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## Approximation with polynomial kernels and SVM classifiers

### Advances in Computational Mathematics (2006-07-01) 25: 323-344 , July 01, 2006

This paper presents an error analysis for classification algorithms generated by regularization schemes with polynomial kernels. Explicit convergence rates are provided for support vector machine (SVM) soft margin classifiers. The misclassification error can be estimated by the sum of sample error and regularization error. The main difficulty for studying algorithms with polynomial kernels is the regularization error which involves deeply the degrees of the kernel polynomials. Here we overcome this difficulty by bounding the reproducing kernel Hilbert space norm of Durrmeyer operators, and estimating the rate of approximation by Durrmeyer operators in a weighted *L*^{1} space (the weight is a probability distribution). Our study shows that the regularization parameter should decrease exponentially fast with the sample size, which is a special feature of polynomial kernels.

## Simulation of liquid crystal elastomers using Chebyshev spectral method with a new preconditioner

### Advances in Computational Mathematics (2015-08-01) 41: 853-879 , August 01, 2015

Liquid crystal elastomers (LCEs) are soft complex materials of potential technological importance because of their remarkable responsiveness to excitations. In the previous work (Zhu et al. Phys. Rev. E *83*, 051703 2011), we proposed a non-local continuum model to explore the dynamical behaviors of LCEs. The preliminary simulation demonstrated that the model can successfully capture the shape changing phenomena and other features of LCEs that were observed in real experiments (Camacho-Lopez et al. Nat. Mat. *3*, 307–310 2004). However, due to the complexity of the governing equations, especially the velocity equation, the simulation is very time-consuming and thus efficient methods are imperatively needed. In this work, we propose a novel preconditioner for solving the velocity equation using Chebyshev spectral collocation method. Different from the well-known finite difference preconditioning (Orszag J. Comput. Phys. *37*, 70–92 1980), the proposed preconditioner is constructed directly from the coefficient matrix of solving the velocity equation, without resorting to other operators such as the finite difference operator. With this preconditioner, very few inner iterations are needed when solving the resulting large scale linear system using GMRES method (Saad and Schultz SIAM J. Sci. STAT. Comput. *7*(3), 856–869 1986). The experiments validate the efficiency of the proposed preconditioner.

## On extensions of wavelet systems to dual pairs of frames

### Advances in Computational Mathematics (2016-04-01) 42: 489-503 , April 01, 2016

It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.

## Characterization of compactly supported refinable splines

### Advances in Computational Mathematics (1995-01-01) 3: 137-145 , January 01, 1995

We prove that a compactly supported spline function*φ* of degree k satisfies the scaling equation
$$
\phi (x) = \sum _{n = 0}^N c(n)\phi (mx - n)
$$
for some integer*m* ≥ 2, if and only if
$$
\phi (x) = \sum _n p(n)B_k (x - n)
$$
where*p*(*n*) are the coefficients of a polynomial*P*(*z*) such that the roots of*P*(*z*)(*z* - 1)^{k+1} TM are mapped into themselves by the mapping*z* →*z*^{m}, and*B*_{k} is the uniform B-spline of degree*k*. Furthermore, the shifts of*φ* form a Riesz basis if and only if*P* is a monomial.

## Cardinal interpolation with general multiquadrics

### Advances in Computational Mathematics (2016-10-01) 42: 1149-1186 , October 01, 2016

This paper studies the cardinal interpolation operators associated with the general multiquadrics, *ϕ*_{α, c}(*x*)=(∥*x*∥^{2} + *c*^{2})^{α},
$x\in \mathbb {R}^{d}$
. These operators take the form
$$\mathcal{I}_{\alpha,c}\mathbf{y}(x) = \sum\limits_{j\in\mathbb{Z}^{d}}y_{j}L_{\alpha,c}(x-j),\quad\mathbf{y}=(y_{j})_{j\in\mathbb{Z}^{d}},\quad x\in\mathbb{R}^{d}, $$
where *L*_{α, c} is a fundamental function formed by integer translates of *ϕ*_{α, c} which satisfies the interpolatory condition
$L_{\alpha ,c}(k) = \delta _{0,k},\; k\in \mathbb {Z}^{d}$
. We consider recovery results for interpolation of bandlimited functions in higher dimensions by limiting the parameter
$c\to \infty $
. In the univariate case, we consider the norm of the operator
$\mathcal {I}_{\alpha ,c}$
acting on *ℓ*_{p} spaces as well as prove decay rates for *L*_{α, c} using a detailed analysis of the derivatives of its Fourier transform,
$\widehat {L_{\alpha ,c}}$
.

## Interpolation error estimates for mean value coordinates over convex polygons

### Advances in Computational Mathematics (2013-08-01) 39: 327-347 , August 01, 2013

In a similar fashion to estimates shown for Harmonic, Wachspress, and Sibson coordinates in Gillette et al. (Adv Comput Math 37(3), 417–439, 2012), we prove interpolation error estimates for the mean value coordinates on convex polygons suitable for standard finite element analysis. Our analysis is based on providing a uniform bound on the gradient of the mean value functions for all convex polygons of diameter one satisfying certain simple geometric restrictions. This work makes rigorous an observed practical advantage of the mean value coordinates: unlike Wachspress coordinates, the gradients of the mean value coordinates do not become large as interior angles of the polygon approach *π*.

## An acceleration method for integral equations by using interpolation post-processing

### Advances in Computational Mathematics (1998-09-01) 9: 117-129 , September 01, 1998

Two post-processing techniques are widely used in the literature in the context of convergence acceleration. One of them is an interpolation technique, used for partial differential equations and integral differential equations, and the other is an iteration technique used for integral equations. These two techniques, interpolation and iteration, are quite different, and the former is simpler. Can we use the interpolation technique for integral equations instead of using the second iteration technique? This report gives a positive answer.

## An adaptive residual local projection finite element method for the Navier–Stokes equations

### Advances in Computational Mathematics (2014-12-01) 40: 1093-1119 , December 01, 2014

This work proposes and analyses an adaptive finite element scheme for the fully non-linear incompressible Navier-Stokes equations. A residual a posteriori error estimator is shown to be effective and reliable. The error estimator relies on a Residual Local Projection (RELP) finite element method for which we prove well-posedness under mild conditions. Several well-established numerical tests assess the theoretical results.

## Learning theory: stability is sufficient for generalization and necessary and sufficient for consistency of empirical risk minimization

### Advances in Computational Mathematics (2006-07-01) 25: 161-193 , July 01, 2006

Solutions of learning problems by Empirical Risk Minimization (ERM) – and almost-ERM when the minimizer does not exist – need to be *consistent*, so that they may be predictive. They also need to be well-posed in the sense of being *stable*, so that they might be used robustly. We propose a statistical form of stability, defined as *leave-one-out* (LOO) *stability*. We prove that for bounded loss classes LOO stability is (a) *sufficient for generalization*, that is convergence in probability of the empirical error to the expected error, for any algorithm satisfying it and, (b) *necessary and sufficient for consistency of ERM*. Thus LOO stability is a weak form of stability that represents a sufficient condition for generalization for symmetric learning algorithms while subsuming the classical conditions for consistency of ERM. In particular, we conclude that a certain form of well-posedness and consistency are equivalent for ERM.

## Efficient preconditioning of linear systems arising from the discretization of hyperbolic conservation laws

### Advances in Computational Mathematics (2001-01-01) 14: 49-73 , January 01, 2001

In this paper, we describe a novel formulation of a preconditioned BiCGSTAB algorithm for the solution of ill-conditioned linear systems *Ax*=*b*. The developed extension enables the control of the residual *r*_{m}=*b*−*Ax*_{m} of the approximate solution *x*_{m} independent of the specific left, right or two-sided preconditioning technique considered. Thereby, the presented modification does not require any additional computational effort and can be introduced directly into existing computer codes. Furthermore, the proceeding is not restricted to the BiCGSTAB method, hence the strategy can serve as a guideline to extend similar Krylov sub-space methods in the same manner. Based on the presented algorithm, we study the behavior of different preconditioning techniques. We introduce a new physically motivated approach within an implicit finite volume scheme for the system of the Euler equations of gas dynamics which is a typical representative of hyperbolic conservation laws. Thereupon a great variety of realistic flow problems are considered in order to give reliable statements concerning the efficiency and performance of modern preconditioning techniques.