Exponential Basis Research Articles

Predicting Pulmonary Distension in a Virtual Patient Model for Mechanical Ventilation

Abstract Recruitment maneuvers (RMs) following with positive-end-expiratory-pressure (PEEP) have proved effective in recruiting lung volume and preventing alveoli collapse. To date, standards for optimal patient-specific PEEP are unknown, resulting in variability in care and reduced outcomes, both indicating the need for personalized care. This research extends a well-validated virtual patient model by adding novel elements to model, which is able to utilize bedside available respiratory data, without increasing modelling complexity, to predict patient-specific lung distension and thus to minimise barotrauma risk. Prediction accuracy and robustness are validated against clinical data from 18 volume controlled ventilation (VCV) patients at 7 different baseline PEEP levels (0 to 12cmH2O), where predictions were made up to 12cmH2O of PEEP ahead. Using an exponential basis function set for prediction yields an absolute median peak inspiratory pressure prediction error of 1.50cmH2O for 623 prediction cases. Comparing predicted and clinically measured distension prediction in VCV demonstrated consistent, robust high accuracy with R2=0.90 (623 predictions), which is a measurable improvement in prediction error compared to predictions without using the proposed distension function (R2=0.82). Moreover, the R2 value increases to 0.93-0.95 if only clinically relevant ΔPEEP steps (2-6cmH2O) are considered with an overall median absolute error in peak pressure prediction of 1.04cmH2O. Overall, the results demonstrate the potential and significance for accurately capturing distension mechanics, allowing better risk assessment, as well as extending and more fully validating this virtual mechanical ventilation patient model.

Impact of Two Lung Elastance Identification Methods on Pulmonary Mechanics Prediction

Abstract Positive-end-expiratory-pressure (PEEP) have proved effective in recruiting lung volume and keeping alveoli open. However, there is no standard means to find an optimal patient-specific PEEP, creating variability in care and outcomes. There is thus a need for personalized approaches to find the best PEEP and optimise care. This research extends a well-validated virtual patient model with a newly proposed function to predict lung distension, while the impact on outcome of two different elastance identification strategies are discussed and compared. A prior studied and effective exponential basis function set is used as the general model, while elastance are identified using overlapped and separate methods, respectively. In this approach, model with overlapped elastance identification and proposed distension function yields an absolute median peak inspiratory pressure (PIP) prediction error of 1.50cmH2O for 623 prediction cases. Comparison between clinically measurement and model prediction for PIP yields R2=0.90 across 623 predictions in total, while R2=0.87 with separate elastance identification. Furthermore, both elastance identification methods are an improvement compared to predictions without proposed distension function (R2=0.82). Validation is fulfilled with 18 volume controlled ventilation patients respiratory data at 7 different baseline PEEP levels (0-12cmH2O) with a maximal PEEP prediction interval of 12cmH2O. Overall, the results demonstrate the impact of elastance identification methods, as well as the potential and significance for accurately capturing distension mechanics, which thus providing guidance for clinical care and insights for predictive lung mechanics modelling.

An efficient time-space formulation for dynamic transient analyses: Application to the beam assemblies subjected to moving loads and masses

In this study, the formulation of a time marching method has been developed for the dynamic analysis of beam-type structures in the presence of moving loads/masses. By considering each flexural member of the system as a beam element, the introduced idea of the time-weighted residual method in our recent study “Int. J. Numer. Methods Eng., Vol. 114(7); pp. 719–748″, has been evoked to express the beam response in terms of a series of exponential basis functions. The solution advances in time by modifying the series coefficients through employing a pre-integration process as well as the governing equilibrium equation. The presented method needs neither discretization of the members nor any numerical integration processes so the solution can be obtained by the minimum numbers of required degrees of freedom. Six sample problems have been provided to show the capabilities of the proposed method in the solution of a variety of single and multi-spans beam under constant and accelerated moving loads/masses.

Electronic structure calculations of compressed Li atom using composite technique under Ritz variational framework

AbstractThe structural properties of Li atom in 1s2nl [n = 2–5, l = 0–4] state inside an impenetrable cavity is studied. The system is assumed to be composed of two parts namely the core (1s2) electrons and the valence electron (nl). A model potential is considered to mimic the interaction between the outer electron with the core. The wavefunction of the core electrons is expanded in explicitly correlated multi‐exponent Hyllerass type basis set while the wavefunction of the valence electron is expanded in pure exponential basis set. Both the wavefunctions are consistent with Dirichlet's boundary conditions. The energy levels are determined by using Ritz variational method and are pushed towards continuum as the effect of confinement increases. The results are in reasonable agreement with the existing numerical estimates. We noticed incidental degeneracy and the subsequent level‐crossing phenomena along with evolution of quasibound states of confined Li atom. The thermodynamic pressures felt by the Li atom, Li+ and Li2+ ions inside the impenetrable cavity.

On self-similar spectral measures

A probability measure μ on R is called a spectral measure if it has an exponential orthogonal basis for L2(μ). In this paper, we study the spectrality of the self-similar measure μρ,D generated by an iterated function system {τd(⋅)=ρ(⋅+d)}d∈D associated with a real number 0<ρ<1 and a finite set D⊂R. It can also be expressed asμρ,D=δρD⁎δρ2D⁎δρ3D⁎⋯=μk⁎μρ,D(ρ−k⋅), where μk is the convolutional product of the first k discrete measures. Until now, all known self-similar spectral measures are obtained from ρ−1∈N and spectral measures μk. We will show that these two conditions are also necessary under some natural assumptions. It improves significantly many results studied by recent research. As an application, we characterize a self-similar spectral measure associated with an integer tile.

Wave propagation in two dimensional structures: An efficient solution method in time domain using exponential basis functions

In this paper, an efficient time domain formulation is proposed for computing time history dynamic responses of 2D structures, i.e. various types of trusses and frames. Unlike the methods using frequency domain, the proposed method employs the time weighted residual approach to solve the governing differential equations of the axial and flexural wave propagation problems, directly in time. Upon choosing the time step, the solution procedure begins with expressing the variation of axial and transverse displacement components, through the member’s length, as the summations of Fourier exponential basis functions in each time step. The unknown coefficients of the next steps are computed by the implementation of recurrent relations obtained based on a pre-integration process. The equilibrium and necessary continuity conditions at the common nodes/joints of adjacent elements are also fulfilled by introducing an innovative concept utilizing source functions. The role of these source functions is to simulate the boundary effects of the axial and flexural waves propagating within the structural members, simultaneously. The efficiency of the method is demonstrated in the dynamic analysis of some samples of truss and framed structures in terms of accuracy and computational time.

Low complexity joint equalization and interference compensation in fractional Fourier domain for DCT-OFDM system with CFO

Orthogonal frequency division multiplexing (OFDM) systems based on discrete fractional Fourier transform (DFRFT) chirp basis replacing the complex exponential basis of discrete Fourier transform (DFT) is extensively studied in the literature. Authors of this paper have recently proposed fractional Fourier domain equalization (FFDE) for an OFDM system with a discrete cosine transform (DCT) multicarrier basis. The proposed DCT-OFDM system with FFDE nearly achieves the performance of the DFRFT-OFDM system without requiring any chirp rate feedback to the transmitter. This is enabled by separately performing the tasks of multicarrier modulation/demodulation with IDCT/DCT kernels at transmitter/receiver and FFDE using DFRFT/IDFRFT kernels both at the receiver. However, in the presence of a high value of CFO and also with higher modulation order, the residual interference power existing after channel equalization in fractional Fourier domain (FFD) using the channel decomposed with DFRFT significantly degrades the error rate performance. In this paper, the equalization and interference compensation is treated as a joint and linear problem in FFD after DFRFT at the receiver. Full compensation by performing direct matrix inverse can ensure the performance free from CFO for the DCT-OFDM system with both FDE and FFDE, but will drastically increase the computational cost to the O(N3) for implementing the joint equalizer at the receiver for N subcarriers. Therefore, low complexity joint equalization and interference compensation (JEIC) schemes are proposed for the DCT-OFDM system with FFD decomposition utilizing the banded implementation of joint ICI matrix after DFRFT at the receiver. The banded JEIC can closely preserve the interference-free performance and only requires operations in the O(N) to implement the joint equalizer.

Boundary shape functions methods for solving the nonlinear singularly perturbed problems with Robin boundary conditions

Abstract For a second-order nonlinear singularly perturbed boundary value problem (SPBVP), we develop two novel algorithms to find the solution, which automatically satisfies the Robin boundary conditions. For the highly singular nonlinear SPBVP the Robin boundary functions are hard to be fulfilled exactly. In the paper we first introduce the new idea of boundary shape function (BSF), whose existence is proven and it can automatically satisfy the Robin boundary conditions. In the BSF, there exists a free function, which leaves us a chance to develop new algorithms by adopting two different roles of the free function. In the first type algorithm we let the free functions be the exponential type bases endowed with fractional powers, which not only satisfy the Robin boundary conditions automatically, but also can capture the singular behavior to find accurate numerical solution by a simple collocation technique. In the second type algorithm we let the BSF be solution and the free function be another variable, such that we can transform the boundary value problem to an initial value problem (IVP) for the new variable, which can quickly find accurate solution for the nonlinear SPBVP through a few iterations.

A Kernel-Based explicit unconditionally stable scheme for Hamilton-Jacobi equations on nonuniform meshes

In [11], the authors developed a class of high-order numerical schemes for the Hamilton-Jacobi (H-J) equations, which are unconditionally stable, yet take the form of an explicit scheme. This paper extends such schemes, so that they are more effective at capturing sharp gradients, especially on nonuniform meshes. In particular, we modify the weighted essentially non-oscillatory (WENO) methodology in the previously developed schemes by incorporating an exponential basis and adapting the previously developed nonlinear filters used to control oscillations. The main advantages of the proposed schemes are their effectiveness and simplicity, since they can be easily implemented on higher-dimensional nonuniform meshes. We perform numerical experiments on a collection of examples, including H-J equations with linear, nonlinear, convex and non-convex Hamiltonians. To demonstrate the flexibility of the proposed schemes, we also include test problems defined on non-trivial geometry.

Fitted second order numerical method for a singularly perturbed Fredholm integro-differential equation

In this paper, we consider the linear first order singularly perturbed Fredholm integro-differential equation. For the solution of this problem, fitted difference scheme is constructed on a Shishkin mesh. The method is based on the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. The method is proved to be second-order convergent in the discrete maximum norm. Also, numerical results are given to support theoretical analysis.

Spectrality of Moran measures with finite arithmetic digit sets

Let [Formula: see text] be a prime and [Formula: see text] be a sequence of finite arithmetic digit sets in [Formula: see text] with [Formula: see text] uniformly bounded, and let [Formula: see text] be the discrete probability measure on the finite set [Formula: see text] with equal distribution. For [Formula: see text], the infinite Bernoulli convolution [Formula: see text] converges weakly to a Borel probability measure (Moran measure). In this paper, we study the existence of exponential orthonormal basis for [Formula: see text].

On dynamic behavior of composite plates using a higher-order Zig-Zag theory and exponential basis functions

In this paper, a boundary node method is presented to study wave propagation in laminated composite plates. The Zig-Zag theory of Cho and Parmerter is used for deriving the governing equations of laminated plates. To the best of the authors’ knowledge, this study is the first one employing the theory in non-stationary dynamic plate problems addressing wave propagation issues. For this theory, there is no information about the Green’s functions and thus the presented method can be considered as an alternative to the boundary integral method. With the use of exponential basis functions (EBFs), the response of the structure is first found in the frequency domain and finally the time-domain response is obtained using inverse Fourier transformation. The EBFs are found so that they satisfy the governing equations in the frequency domain. For the first time, in this paper, we shall present explicit relations for the EBFs in the frequency domain for Cho and Parmerter’s Zig-Zag theory. The coefficients of the EBFs are found through the collocation of the boundary conditions. The dynamic analysis of some composite laminated plates is presented, and the results are compared to those obtained from the dynamic analysis of two-/three-dimensional finite element method (FEM). We shall discuss the capabilities and limitations of the theory and the solution method. The capability of the method, in the analysis of problems excited by high-frequency loads, is shown in the solution of wave propagation problems for which the FEM needs excessive number of elements and thus it is practically impossible to be applied.

Linear signal combination T2 spectroscopy

A technique is presented for performing T2 spectroscopy in magnetic resonance imaging (MRI). It is based on a weighted linear combination of T2 decay data. The data is combined in a manner that acts like a filter on the T2 spectrum. The choice of weighting coefficients determines the filter specifications (e.g. passband/stopband locations, stopband suppression factors). To perform spectroscopy, a series of filters are designed with narrow passbands centered about consecutive regions of the T2 spectrum. This provides an estimate of every region of the spectrum. Taken together, an initial estimate of the full T2 spectrum is thus obtained. However, the filtering process causes a distortion of the estimate relative to the true spectrum. To reduce this distortion, deconvolution is performed. The characteristics of the technique are first evaluated through simulation. The technique is then applied to experimental MRI data to demonstrate practical feasibility. T2 spectroscopy falls into a class of problems requiring inverse transformation with a set of exponential basis functions (i.e. the Laplace Transform). It is demonstrated how the present technique may be applied to problems involving non-exponential basis functions as well.

Solving a nonlinear inverse system of Burgers equations

By applying finite difference formula to time discretization and the cubic B-splines for spatial variable, a numerical method for solving the inverse system of Burgers equations is presented. Also, the convergence analysis and stability for this problem are investigated and the order of convergence is obtained. By using two test problems, the accuracy of presented method is verified. Additionally, obtained numerical results of the cubic B-spline method are compared to trigonometric cubic B-spline method, exponential cubic B-spline method and radial basis function method. Implementation simplicity and less computational cost are the main advantages of proposed scheme compared to previous proposals.

Generalization of the exponential basis for tensor network representations of long-range interactions in two and three dimensions

In one dimension (1D), a general decaying long-range interaction can be fit to a sum of exponential interactions $e^{-\lambda r_{ij}}$ with varying exponents $\lambda$, each of which can be represented by a simple matrix product operator (MPO) with bond dimension $D=3$. Using this technique, efficient and accurate simulations of 1D quantum systems with long-range interactions can be performed using matrix product states (MPS). However, the extension of this construction to higher dimensions is not obvious. We report how to generalize the exponential basis to 2D and 3D by defining the basis functions as the Green's functions of the discretized Helmholtz equation for different Helmholtz parameters $\lambda$, a construction which is valid for lattices of any spatial dimension. Compact tensor network representations can then be found for the discretized Green's functions, by expressing them as correlation functions of auxiliary fermionic fields with nearest neighbor interactions via Grassmann Gaussian integration. Interestingly, this analytic construction in 3D yields a $D=4$ tensor network representation of correlation functions which (asymptotically) decay as the inverse distance ($r^{-1}_{ij}$), thus generating the (screened) Coulomb potential on a cubic lattice. These techniques will be useful in tensor network simulations of realistic materials.

Generalized B functions applied to atomic calculations

During the past years, there has been an increasing interest in the construction of alternative basis sets for electronic systems. .In this work, by the use of generalized exponential functions approach, new efficient exponential type basis sets which include the B functions in the radial part have been constructed. Additional variational flexibility for the radial part of B functions is obtained by replacing the radial variable by rμ, with μ optimisable. Thus, the new variational parameter μ in generalized B functions greatly improves the quality of atomic orbitals in the LCAO calculations. Results obtained using generalized B function basis sets within the minimal basis framework are compared to the results of the standard B functions basis sets and numerical Hartree-Fock values for atomic systems.

Uniform difference method for singularly pertubated delay Sobolev problems

In this paper, a numerical study is made of an initial-boundary value problem for a singularly perturbed delay Sobolev equations (SPDSEs). Here we propose an exponentially fitted method based on finite differences to solve an SPDSE. An exponentially-fitted difference scheme on a uniform mesh, which is accomplished by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form, is presented. We calculate the fitting parameter for an exponentially fitted finite difference scheme corresponding to the problem and establish an error estimate which shows that the method has order of convergence 2 in space and time, independently of the perturbation parameter to the solution of the problem. The stability of the method is discussed. Numerical experiments are performed to support the theoretical results.

Exponential basis and exponential expanding grids third (fourth)-order compact schemes for nonlinear three-dimensional convection-diffusion-reaction equation

This paper addresses exponential basis and compact formulation for solving three-dimensional convection-diffusion-reaction equations that exhibit an accuracy of order three or four depending on exponential expanding or uniformly spaced grid network. The compact formulation is derived with three grid points in each spatial direction and results in a block-block tri-diagonal Jacobian matrix, which makes it more suitable for efficient computing. In each direction, there are two tuning parameters; one associated with exponential basis, known as the frequency parameter, and the other one is the grid ratio parameter that appears in exponential expanding grid sequences. The interplay of these parameters provides more accurate solution values in short computing time with less memory space, and their estimates are determined according to the location of layer concentration. The Jacobian iteration matrix of the proposed scheme is proved to be monotone and irreducible. Computational experiments with convection dominated diffusion equation, Schrödinger equation, Helmholtz equation, nonlinear elliptic Allen–Cahn equation, and sine-Gordon equation support the theoretical convergence analysis.

A numerical solution study on singularly perturbed convection-diffusion nonlocal boundary problem

This important numerical method is given for the numerical solution of singularly perturbed convection-diffusion nonlocal boundary value problem. First, the behavior of the exact solution is analyzed, which is needed for analysis of the numerical solution in later sections. Next, uniformly convergent finite difference scheme on a Shishkin mesh is established, which is based on the method of integral identities with the use exponential basis functions and interpolating quadrature rules with weight and remainder term in integral form. It is shown that the method is first order accurate expect for a logarithmic factor, in the discrete maximum norm. Finally, the numerical results are presented in table and graphs, and these results reveal the validity of the theoretical results of our method.

A spacetime collocation Trefftz method for solving the inverse heat conduction problem

In this article, a novel spacetime collocation Trefftz method for solving the inverse heat conduction problem is presented. This pioneering work is based on the spacetime collocation Trefftz method; the method operates by collocating the boundary points in the spacetime coordinate system. In the spacetime domain, the initial and boundary conditions are both regarded as boundary conditions on the spacetime domain boundary. We may therefore rewrite an initial value problem (such as a heat conduction problem) as a boundary value problem. Hence, the spacetime collocation Trefftz method is adopted to solve the inverse heat conduction problem by approximating numerical solutions using Trefftz base functions satisfying the governing equation. The validity of the proposed method is established for a number of test problems. We compared the accuracy of the proposed method with that of the Trefftz method based on exponential basis functions. Results demonstrate that the proposed method obtains highly accurate numerical solutions and that the boundary data on the inaccessible boundary can be recovered even if the accessible data are specified at only one-fourth of the overall spacetime boundary.