Hermite Basis Research Articles

A minimisation approach for computing the ground state of Gross–Pitaevskii systems

In this paper, we present a minimisation method for computing the ground state of systems of coupled Gross–Pitaevskii equations. Our approach relies on a spectral decomposition of the solution into Hermite basis functions. Inserting the spectral representation into the energy functional yields a constrained nonlinear minimisation problem for the coefficients. For its numerical solution, we employ a Newton-like method with an approximate line-search strategy. We analyse this method and prove global convergence. Appropriate starting values for the minimisation process are determined by a standard continuation strategy. Numerical examples with two- and three-component two-dimensional condensates are included. These experiments demonstrate the reliability of our method and nicely illustrate the effect of phase segregation.

Studies on Barlow points, Gauss points and Superconvergent points in 1D with Lagrangian and Hermitian finite element basis

In this work, we consider the superconvergence property of the finite element derivative for Lagrange's and Hermite's Family elements in the one dimensional interpolation problem. We also compare the Barlow points, Gauss points and Superconvergence points in the sense of Taylor's Series, confirming that they are not the same as believed before. We prove a not evident and new superconvergence property of Hermite's basis as well which shows that the centroid is not only a superconvergent for u'h but an O(h5) accuracy point.

Studies on Barlow points, Gauss points and Superconvergent points in 1D with Lagrangian and Hermitian finite element basis

In this work, we consider the superconvergence property of the finite element derivative for Lagrange's and Hermite's Family elements in the one dimensional interpolation problem. We also compare the Barlow points, Gauss points and Superconvergence points in the sense of Taylor's Series, confirming that they are not the same as believed before. We prove a not evident and new superconvergence property of Hermite's basis as well which shows that the centroid is not only a superconvergent for u'hbut an O(h5) accuracy point.

Bridging Scales: A Three-Dimensional Electromechanical Finite Element Model of Skeletal Muscle

This paper introduces a framework for skeletal muscles that couples outputs from a detailed biophysically based electrophysiological cell model to a three-dimensional continuum-based finite element model of muscle mechanics. Due to the unique manner in which a skeletal muscle is activated, specifically the fact that neighboring fibers are electrically isolated and can act independently of each other, a completely new and novel coupling framework has been created. Within this framework, the electrical activity within a fiber is modeled with a biophysically based cell model, which is itself an amalgamation of several existing cell models. From this amalgamated cell model, specific output parameters that describe the level of crossbridge activity are computed and stored within a lookup table. This lookup table is then used to map the appropriate level of activity to all fibers within the muscle. To link the level of activity to a three-dimensional finite flement model of a skeletal muscle, which is based on principles of continuum mechanics, an upscaling method is introduced to compensate for the fact that the finite element mesh does not attempt to separately represent each individual fiber. This upscaling method allows the stress equilibrium equations to be computed at each Gauss point based on different values of the cell model outputs in all the neighboring cells. Since adjacent fibers can operate independently, the cell model outputs used in the finite element solution of the finite elasticity equations are discontinuous. The behavior and performance of the entire coupling framework is carefully analyzed in some simple test cases analyzing the reduction of the discretization error with respect to a sequence of uniformly refined meshes and different activation patterns. The results show that the error-reduction factors obtained from the electromechanical framework using triquadratic Lagrange and tricubic Hermite basis functions in solving the Galerkin finite element stress equilibrium equations are very similar to those obtained from a mechanics-only continuum-based model. Following this, an example of this process applied to the lateral pterygoid muscle is presented. The proposed framework can be used, for example, to investigate the mechanical effects with respect to cellular changes or to analyze the effects of different neuromuscular activation patterns on the tissue response.

Calculating molecular vibrational spectra beyond the harmonic approximation

We present a new approach for calculating anharmonic corrections to vibrational frequency calculations. The vibrational wavefunction is modelled using translated Hermite functions thus allowing anharmonic effects to be incorporated directly into the wavefunction whilst still retaining the simplicity of the Hermite basis. We combine this new method with an optimised finite-difference grid for computing the necessary third and fourth nuclear derivatives of the energy. We compare our combined approach to existing anharmonic models—vibrational self-consistent field theory (VSCF), vibrational perturbation theory (VPT), and vibrational configuration interaction theory (VCI)—and find that it is more cost effective than these alternatives. This makes our method well-suited to computing anharmonic corrections for frequencies in medium-sized molecules.

A Hermite pseudo-spectral method for solving systems of Gross-Pitaevskii equations

We propose and analyze discretization methods for solving finite systems of nonlinearly coupled Schrodinger equations, which arise as an asymptotic limit of the three-dimensional Gross-Pitaevskii equation with strongly anisotropic potential. A pseudo-spectral method with Hermite basis functions combined with a Crank-Nicolson type method is introduced. Numerical experiments are presented, including a comparison with an alternative discretization approach.

Spectral convergence of the Hermite basis function solution of the Vlasov equation: The free-streaming term

Spectral convergence of the Hermite basis function solution of the Vlasov equation: The free-streaming term

Computation of force density inside the channel of an electromagnetic pump by Hermite projection

This paper presents a coupled model of linear induction pump for liquid metal. The model computes the magnetic field in the motionless domains by a finite-element method and calculates the field inside the moving parts by Fourier transform and spectral projection on a Hermite basis. The paper focuses on the analysis of the force density according to the current supply and speed variations.

Non-invasive method for patient-specific virtual heart based on fiber-fluid model

In this paper, we present a non-invasive methodology in constructing a patient-specific virtual human heart based on fiber-fluid model. We applied digital image processing techniques on patient-specific MR images for obtaining the geometry information of human heart. The techniques include: acquisition, image visualization, image enhancement and segmentation. We incorporate cubic Hermite basis functions in our epicardium surface interpolation algorithm. We formularized a three-dimensional rule based fiber reconstruction mechanism to reconstruct the cardiac fiber architecture. A fiber model has been constructed which consists of 1,038 fibers with 371,239 fiber points. This model describes the ventricular three-dimensional geometry. Immersed Boundary Methods is used in constructing fiber-fluid cardiac simulation. The simulation of early ejection (from 0ms to 0.5ms) for the Left Ventricle (LV) has been implemented in SGI workstation. Simulation results for cardiac fiber and blood flow are presented in three-dimensional (3- D). Open GL-based animated visualization programs are developed to serve three purposes: (1) to demonstrate the interpolation and rule-based fiber reconstruction process. (2) To visualize the simulation results of cardiac fiber and blood flow; (3) To analyse the dynamics of the epicardium fibers as well as the blood flow in the cavity of the LV.

The use of sparse CT datasets for auto-generating accurate FE models of the femur and pelvis

The finite element (FE) method when coupled with computed tomography (CT) is a powerful tool in orthopaedic biomechanics. However, substantial data is required for patient-specific modelling. Here we present a new method for generating a FE model with a minimum amount of patient data. Our method uses high order cubic Hermite basis functions for mesh generation and least-square fits the mesh to the dataset. We have tested our method on seven patient data sets obtained from CT assisted osteodensitometry of the proximal femur. Using only 12 CT slices we generated smooth and accurate meshes of the proximal femur with a geometric root mean square (RMS) error of less than 1 mm and peak errors less than 8 mm. To model the complex geometry of the pelvis we developed a hybrid method which supplements sparse patient data with data from the visible human data set. We tested this method on three patient data sets, generating FE meshes of the pelvis using only 10 CT slices with an overall RMS error less than 3 mm. Although we have peak errors about 12 mm in these meshes, they occur relatively far from the region of interest (the acetabulum) and will have minimal effects on the performance of the model. Considering that linear meshes usually require about 70–100 pelvic CT slices (in axial mode) to generate FE models, our method has brought a significant data reduction to the automatic mesh generation step. The method, that is fully automated except for a semi-automatic bone/tissue boundary extraction part, will bring the benefits of FE methods to the clinical environment with much reduced radiation risks and data requirement.

Quantum dynamics of inelastic scattering with a moving grid

AbstractThe time‐dependent discrete variable representation (TDDVR) of a wave function with grid points defined by the Hermite part of the Gauss–Hermite (G‐H) basis set introduces quantum corrections to classical mechanics. The grid points in this method follow classical trajectory and the approach converges to the exact quantum formulation with sufficient trajectories (TDDVR points) but just with a single grid point; only classical mechanics performs the dynamics. This newly formulated approach (developed for handling time‐dependent molecular quantum dynamics) has been explored to calculate vibrational transitions in the inelastic scattering processes. Traditional quantum mechanical results exhibit an excellent agreement with TDDVR profiles during the entire propagation when enough grid points are included in the quantum‐classical dynamics. © 2005 Wiley Periodicals, Inc. Int J Quantum Chem, 2005

Refinable bivariate quartic $C^2$-splines for multi-level data representation and surface display

In this paper, a second-order Hermite basis of the space of C 2 -quartic splines on the six-directional mesh is constructed and the refinable mask of the basis functions is derived. In addition, the extra parameters of this basis are modified to extend the Hermite interpolating property at the integer lattices by including Lagrange interpolation at the half integers as well. We also formulate a compactly supported super function in terms of the basis functions to facilitate the construction of quasi-interpolants to achieve the highest (i.e., fifth) order of approximation in an efficient way. Due to the small (minimum) support of the basis functions, the refinable mask immediately yields (up to) four-point matrix-valued coefficient stencils of a vector subdivision scheme for efficient display of C 2 -quartic spline surfaces. Finally, this vector subdivision approach is further modified to reduce the size of the coefficient stencils to two-point templates while maintaining the second-order Hermite interpolating property.

Detecting acute myocardial infarction in the 12-lead ECG using Hermite expansions and neural networks

We use artificial neural networks (ANNs) to detect signs of acute myocardial infarction (AMI) in ECGs. The 12-lead ECG is decomposed into Hermite basis functions, and the resulting coefficients are used as inputs to the ANNs. Furthermore, we present a case-based method that qualitatively explains the operation of the ANNs, by showing regions of each ECG critical for ANN response. Key ingredients in this method are: (i) a cost function used to find local ECG perturbations leading to the largest possible change in ANN output and (ii) a minimization scheme for this cost function using mean field annealing. Our approach was tested on 2238 ECGs recorded at an emergency department. The obtained ROC areas for ANNs trained with the Hermite representation and standard ECG measurements were 83.4 and 84.3% ( P=0.4), respectively. We believe that the proposed method has potential as a decision support system that can provide good advice for diagnosis, as well as providing the physician with insight into the reason underlying the advice.

A QUANTUM-CLASSICAL APPROACH TO MOLECULAR DYNAMICS

We present a new method for treating the dynamics of molecular systems. The method has been named "quantum dressed" classical mechanics and is based on an expansion of the wave function in a time-dependent basis-set, the Gauss–Hermite basis-set. From here it is possible to proceed in two ways, one is in principle exact and the other approximate. In the exact approach one constructs a discrete variable representation (DVR) in which the grid points are defined by the Hermite part of the Gauss–Hermite basis set. In the approximate method a second order expansion of the potential around the classical trajectories is introduced and the quantum dymamics solved in a second quantization rather than a wave-function representation.

Quantum-dressed classical mechanics: Theory and application

A new method called “quantum dressed” classical mechanics has been formulated for treating problems within molecular dynamics, i.e. inelastic and reactive collisions, photodissociation, molecule–surface dynamics, non-adiabatic transitions etc. The method is based on an expansion of the wavefunction in a time-dependent basis set, the Gauss–Hermite basis set. From here it is possible to construct a discrete variable representation in which the grid points are defined by the Hermite part of the Gauss–Hermite basis set. The formulation introduces a set of grid points which follow the classical trajectory in space. With enough grid points the method approaches the exact quantum mechanical formulation. With just a single grid point in each dimension, we recover classical mechanics. Hence the new method adds a quantum option to ordinary Newtonian mechanics.

Shape Reconstruction and Weak Lensing Measurement with Interferometers: A Shapelet Approach

We present a new approach for image reconstruction and weak lensing measurements with interferometers. Based on the shapelet formalism presented in Refregier (2001), object images are decomposed into orthonormal Hermite basis functions. The shapelet coefficients of a collection of sources are simultaneously fit on the uv plane, the Fourier transform of the sky brightness distribution observed by interferometers. The resulting chi-square fit is linear in its parameters and can thus be performed efficiently by simple matrix multiplications. We show how the complex effects of bandwidth smearing, time averaging and non-coplanarity of the array can be easily and fully corrected for in our method. Optimal image reconstruction, co-addition, astrometry, and photometry can all be achieved using weighted sums of the derived coefficients. As an example we consider the observing conditions of the FIRST radio survey (Becker et al. 1995; White et al. 1997). We find that our method accurately recovers the shapes of simulated images even for the sparse uv sampling of this snapshot survey. Using one of the FIRST pointings, we find our method compares well with CLEAN, the commonly used method for interferometric imaging. Our method has the advantage of being linear in the fit parameters, of fitting all sources simultaneously, and of providing the full covariance matrix of the coefficients, which allows us to quantify the errors and cross-talk in image shapes. It is therefore well-suited for quantitative shape measurements which require high-precision. In particular, we show how our method can be combined with the results of Refregier & Bacon (2001) to provide an accurate measurement of weak lensing from interferometric data.

Combined Hermite spectral-finite difference method for the Fokker-Planck equation

The convergence of a class of combined spectral-finite difference methods using Hermite basis, applied to the Fokker-Planck equation, is studied. It is shown that the Hermite based spectral methods are convergent with spectral accuracy in weighted Sobolev space. Numerical results indicating the spectral convergence rate are presented. A velocity scaling factor is used in the Hermite basis and is shown to improve the accuracy and effectiveness of the Hermite spectral approximation, with no increase in workload. Some basic analysis for the selection of the scaling factors is also presented.

Quantum dressed classical mechanics

We have formulated a new way of making quantum corrections to classical mechanics. The method is based on a time-dependent discrete variable representation (DVR) of the wave function with grid points defined by the Hermite part of a basis set, the Gauss–Hermite basis set. The formulation introduces a set of grid points which follow the classical trajectory in space. With enough trajectories (DVR-points) the method approaches the exact quantum formulation. With just a single grid point in each dimension, we recover classical mechanics.

Reactive scattering within a time‐dependent discrete variable representation

AbstractWe have formulated a theory in which the quantum corrections to classical mechanics are easy to introduce. The method is based upon a time‐dependent discrete variable representation (DVR) of the wavefunction with grid points defined by the Hermite part of a basis set, the so‐called Gauss–Hermite basis set. The formulation introduces a set of grid points which follow the classical trajectory in space. With enough trajectories (DVR points), the method approaches the exact quantum formulation. With just a single grid point in a given degree of freedom, we have a classical mechanical description. © 2001 John Wiley & Sons, Inc. Int J Quant Chem, 2001

A time-dependent discrete variable representation method

We have developed a novel discrete variable representation (DVR) method where not only the amplitudes of the wave function at the DVR grid points can change but also the positions of these grid points can move as a function of time. Since the Gauss–Hermite basis set is used as the primitive basis functions (PBF) to construct the DVR basis set, the method appears as a semiclassical one with a small number of PBF but converges very fast to the quantum with an increasing PBF. We have investigated the dynamics of a reaction coordinate with or without coupling to a heat bath of harmonic oscillators to demonstrate the validity of the proposed method. The excellent agreement of the calculated tunneling probabilities with numbers obtained by traditional quantum grid method (FFT) and the fast computability of the present method compared to the latter are remarkable.