Computation of connection coefficients and measure modifications for orthogonal polynomials

Narayan, Akil; Hesthaven, Jan S.

doi:10.1007/s10543-011-0363-z

Cited by 9 publications

(9 citation statements)

References 30 publications

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“…The statistical significance of main effects and interactions was assessed using Wald statistics. We used orthogonal polynomials, computed via the Cholesky decomposition to avoid high correlation between original “time” scale variables, which can cause estimation problems (Narayan & Hesthaven, ). SAS 9.3 (SAS Institute Inc., Cary, NC, USA) was used for the analysis.…”

Section: Methodsmentioning

confidence: 99%

Telehealth Monitoring of Patients with Schizophrenia and Suicidal Ideation

Kasckow

Gao

Hanusa

et al. 2015

Suicide & Life Threat Behav

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A telehealth system was developed to monitor risk following hospitalization for suicidal ideation. We hypothesized that 3 months of telehealth monitoring will result in a greater reduction in suicidal ideation. Veterans with schizophrenia admitted with recent suicidal ideation and/or a suicidal attempt were recruited into a discharge program of VA Usual Care with daily Health Buddy© monitoring (HB) or Usual Care (UC) alone. Fifteen of 25 were randomized to HB and 10 received UC. Daily adherence in the use of the HB system during months 1-3 was, respectively, 86.9%, 86.3%, and 84.1%. There were significant improvements in Beck Scale for Suicide Ideation scores in HB participants. There were no changes in depressive symptoms. Telehealth monitoring for this population of patients appears to be feasible.

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Section: Methodsmentioning

confidence: 99%

Telehealth Monitoring of Patients with Schizophrenia and Suicidal Ideation

Kasckow

Gao

Hanusa

et al. 2015

Suicide & Life Threat Behav

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“…We note that O(N log(N )) fast algorithms can be applied to obtain B k.j ; see, e.g., [2] for β = δ = 0 and [27,32] for β, δ > −1 when x j are Gauss-Chebyshev quadrature points δ = γ = −1/2.…”

Section: Regularitymentioning

confidence: 99%

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Zhang¹,

Zeng²,

Karniadakis³

2015

SIAM J. Numer. Anal.

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We present optimal error estimates for spectral Petrov-Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov-Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates. Introduction.Numerical methods for fractional differential equations have been investigated for decades; see, e.g., [8,12,33]. However, spectral methods for these equations have a rather short history since the solutions usually have some singular lower-order derivatives. The recent investigation of spectral methods for these singular problems is motivated by the following facts. First, all numerical methods for these problems are nonlocal as spectral methods are. For example, in finite difference methods, see, e.g., [33], and in finite element methods, see, e.g., [13,18], data at almost all grids or all elements are exploited to approximate the integral operators at one grid or element. These methods are still nonlocal, even when a "fixed memory principle" is applied to reduce the use of all data; see, e.g., [15,19]. Second, spectral methods lead to high-order accuracy when the solutions are smooth; see, e.g., [7,24] for integerorder differential equations. For fractional differential equations with weakly singular kernels, solutions can be smooth even when some inputs of the equations are singular.Consider the following fractional ordinary differential equation (FODE) over the interval I = (−1, 1):

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“…where ppxq is a polynomial, non-negative on the support of µ. This is a well-studied problem [10,6,7,19]. In particular, one may reduce the problem to iterating over modifications by linear and quadratic polynomials.…”

Section: Measure Modificationsmentioning

confidence: 99%

“…, M`A`2pn´jq with modification factor pu´px j,n´x qq 2 . (19), and return p F c n`x ; µ pα,ρq˘g iven by (20). Algorithm 3: Computation of p F c n pxq for µ pα,ρq HF corresponding to a half-line Freud weight.…”

Section: Computing Pmentioning

confidence: 99%

Computation of induced orthogonal polynomial distributions

Narayan¹

2018

etna

Self Cite

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We provide a robust and general algorithm for computing distribution functions associated to induced orthogonal polynomial measures. We leverage several tools for orthogonal polynomials to provide a spectrally-accurate method for a broad class of measures, which is stable for polynomial degrees up to at least degree 1000. Paired with other standard tools such as a numerical root-finding algorithm and inverse transform sampling, this provides a methodology for generating random samples from an induced orthogonal polynomial measure. Generating samples from this measure is one ingredient in optimal numerical methods for certain types of multivariate polynomial approximation. For example, sampling from induced distributions for weighted discrete least-squares approximation has recently been shown to yield convergence guarantees with a minimal number of samples. We also provide publicly-available code that implements the algorithms in this paper for sampling from induced distributions.

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Computation of connection coefficients and measure modifications for orthogonal polynomials

Cited by 9 publications

References 30 publications

Telehealth Monitoring of Patients with Schizophrenia and Suicidal Ideation

Telehealth Monitoring of Patients with Schizophrenia and Suicidal Ideation

Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Computation of induced orthogonal polynomial distributions

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