Minimax Approximation Research Articles

A minmax approach to adaptive matched field processing in an uncertain propagation environment

Adaptive array processing algorithms have achieved widespread use because they are very effective at rejecting unwanted signals (i.e., controlling sidelobe levels) and in general have very good resolution (i.e., have narrow mainlobes). However, many adaptive high-resolution array processing algorithms suffer a significant degradation in performance in the presence of environmental mismatch. This sensitivity to environmental mismatch is of particular concern in problems such as long-range acoustic array processing in the ocean where the array processor's knowledge of the propagation characteristics of the ocean is imperfect. An adaptive minmax matched field processor is formulated which combines adaptive matched field processing and minmax approximation techniques to achieve the effective interference rejection characteristic of adaptive processors, while limiting the sensitivity of the processor to environmental mismatch. An efficient implementation and alternative interpretation of the processor are developed. The performance of the processor is analyzed using numerical simulations. >

Linear approximation of simple objects

Let S be a family of m convex polygons in the plane with a total number of n vertices and let each polygon have a positive weight associated with it. This paper presents algorithms to solve the weighted minmax approximation and the weighted minsum approximation problems. For the first problem, a line minimizing the maximum weighted orthogonal Euclidean distance to the polygons can be found in O( n 2 log n) time and O( n 2 ) space. The time and space complexities can be reduced to O( n log n) and O( n), respectively, when the weights are equal. For the second problem, a line minimizing the sum of the weighted distances to the polygons can be found in O( nm log m) time and O( n) space. For both problems, we also consider constrained versions of these problems.

On the orthogonality of residual polynomials\newline of minimax polynomial preconditioning

This paper studies polynomials used in polynomial preconditioning for solving linear systems of equations. Optimum preconditioning polynomials are obtained by solving some constrained minimax approximation problems. The resulting residual polynomials are referred to as the de Boor-Rice and Grcar polynomials. It will be shown in this paper that the de Boor-Rice and Grcar polynomials are orthogonal polynomials over several intervals. More specifically, each de Boor-Rice or Grcar polynomial belongs to an orthogonal family, but the orthogonal family varies with the polynomial. This orthogonality property is important, because it enables one to generate the minimax preconditioning polynomials by three-term recursive relations. Some results on the convergence properties of certain preconditioning polynomials are also presented.

Design of two-dimensional FIR filter using hybrid domain approximation

This paper proposes a zero-phase, two-dimensional (2-D) FIR filter design using a hybrid domain where a spatial variable and frequency variable are mixed. For a desired amplitude response given in 2-D frequency plane, the approximation method of 1-D FIR filter design is applied in two steps and the desired response is approximated. First, the approximation coefficients in the hybrid domain are obtained by min-max approximation for one of the frequency variables. Then filter coefficients are obtained by the frequency sampling method or inconsistent Čhebyshev approximation method for another of the variables. Moreover, the error distribution of the designed 2-D filter is evaluated; and, when the magnitude of the error is uneven in the frequency domain, then the method of redesigning is proposed to obtain good approximation. Since this method decomposes the calculation of the 2-D filter into the 1-D case, its design is very simple. Moreover, this method has the advantage of free setting the magnitude of error in the passband and stopband of filter.

Constrained minimax approximation and optimal preconditioned for Toeplitz matrices

A good preconditioner is extremely important in order for the conjugate gradients method to converge quickly. In the case of Toeplitz matrices, a number of recent studies were made to relate approximation of functions to good preconditioners. In this paper, we present a new result relating the quality of the Toeplitz preconditionerC for the Toeplitz matrixT to the Chebyshev norm ∥(f− g)/f∥∞, wheref and g are the generating functions forT andC, respectively. In particular, the construction of band-Toeplitz preconditioners becomes a linear minimax approximation problem. The case whenf has zeros (but is nonnegative) is especially interesting and the corresponding approximation problem becomes constrained. We show how the Remez algorithm can be modified to handle the constraints. Numerical experiments confirming the theoretical results are presented.

Minimax approximation of the inverse of a transform function of measuring devices

Minimax approximation of the inverse of a transform function of measuring devices

Rational Function Approximations for Fermi-Dirac Integrals

Rational function minimax approximations are given for the complete Fermi-Dirac integrals of orders -1/2, 1/2, 3/2, and 5/2. In each case, three sets of approximations are provided with maximum relative error ≃10 -4 , 10 -8 , and 10 -12 , respectively. These approximations can be used to compute the Fermi-Dirac integrals efficiently over the entire range. Approximations to the corresponding inverse functions with an accuracy of ≃10 -4 and 10 -8 are also obtained

A note on planar minimax arc splines

Arc splines are planar, tangent continuous, piecewise curves made of circular arcs and straight line segments. They are important because they can be used as the cutting paths for numerically controlled cutting machinery. This paper presents an algorithm for finding an arc spline that is a minimax approximation to discrete data.

On the minimax approximation in the class of the univariate piecewise constant functions

The paper is concerned with the best approximation of a piecewise constant univariate function with n constant pieces by a similar function with at most k constant pieces, where k < n. We show that for the case of the maximum norm, the general dynamic programming algorithm of complexity O( kn 2) can be considerably improved.

Minimax approximation to stress intensity factors with mathematica

The use of minimax approximation is suggested as the best possible approach during the evaluation of stress intensity factors from the respective integral including the loading distribution and the corresponding weight/Green function. This kind of approximation is generally better than alternative methods. The computer algebra system MATHEMATICA has been used for the derivation of the related SAN (semi-analytical/numerical) results in the case of a single crack under exponential loading distribution (including a symbolic parameter). A simple related procedure (in the MATHEMATICA language) is also presented.

Advection-dispersion simulation by minimization characteristics and alternate direction-explicit methods

An accurate, stable and explicit split-operator scheme is developed for the solution of the unsteady advection-dispersion equation in two dimensions. The advection step is treated by the minimization characteristics method, which generates a class of finite difference schemes through L p -error-norm minimization of the numerical approximation to the exact solution obtained by the method of characteristics. The scheme obtained from the minimax approximation ( p = ∞) is found to he superior to that obtained from the least squares approximation ( p = 2 ) and that from the minimum absolute difference approximation ( p = 1 ). The scheme is also more accurate than the other commonly used backward characteristics schemes based on the same computational effort for comparison. The dispersion step is solved by using the alternate direction-explicit difference to the axial dispersions and the forward-centered difference to the cross-dispersion. In the absence of the cross-dispersion the above scheme is unconditionally stable. The resulting split operator scheme is accurate, computationally efficient, and easy to program and requires small core storage.

Minimax Polynomial Preconditioning for Hermitian Linear Systems

This paper explores the use of polynomial preconditioning for Hermitian positive definite and indefinite linear systems $Ax = b$. Unlike preconditioners based on incomplete factorizations, polynomial preconditioners are easy to employ and well suited to vector and/or parallel architectures. It is shown that any polynomial iterative method may be used to define a preconditioning polynomial, and that the optimum polynomial preconditioner is obtained from a minimax approximation problem. A variety of preconditioning polynomials are then surveyed, including the Chebyshev, de Boor and Rice, Grcar, and bilevel polynomials. Adaptive procedures for each of these polynomials are also discussed, and it is shown that the new bilevel polynomial is particularly well suited for use in adaptive CG algorithms.

On the design of 1D IIR digital filters with coefficients of finite word length

AbstractThis paper presents new linear programming techniques for the design of optimal 1D IIR digital filters. the first method is based on a linear minimax criterion and leads to a linear programming approximation problem whose optimal solution is attained by a new fast algorithm. Next, the linear minimax approximation problem is extended and formulated as two linear integer programming problems which permit the design of 1D IIR digital filters with coefficients of finite word length. the first of these methods is formulated as a 0‐1 integer linear programming problem and its optimal solution is attained rapidly by a new algorithm. This method appears to be suitable for the design of high‐order digital filters. the second method is more general and is formulated as a mixed integer linear programming problem. to solve the design problem efficiently, a new integer tree search algorithm is introduced. the feasibility of the proposed algorithms is illustrated with detailed solutions or numerical examples.

An algorithm for simultaneous magnitude and phase approximation of bivariate rational polynomials

An algorithm is presented for simultaneous magnitude and phase approximation of bivariate rational polynomials. The algorithm is reminiscent of the original differential correction (ODC) algorithm (Barrodale (1972)) for real rational minimax approximation, however the basic iteration of the presented algorithm is not linear but quadratic in the optimization parameters. The stability and the convergence of the algorithm are proven and its application is exemplified.

A new optimal window (signal processing)

An optimal window that can provide the best tradeoff between the peak sidelobe level and the total sidelobe level is presented. Necessary and sufficient conditions are established for the optimal window. A design algorithm is presented, and examples are discussed. The examples demonstrate that the window is capable of much better performance than previously available windows, such as the Hamming window. Relationships between the maximum error and the mean-squared error are explored for the window optimization problem. It is concluded that the minimax and least squares approximations are both fundamentally inefficient. This conclusion is supported by experimental evidence and mathematical analyses. >

Advection simulation by mlnimax‐Characteristics Method

The advective transport of a scalar is simulated by using the minimax-characteristics method, which is an explicit and efficient finite difference scheme derived from the local minimax approximation of the exact solution of the pure advection equation. Fourier mode analysis shows that the method is unconditionally stable, and produces relatively small celerity error and little amplitude dissipation. The scheme compares favorably with the other commonly used backward characteristics schemes: it is better than the scheme using quadratic interpolation (higher accuracy, approximately equal computational effort), and is better than the schemes using cubic interpolation (approximately equal accuracy, less computational effort); it is less accurate than the scheme using Hermitian cubic interpolation, but requires only one-half the computational effort. By interpreting the scheme as a backward characteristics scheme with quadratic approximation of the exact solution over four nodes, the extension of the method in a split-operator approach for advection-dispersion and hydrodynamics modeling in two or three dimensions is straightforward.

Solving nonlinear systems of equations with only one nonlinear variable

We describe a method for solving systems of N + 1 nonlinear equations in N + 1 unknowns y \ e; R and z \ e; R N of the form A( y) z + b( y) = 0, where the ( N + 1) × N matrix A( y) and vector b( y) are functions of y alone. Such equations arise in minimax approximation. We reduce the problem to one equation in y only. An efficient quadratically convergent numerical technique based on Newton's method in one variable is used to solve this equation. Computational details and results are provided, and two generalizations are discussed.

Optimal FIR and IIR Hilbert transformer design via LS and minimax fitting

A novel method for the design of digital finite impulse response (FIR) and infinite impulse response (IIR) Hilbert transformers using the least squares (LS) and the minimax criteria is presented. The LS approximation is performed in the complex domain. Also presented is an iterative extension of the algorithm, which results in a minimax (Chebyshev) approximation, also in the complex domain. For FIR filters the results are the same as those of the optimal methods known from the literature. For the same task, stable IIR filters have also been successfully designed. The procedures proposed are usable for the design of digital filters other than Hilbert transformers, since the desired frequency response can be given point by point. >

Adaptive polynomial preconditioning for hermitian indefinite linear systems

This paper explores the use of polynomial preconditioned CG methods for hermitian indefinite linear systems,Ax=b. Polynomial preconditioning is attractive for several reasons. First, it is well-suited to vector and/or parallel architectures. It is also easy to employ, requiring only matrix-vector multiplication and vector addition. To obtain an optimum polynomial preconditioner we solve a minimax approximation problem. The preconditioning polynomial,C(λ), is optimum in that it minimizes a bound on the condition number of the preconditioned matrix,C(A)A. We also characterize the behavior of this minimax polynomial, which makes possible a thorough understanding of the associated CG methods. This characterization is also essential to the development of an adaptive procedure for dynamically determining the optimum polynomial preconditioner. Finally, we demonstrate the effectiveness of polynomial preconditioning in a variety of numerical experiments on a Cray X-MP/48. Our results suggest that high degree (20–50) polynomials are usually best.

A program which implements a new algorithm for the solution of the linear minimax approximation problem

This paper presents a Turbo-Basic program that implements a revised version of a new algorithm for the fast and optimum solution of the general linear minimax approximation problem. The algorithm proceeds iteratively and terminates when the global minimum value of the objective function is reached. The initial point may be selected arbitrarily or it may be optimally determined through a linear method to speed up algorithmic convergence. This algorithm is much faster and requires less storage than other approximation techniques. The program is implemented on an IBM-PC AT and tested by a number of approximation problems. Analytical examples are presented to illustrate how the program is used and the effectiveness of the algorithm.