Karlis Kaugars

Loop integration results using numerical extrapolation for a non-scalar integral

Abstract Loop integration results have been obtained using numerical integration and extrapolation. An extrapolation to the limit is performed with respect to a parameter in the integrand which tends to zero. Results are given for a non-scalar four-point diagram. Extensions to accommodate loop integration by existing integration packages are also discussed. These include: using previously generated partitions of the domain and roundoff error guards.

On Iterated Numerical Integration

We revisit the iterated numerical integration method and show that it is extremely efficient in solving certain classes of problems. A multidimensional integral can be approximated by a combination of lower-dimensional or one-dimensional adaptive methods iteratively. When an integrand contains sharp ridges which are not parallel with any axis, iterated methods often outperform adaptive cubature methods in low dimensions. We use examples to support our analysis.

Grid-based numerical integration and visualization

A grid service, called integration service (PI), is used to solve numerical integration problems which are computationally intensive. Remote visualization helps monitor the progress of the computation, and gives a vivid view of the shape of a multidimensional function. The data are filtered by the server and transferred to the client, which is responsible for visualization mapping and rendering. Both numerical integration and its visualization are Web service-based to ensure interoperability.

Performance and Irregular Behavior of Adaptive Task Partitioning

We study the effect of irregular function behavior and dynamic task partitioning on the parallel performance of the adaptive multivariate integration algorithm currently incorporated in PARINT. In view of the implicit hot spots in the computations, load balancing is essential to maintain parallel efficiency. A convergence model is given for a class of singular functions. Results are included for the computation of the cross section of a particle interaction. The adaptive meshes produced by PARINT for these problems are represented using the PARVIS visualization tool.

Dimensional recursion for multivariate adaptive integration

We consider multivariate integrals which can be expressed as iterated integrals over product regions. The iteration over the dimensions is applied recursively for a numerical evaluation. We evaluate a scheme for setting the tolerated error in the interface between the integration levels and address the efficiency of the resulting method with respect to time and space requirements.

DISTRIBUTED ADAPTIVE MULTIVARIATE FUNCTION VISUALIZATION

In this article, we introduce a new function visualization method and demonstrate that numerical integration and visualization of multi-dimensional functions are closely related. Adaptive numerical integration is utilized to reduce the number of function evaluations, and generate time series data. The integration region is partitioned into a uniform grid. A grid cell can be sampled many times, or is not sampled at all, depending on the function properties and the integration rule. Function properties are extracted during the process of function evaluation. An aging technique helps visualize functions by retaining the most recently sampled areas and making the older ones transparent. This also results in giving the non-smooth areas more attention than the smooth areas. The new function visualization method gives a view of the whole function while elaborating on important areas such as ridges and troughs, which are critical in many fields, including numerical integration. A Grid service, called Integration Service, is used to solve computationally intensive integration problems. Remote visualization based on the adaptive method helps monitor the progress of a computation, and can be utilized for computational steering. The data are filtered by the server and transferred to the client, which is responsible for visualization mapping and rendering.

A fast integration method and its application in a medical physics problem

A numerical integration method is proposed to evaluate a very computationally expensive integration encountered in the analysis of the optimal dose grid size for the intensity modulated proton therapy (IMPT) fluence map optimization (FMO). The resolution analysis consists of obtaining the Fourier transform of the 3-dimensional (3D) dose function and then performing the inverse transform numerically. When the proton beam is at an angle with the dose grid, the Fourier transform of the 3D dose function contains integrals involving oscillatory sine and cosine functions and oscillates in all of its three dimensions. Because of the oscillatory behavior, it takes about 300 hours to compute the integration of the inverse Fourier transform to achieve a relative accuracy of 0.1 percent with a 2 GHz Intel PC and using an iterative division algorithm. The proposed method (subsequently referred to as table method) solves integration problems with a partially separated integrand by integrating the inner integral for a number of points of the outer integrand and finding the values of other evaluation points by interpolation. The table method reduces the computational time to less than one percent for the integration of the inverse Fourier transform. This method can also be used for other integration problems that fit the method application conditions.

Interdisciplinary applications of mathematical modeling

We demonstrate applications of numerical integration and visualization algorithms in diverse fields including psychological modeling (biometrics); in high energy physics for the study of collisions of elementary particles; and in medical physics for regulating the dosage of proton beam radiation therapy. We discuss the problems and solution methods, as supported by numerical results.