric disturbances: Sensitivity to forcing, topography, and friction

This study examines high‐frequency sea level variations forced by changes in surface atmospheric pressure and wind and their sensitivity to different forcing mechanisms, bottom topography resolution, and amount of friction in a barotropic ocean model. Optimal model performance, defined in terms of the explained variance in satellite altimeter and bottom pressure data, is found when using relatively strong friction, equivalent to a damping timescale of only a few days over the deep ocean, and topography with minimal smoothing. Spatial variations of the optimal friction parameter seem to reflect the roughness of bottom topography. The model demonstrates skill in simulating the wind‐driven response as well as the nonequilibrium response to atmospheric pressure variations.

A framework for generalized conjugate gradient methods—with special emphasis on contributions by Rüdiger Weiss

We discuss a general framework for generalized conjugate gradient methods, that is, iterative methods (for solving linear systems) that are based on generating residuals that are formally orthogonal to each other with respect to some true or formal inner product. This includes methods that generate residuals that are minimal with respect to some norm based on an inner product. A similar, even more general framework was introduced and extensively discussed by Weiss in his work. Weiss also emphasized in his work the relationship between certain pairs of orthogonal and minimal residual methods, where the results of the second method can be generated from those of the first method by applying the minimal residual smoothing process. Examples of such pairs are the conjugate gradient (CG) and the conjugate residual (CR) methods, the full orthogonalization method (FOM) and the generalized minimal residual (GMR es) method, as well as the CGNE and CGNR versions of applying CG to the normal equations.

Acceleration and stabilization properties of minimal residual smoothing technique in multigrid

We analyze the standard multigrid method accelerated by a minimal residual smoothing (MRS) technique. We show that MRS can accelerate the convergence of the slow residual components, thus accelerates the overall multigrid convergence. We prove that, under certain hypotheses, MRS stabilizes the divergence of certain slow residual components and thus stabilizes the divergent multigrid iteration. The analysis is customarily conducted on the two-level method.

Multi-level minimal residual smoothing: a family of general purpose multigrid acceleration techniques

We employ multi-level minimal residual smoothing (MRS) as a pre-optimization technique to accelerate standard multigrid convergence. The MRS method is used to improve the current multigrid iterate by smoothing its corresponding residual before the latter is projected to the coarse grid. We develop different schemes for implementing MRS technique on the finest grid and on the coarse grids, and several versions of the inexact MRS technique. Numerical experiments are conducted to show the efficiency of the multi-level and inexact MRS techniques.

Two-grid analysis of minimal residual smoothing as a multigrid acceleration technique

We analyze the two-level method accelerated by a minimal residual smoothing (MRS) technique. The two-grid analysis is sufficient for our purpose because our MRS acceleration scheme is only applied on the finest level of the multigrid method. We prove that the MRS acceleration scheme is a semi-iterative method with respect to the underlying two-level iteration and that the MRS accelerated two-level method is a polynomial acceleration of first order. We explain why MRS may not effectively accelerate standard multigrid method for solving Poisson-like problems. The iteration matrices for the MRS accelerated coarse-grid-correction operator and the MRS accelerated two-level operator are obtained. We give bounds for the residual reduction rates of the accelerated two-level method. Numerical experiments are employed to support the analytical results.

Multigrid with inexact minimal residual smoothing acceleration

We introduce some inexact versions of the minimal residual smoothing (IMRS) technique to accelerate the standard multigrid convergence. These are modified versions of the minimal residual smoothing (MRS) technique introduced in a recent paper (Zhang, to appear). The IMRS acceleration schemes minimize the residual norm of the multigrid iterate in a subspace and reduce the cost of the standard MRS acceleration by about 40% for two-dimensional problems and frequently achieve even faster convergence. Numerical experiments are employed to compare the performance of the exact and the inexact minimal residual smoothing schemes.

Minimal residual smoothing in multi-level iterative method

A minimal residual smoothing (MRS) technique is employed to accelerate the convergence of the multi-level iterative method by smoothing the residuals of the original iterative sequence. The sequence with smoothed residuals is re-introduced into the multi-level iterative process. The new sequence generated by this acceleration procedure converges much faster than both the sequence generated by the original multi-level method and the sequence generated by MRS technique. The cost of this acceleration scheme is independent of the original operator and in many cases is negligible. The emphasis of this paper is on the practical implementation of MRS acceleration techniques in the multi-level method. The discussions are focused on the two-level method because the acceleration scheme is only applied on the finest level of the multi-level method. Numerical experiments using the proposed MRS acceleration scheme to accelerate both the two-level and multi-level methods are conducted to show the efficiency and the cost-effectiveness of this acceleration scheme.

Accelerated multigrid high accuracy solution of the convection-diffusion equation with high Reynolds number

A fourth-order compact finite difference scheme is employed with the multigrid algorithm to obtain highly accurate numerical solution of the convection-diffusion equation with very high Reynolds number and variable coefficients. The multigrid solution process is accelerated by a minimal residual smoothing (MRS) technique. Numerical experiments are employed to show that the proposed multigrid solver is stable and yields accurate solution for high Reynolds number problems. We also show that the MRS acceleration procedure is efficient and the acceleration cost is negligible. © 1997 John Wiley & Sons, Inc.

Creep equations for high temperature alloys on the basis of a parametric assessment of multi-heat data

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Comparison of sensors and techniques for crop yield mapping

The implementation of site-specific crop management is dependent on the variations in yield and yield potential within a field. Crop yield maps are important for both the implementation and evaluation of site-specific crop management strategies. Management decisions and evaluations based on yield maps must take into consideration the accuracy and resolution of the maps. An impact-based yield monitor and a volumetric yield monitor were compared. The effect of different dynamic models of combine grain flow on the calculated instantaneous yields were investigated. Both simple time delay models and first order models could be used to model the grain flow. In general, a simple time delay model with minimal smoothing provided the best yield maps. Yield maps developed using different methods of Kriging and other mapping techniques were compared. The maps showed the same general trends. However, localized yield features were represented differently due to the methods used for developing the maps and the degree of smoothing.

Wavelet analysis of inhomogeneous data with application to the cosmic velocity field

An account is given of a method of smoothing spatial inhomogeneous data sets by using wavelet reconstruction on a regular grid in an auxilliary space onto which the original data is mapped. In a previous paper by the present authors, the authors devised a method for inferring the velocity potential from the radial component of the cosmic velocity field assuming an ideal sampling. Unfortunately the sparseness of the real data (the peculiar velocities of galaxies) as well as errors of measurement requires that the velocity field is first smoothed as observed on a three-dimensional support (i.e. the galaxy positions) inhomogeneously distributed throughout the sampled volume. The wavelet formalism permits a minimal smoothing procedure to be introduced that is characterized by the variation in size of the smoothing window function. Moreover, the output-smoothed radial velocity field can be shown to correspond to a well defined theoretical quantity as long as the spatial sampling support satisfies certain criteria. The authors also argue that one should be very cautious when comparing the velocity potential derived from such a smoothed radial component of the velocity field with related quantities derived from other studies (e.g. of the density field).

Residual Smoothing Techniques for Iterative Methods

An iterative method for solving a linear system $Ax = b$ produces iterates $\{ x_k \} $ with associated residual norms that, in general, need not decrease “smoothly” to zero. “Residual smoothing” techniques are considered that generate a second sequence $\{ y_k \} $ via a simple relation $y_k = (1 - \eta _k )y_{k - 1} + \eta _k x_k $. The authors first review and comment on a technique of this form introduced by Schonauer and Weiss that results in $\{ y_k \} $ with monotone decreasing residual norms; this is referred to as minimal residual smoothing Certain relationships between the residuals and residual norms of the biconjugate gradient (BCG) and quasi-minimal residual (QMR) methods are then noted, from which it follows that QMR can be obtained from BCG by a technique of this form; this technique is extended to generally applicable quasi-minimal residual smoothing. The practical performance of these techniques is illustrated in a number of numerical experiments.

Synthesis of English monosyllables by demisyllable concatenation

The use of demisyllables plus voice‐assimilated phonetic affixes (/− s, − z, − t/, etc.), rather than segments or syllables, as the basic units in speech synthesis would have important advantages [O. Fujimura, J. Acoust. Soc. Am. 59, S55(A) (1976)]. We have tested the practicality of such a model for monosyllabic words of English, using an inventory of demisyllables with a variety of nonaffixed clusters and syllable nuclei. Each initial demisyllable is prepared so as to include a constant length of the transitional portion, and nucleus length variations dependent on the final consonant(s) are automatically included in the “vowel” portion of the final demisyllable. Syllable nucleus “coloration” by postvocalic nasals, /r/, or /l/ is also found to be automatically accounted for in this scheme. For example, the initial demisyllable excised from CVC may be joined with − VN to make a natural‐sounding CVN syllable. These experiments used both a parallel‐type formant analyzer—synthesizer [J.P. Olive and M.J. Macchi, J. Acoust. Soc. Am. 58, S23(A) (1975)] and an LPC processor, with some retouching, and with minimal smoothing across the concatenative boundary of the parameters.

Influence of singularities on perturbation theory for the effective Hamiltonian

The energy-independent effective Hamiltonian $\mathcal{H}(z)$ is considered as an analytic function of the coupling parameter $z$. It is shown that a point ${z}_{b}$ is not a singularity of $\mathcal{H}(z)$ unless the energy of some state that is selected for representation in the model space (${L}_{P}$) coincides, at $z={z}_{b}$, with the energy of a state that is excluded from representation in ${L}_{P}$. This result establishes the correctness of previous conjectures by Schucan and Weidenm\uller, and implies that perturbation theory for $\mathcal{H}(z)$ will generally have a larger radius of convergence than perturbation theory for the individual energies. For the case of a cut (intruder-state cut) joining an isolated pair of branch points close to the real axis, the singular behavior of $\mathcal{H}(z)$ is examined in detail. The residue of the cut is expressed in terms of quantities that can be calculated by diagonalization of a real, symmetric modification (minimal smoothing) of the Hamiltonian matrix. A formula is given for the contribution of an intruder-state cut to the error incurred by $n\mathrm{th}$ order perturbation theory for the physical quantity $\mathcal{H}(1)$. Consideration of a numerical example shows that if the values of sufficiently high orders of perturbation theory are known, the residues of intruder-state cuts may be evaluated. This allows estimation of the errors that intruder-state cuts produce in perturbation theory summed to finite order, and yields a criterion for the optimal truncation of divergent perturbation series.NUCLEAR STRUCTURE Effective Hamiltonian for $^{18}O{J}^{\ensuremath{\pi}}={0}^{+}$; estimated errors of perturbation theory produced by intruder state.