# Bulletin of the American Physical Society

# Fall 2022 Meeting of the APS Eastern Great Lake Section and the Michigan Section of AAPT: Pushing Boundaries in Physics and Education

## Volume 67, Number 16

## Friday–Saturday, October 21–22, 2022; Lawrence Technological University, Southfield, Michigan

### Session J04: Astro and Mathematical Physics

8:15 AM–9:30 AM,
Saturday, October 22, 2022

Lawrence Technological University
Room: S217

Chair: Cynthia Aku-Leh, ISciences

### Abstract: J04.00001 : Numerical Methods for Computing Forward and Inverse Laplace Transform For discrete and continuous signals*

8:15 AM–8:30 AM

#### Presenter:

Yueyang Shen

(University of Michigan)

#### Authors:

Yueyang Shen

(University of Michigan)

Yupeng Zhang

(University of Wisconsion-Madison)

Ivo D Dinov

(University of Michigan)

#### Collaboration:

SOCR (Statistical Online Computation Resource)

The classical Laplace transform may also be viewed as a bounded and invertible linear operator, a unitary isomorphism between positive real square integrable functions and the complex Hardy space on the right half plane. A proper decay factor γ may be introduced to enforce square integrability e

^{-γt}f(t)∈L

^{2}([0,∞)) and the ILT can be computed exactly via the Bromwhich integral L

^{-1}(F(s))=∫

^{γ+i∞}

_{γ-i∞ }e

^{st}F(s)ds. However, for some functions and observed discrete signals, the precise function class is often intractable. In this work, we propose a numerical LT-ILT computing framework, implemented in R, which is approximately invertible for a large class of signals, exhibits sufficient robustness, and facilitates numerical computations through appropriate parameter estimations and signal approximations.

In addition, we will present various challenges and open problems relating to random matrix theory, harmonic analysis, formulation of LT-ILT on groups, and a Clifford algebra approach to define Laplace transform to higher spacetime dimensions. Theoretical aspects, related conjectures, and empirical evaluation of the algorithm will be discussed.

*The list of complete funding resource is documented on https://socr.umich.edu/html/SOCR_Funding.html

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