Summary of Research Interests
My research interests include partial differential equations (PDEs), integral equations, and numerical analysis, which together lay the theoretical and computational framework for modeling a wide array of problems in the physical and biological sciences, and also economics and finance.
One of the major themes of my work is the study of nonlocal models, which, unlike standard PDE models, look to capture long-range interactions. The particular models I have studied accomplish this by using integration, and the reason for using such models is to meaningfully track physical phenomena that are inherently discontinuous, such as the spontaneous formation of cracks in a solid. I routinely use tools from functional analysis and the calculus of variations to study the mathematical properties of such problems, such as the existence and uniqueness of solutions, as well as the qualitative comparison of nonlocal and classical PDE models through Gamma-convergence and related techniques. Specific classes of problems I have studied include optimal control and optimal design problems, where the constraints involve a nonlocality.
A second major theme of my work is the use of the finite element method to numerically approximate solutions for problems in the calculus of variations and optimal control, including those which feature nonlocalities. The idea behind these approximations is to create a series of meshes that approximate the domain that the original, "continuous" problem is posed on, and then posing and solving finite-dimensional problems posed on these meshes. Ideally, one can obtain a convergence rate depending on parameters of the family of meshes used, but in some cases this is not possible, including in nonlocal problems where the family of interaction kernels is not assumed to have much structure. More recently, I have begun investigating pointwise error estimates for finite element approximations of optimal recovery problems. These problems involve being given a collection of observations for point values of a function that solves a given PDE (without being provided a boundary condition), and then estimating what the solution looks like on the remainder of the domain.
Simulating these problems using various programming languages and finite element libraries has also been an important aspect of my work. This skill is important not only for verifying theoretical results, but also for giving applied mathematicians, scientists, and engineers a concrete visualization of how to apply the theoretical ideas I develop in practice. To accomplish this goal for various projects, I have gained experience programming in MATLAB, C++, and Python. In particular, I have learned how to use multiple different finite element libraries: Dealii, FreeFEM++, and PyNucleus.
Finally, I have mentored several undergraduates on projects that mostly span analytic number theory and combinatorics, which I consider to be secondary research interests. Speaking from personal experience, I understand the importance of student involvement in research projects very early in their careers as mathematicians and wish to continue mentoring students in the future.
Publication List
Peer-Reviewed Publications
Andrea Bonito, Alan Demlow, Joshua M. Siktar, “On a Pointwise Error Estimate for Riesz Representers in Optimal Recovery Problems” (in preparation).
Javier Cueto, Joshua M. Siktar: “On the Convergence of Solutions for Nonlocal Optimal Control Problems with Varying Fractional Parameter” (in preparation).
Xiaobing Feng, Joshua M. Siktar: "Gamma-Convergence of an Enhanced Finite Element Method for Manià's Problem Exhibiting the Lavrentiev Phenomenon" (submitted).
Tadele Mengesha, Abner J. Salgado, Joshua M. Siktar: “Asymptotically Compatible Schemes for Nonlocal Optimal Design Problems” (submitted).
Joshua M. Siktar: “Existence of Solutions for Fractional Optimal Control Problems with Super-quadratic Controls” (submitted).
Steven J. Miller, Fei Peng, Tudor Popescu, Joshua M. Siktar, Nawapan Wattanawanichkul: “Walking to Infinity Along Some Number Theory Sequences," INTEGERS, Vol. 24, September 2024.
Tadele Mengesha, Abner J. Salgado, Joshua M. Siktar: “On the Optimal Control of a Linear Peridynamics Model,” Applied Mathematics and Optimization, Vol. 88, August 2023.
Amelia Gilson, Hadley Killen, Tamás Lengyel, Steven J. Miller, Nadia Razek, Joshua M. Siktar, Liza Sulkin: “Zeckendorf’s Theorem Using Indices in an Arithmetic Progression,” The Fibonacci Quarterly, Vol. 59 No. 4, December 2021.
Vedant Bonde, Joshua M. Siktar, “On the Combinatorics of Placing Balls into Ordered Bins,” INTEGERS, Vol. 21, August 2021.
Evan Fang, Jonathan Jenkins, Zack Lee, Daniel Li, Ethan Lu, Steven J. Miller, Dilhan Salgado, Joshua M. Siktar: “Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice,” The Fibonacci Quarterly, Vol. 58 No. 3, September 2020.
Eric Chen, Robin Chen, Lucy Guo, Steven Jiang, Steven Miller, Joshua M. Siktar, Peter Yu: “Gaussian Behavior in Zeckendorf Decompositions from Lattices,” The Fibonacci Quarterly, Vol. 57 No. 3, September 2019.
Joshua M. Siktar: “Recasting the Proof of Parseval’s Identity,” Turkish Journal of Inequalities, Issue 3 Vol. 1, July 2019.
PhD Dissertation
Joshua M. Siktar, "Asymptotic Compatibility of Parameterized Nonlocal Optimal Control and Design Problems," PhD Dissertation, University of Tennessee-Knoxville, 2024.