Joseph Kolibal

A localized approach for the method of approximate particular solutions

The method of approximate particular solutions (MAPS) has been recently developed to solve various types of partial differential equations. In the MAPS, radial basis functions play an important role in approximating the forcing term. Coupled with the concept of particular solutions and radial basis functions, a simple and effective numerical method for solving a large class of partial differential equations can be achieved. One of the difficulties of globally applying MAPS is that this method results in a large dense matrix which in turn severely restricts the number of interpolation points, thereby affecting our ability to solve large-scale science and engineering problems. In this paper we develop a localized scheme for the method of approximate particular solutions (LMAPS). The new localized approach allows the use of a small neighborhood of points to find the approximate solution of the given partial differential equation. In this paper, this local numerical scheme is used for solving large-scale problems, up to one million interpolation points. Three numerical examples in two-dimensions are used to validate the proposed numerical scheme.

MALDI-TOF baseline drift removal using stochastic bernstein approximation

Stochastic Bernstein (SB) approximation can tackle the problem of baseline drift correction of instrumentation data. This is demonstrated for spectral data: matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF) data. Two SB schemes for removing the baseline drift are presented: iterative and direct. Following an explanation of the origin of the MALDI-TOF baseline drift that sheds light on the inherent difficulty of its removal by chemical means, SB baseline drift removal is illustrated for both proteomics and genomics MALDI-TOF data sets. SB is an elegant signal processing method to obtain a numerically straightforward baseline shift removal method as it includes a free parameter that can be optimized for different baseline drift removal applications. Therefore, research that determines putative biomarkers from the spectral data might benefit from a sensitivity analysis to the underlying spectral measurement that is made possible by varying the SB free parameter. This can be manually tuned (for constant) or tuned with evolutionary computation (for).

Fractal image error analysis

Abstract Fractal techniques which assess the spatial component of an image use the observation that image complexity is a direct consequence of the operation of many different spatial processes acting to produce images over a wide range of spatial scales. Precisely because fractals inherently involve enormous levels of multiple scale phenomenon, fractal analysis techniques afford new prospects for dealing with spatial images and fractal geometry is now being applied to a variety of spatial problems, and to a limited extent, to remote sensing data. In this study the focus is on assessing the utility of fractal dimension to investigating applications involving imaging of spatial, multi-scale image data. This means understanding quantitatively the effects of discretization errors and signal noise on the determination of the fractal dimension of an image set. The accuracy and efficiency of three methods for estimating the fractal dimension of an image are evaluated: Images of the Cantor set are used to measure accuracy of the isarithm, the triangular prism method and the variogram method. Overall, of these methods, the triangular prism is found to be the most robust in regard to noise rejection and accuracy while at the same time, this algorithm also was computationally the most efficient.

The Novel Stochastic Bernstein Method of Functional Approximation

The stochastic Bernstein method (not to be confused with the Bernstein polynomials) is a novel and significantly improved non-polynomial global method of signal processing that is proving very useful for interpolating and for approximating data. It arose as an obvious extension of the work of Bernstein (it preserves some of the remarkable properties of the Bernstein polynomials). However, this extension means that stochastic interpolation takes on its own properties and additionally can replace the error function by other functions such as the arctangent. The method has a free parameter sigma known as its diffusivity that can be easily optimized with adaptivity and can interpolate or approximate non-uniformly distributed input data - something that is very awkward to set up with other methods. Adaptivity can also reverse engineer the non-uniformly distributed input data that best recovers a function. This short paper provides an introduction to the new mathematical method that should find wide application in many areas of science and engineering

Genetic programming of the stochastic interpolation framework: convection–diffusion equation

The stochastic interpolation (SI) framework of function recovery from input data comprises a de-convolution step followed by a convolution step with row stochastic matrices generated by a mollifier, such as a probability density function. The choice of a mollifier and of how it gets weighted, offers unprecedented flexibility to vary both the interpolation character and the extent of influence of neighbouring data values. In this respect, a soft computing method such as a genetic algorithm or heuristic method may assist applications that model complex or unknown relationships between data by tuning the parameters, functional and component choices inherent in SI. Alternatively or additionally, the input data itself can be reverse engineered to recover a function that satisfies properties, as illustrated in this paper with a genetic programming scheme that enables SI to recover the analytical solution to a two-point boundary value convection–diffusion differential equation. If further developed, this nascent solution method could serve as an alternative to the weighted residual methods, as these are known to have inherent mathematical difficulties.

Engineering presentation of the stochastic interpolation framework and its applications

The paper is an engineering exposition of the Stochastic Interpolation Framework, a novel mathematical approach to data regularization, which recovers a function from input data that is a representation of this data. The framework is an area-based method that comprises a two-step procedure: de-convolution and convolution, involving row-stochastic matrices. Varying the extent of convolution with respect to de-convolution in the framework obtains a gamut of functional recovery ranging from interpolation to approximation, to peak sharpening. Construction of the row stochastic matrices is achieved by means of a mollifier, a positive function which serves as the generator of the row space of these matrices. The properties of the recovered function will depend on the choice of this mollifier. For example, only if the mollifier is differentiable so is the recovered function, and the framework can obtain derivatives anywhere in the domain. The mollifier can be a probability distribution function. Thus, the framework connects interpolation to statistical analysis. Two novel applications in image analysis illustrate the potential of the framework for security applications: as an alternative method of lossy image compression, and as an alternative method to zoom-up an image.

Novel Algorithm for MALDI-TOF Baseline Drift Removal

Baseline drift is an endemic problem in matrix-assisted laser desorption ionization time of flight mass spectrometry (MALDI-TOF), a device frequently used in proteomics investigations and in selected genomics work. Following an explanation of the origin of this baseline drift that sheds light on the inherent difficulty of its removal by chemical means, the stochastic Bernstein function approximation (SB), a new signal processing method, is developed into a procedure to obtain a numerically straightforward baseline shift removal. This is successfully applied to proteomics and genmomics MALDI-TOF spectra. Evolutionary computation (EC) can discover (optimize, tune) aspects of the algorithm, for example, the free parameter σ (x) of the SB method. Since baseline drift affects many other types of instrumentation for poorly understood reasons, EC suggests an approach to customize the baseline removal algorithm.

Image Analysis by Means of the Stochastic Matrix Method of Function Recovery

The recently patented stochastic matrix method of function recovery offers workable alternatives to traditional methods of image analysis. This paper illustrates its application to image compression and its application to image enhancement (image zoom). In the former application, it appears to be competitive with JPEG DCT with respect to file size but with the added advantage that it does not suffer from artifacts of that coder. In the latter application, it appears to be clearly superior to the bi-cubic interpolation that is used by popular commercial graphics packages. An important and characteristic property of the stochastic matrix method (SMM) of function recovery is its free parameter sigma that can be optimized, e.g. by an intelligent system, to change the nature of the image analysis.

Implications of a Novel Family of Stochastic Methods for Function Recovery

We illustrate the use of a novel family of stochastic methods for function recovery and illustrate their applicability to several problems involving one and two dimensional data.

A NUMERICAL METHOD FOR 1D TIME-DEPENDENT SCHRÖDINGER EQUATION USING RADIAL BASIS FUNCTIONS

This paper proposes a new numerical method to solve the 1D time-dependent Schrodinger equations based on the finite difference scheme by means of multiquadrics (MQ) and inverse multiquadrics (IMQ) radial basis functions. The numerical examples are given to confirm the good accuracy of the proposed methods.