#### Event Title

A polynomial based far-field expansion for treecode algorithm

#### Mentor 1

Lei Wang

#### Location

Union Wisconsin Room

#### Start Date

5-4-2019 1:30 PM

#### End Date

5-4-2019 3:30 PM

#### Description

Computers help us save the calculation time and deal with numerous data. Depending on the algorithm we use, the running times are different. The purpose of this research is to find a more accurate and faster way to calculate the data. We have multiple source points and target points which are both large numbers. The kernel-based summations arise in particle simulations in many physics, biology, and engineering fields involving point masses, point charges, and point vortices. Evaluating the sum by direct summation requires O(N^{ 2} ) operations which is prohibitively expensive when N is large. We are interested in developing fast methods to reduce cost. We are using Lagrange interpolation (standard/barycentric) as both far- and near-field expansion and investigating the trade-off between the accuracy and CPU time. For far- and near-field expansion for the source points and target points, we use the Chebyshev points that make the points clustered around the boundary. The interpolation method gives unique polynomials depending on the numbers of Chebyshev points. We change the interpolation points numbers so that we change the polynomials with the highest power. We apply for this work not only 1-Dimensional problems but also 2-Dimensional and 3-Dimensional problems. The relative error between the exact values and approximation values shows us the correlation between the numbers of the interpolation points and the corresponding approximation values.

A polynomial based far-field expansion for treecode algorithm

Union Wisconsin Room

Computers help us save the calculation time and deal with numerous data. Depending on the algorithm we use, the running times are different. The purpose of this research is to find a more accurate and faster way to calculate the data. We have multiple source points and target points which are both large numbers. The kernel-based summations arise in particle simulations in many physics, biology, and engineering fields involving point masses, point charges, and point vortices. Evaluating the sum by direct summation requires O(N^{ 2} ) operations which is prohibitively expensive when N is large. We are interested in developing fast methods to reduce cost. We are using Lagrange interpolation (standard/barycentric) as both far- and near-field expansion and investigating the trade-off between the accuracy and CPU time. For far- and near-field expansion for the source points and target points, we use the Chebyshev points that make the points clustered around the boundary. The interpolation method gives unique polynomials depending on the numbers of Chebyshev points. We change the interpolation points numbers so that we change the polynomials with the highest power. We apply for this work not only 1-Dimensional problems but also 2-Dimensional and 3-Dimensional problems. The relative error between the exact values and approximation values shows us the correlation between the numbers of the interpolation points and the corresponding approximation values.