arXiv subjects

Joshua M. Siktar

Publications and source records attributed to Joshua M. Siktar.

16 recordsLinked to original sources

arXiv2026

On convergence of solutions to nonlocal optimal control problems with quasi-minimization constraints

This paper studies a new class of nonlocal optimal control problems where the constraints involve approximations of minimizers of quasiconvex energies. These problems are parameterized by a fractional parameter $s \in (0, 1)$ and a horizon parameter $δ> 0$, and the energy density depends on a nonlocal fractional gradient. Here, the constraint consists of finding quasi-minimizers with respect to the quasiconvex energy. Despite the fact that the energies of interest may not have unique minimizers, we may prove the existence of solutions to this class of control problems. The constraint in Cueto-Siktar 2026 was finding global minimizers of the energy, and this problem's main limitation was an inability to prove convergence of solutions for the nonlocal control problems to those of a corresponding local, PDE-constrained optimal control problem. While this issue arises from the lack of uniqueness of minimizers for the constraining energy, we get stronger convergence results with our new choice of constraints. Namely, we obtain convergence of minimizers for nonlocal optimal control problems that have a general cost functional depending on the nonlocal gradient.

math.OC↗

arXiv2026

A Mountain-Pass Algorithm for Nonlocal Problems with Super-quadratic Nonlinearities

In this paper we consider a nonlinear equation $-\mathcal{L} u(x) = f(x, u(x))$ with a super-quadratic nonlinearity, $f$, and a nonlocal operator, $\mathcal{L}$, generated by a special class of radially symmetric $L^1$ convolution kernels with finite second moments. The assumptions on this operator are mild and allow for a variety of kernels used in biological and physical applications, including kernels with algebraic decay and sign changing kernels. Using the strong nonlinearities present in the equation, we prove the existence of nontrivial solutions using the classical Mountain Pass Theorem, a central result in minimax theory that equates solutions of our equation to critical points of a corresponding energy functional. This existence result holds with both homogeneous nonlocal Dirichlet and nonlocal Neumann boundary conditions. We supplement these theoretical results with numerical simulations for various nonlinearities with odd maximal degree in the unknown $u$. The numerical scheme exploits the resulting energy landscape which allows one to adapt a gradient descent algorithm.

math.AP↗

arXiv2026

Localization for nonlocal gradient-based optimal control problems

In this paper we consider optimal control problems in the nonlocal function space framework of Bellido-2023, where there are two different parameters: a horizon parameter $δ> 0$; and a fractional parameter $s \in (0, 1)$. The constraints are given in the form of minimizing an energy density, and we will focus on two particular cases: the well-posed case where the underlying energy density is convex and is given by the nonlocal $p$-Laplacian; and a more general poly/quasiconvex energy for which minimizers exist but may not be unique. The study is concluded by analyzing the approximation to local problems in two parallel ways, either taking the fractional parameter $s$ to $1$ or the horizon parameter $δ$ to $0$.

math.OC↗

arXiv2026

Alleviating missing boundary conditions in elliptic partial differential equations using interior point measurements

We consider an optimal recovery problem for the Poisson problem when the boundary data is unknown. Compensating information is provided in the form of a finite number of measurements of the solution. A finite element algorithm for this problem was given in Binev et al. (2024), where measurements were assumed to be either bounded linear functionals of the solution or point measurements at locations lying anywhere in the closure of the computational domain. In contrast, we focus on the case of point measurements at locations lying in the interior of the domain. This lowers the regularity requirements placed on the solution. Also, a key ingredient in the recovery process is the finite element approximation of Riesz representers associated with the measurements. Our main result is a pointwise error estimate for the Riesz representers. We apply this to obtain improved estimates which measure the performance of the recovery algorithm in various norms.

math.NA↗

arXiv2025

$Γ$-convergence of an Enhanced Finite Element Method for Manià's and Foss's Problems Exhibiting the Lavrentiev Gap Phenomenon

It is well-known that numerically approximating calculus of variations problems possessing a Lavrentiev Gap Phenomenon (LGP) is challenging, and the standard numerical methodologies, such as finite element, finite difference, and discontinuous Galerkin methods, fail to give convergent methods because they cannot overcome the gap. This paper is a continuation of a 2016 paper by Feng and Schnake, where a promising enhanced finite element method was proposed to overcome the LGP in the classical Manià's problem. The first goal of this paper is to provide a complete $Γ$-convergence proof for this enhanced finite element method, hence, establishing a theoretical foundation for the method. The crux of the convergence analysis is taking advantage of the regularity of the minimizer and viewing the minimization problem as posed over the fractional Sobolev space $W^{1 + s, p}(0, 1)$ (for $s > 0$) rather than the original admissible space $W^{1, p}(0, 1)$. The second goal is to extend the enhanced finite element method to the two-dimensional Foss's problem from nonlinear elasticity, which is also known to possess the LGP, and to establish its $Γ$-convergence as well.

math.NA↗

arXiv2024

Asymptotic compatibility of parametrized optimal design problems

We study optimal design problems where the design corresponds to a coefficient in the principal part of the state equation. The state equation, in addition, is parameter dependent, and we allow it to change type in the limit of this (modeling) parameter. We develop a framework that guarantees asymptotic compatibility, that is unconditional convergence with respect to modeling and discretization parameters to the solution of the corresponding limiting problems. This framework is then applied to two distinct classes of problems where the modeling parameter represents the degree of nonlocality. Specifically, we show unconditional convergence of optimal design problems when the state equation is either a scalar-valued fractional equation, or a strongly coupled system of nonlocal equations derived from the bond-based model of peridynamics.

math.OC↗

arXiv2024

Composite Numbers in an Arithmetic Progression

One challenge (or opportunity!) that many instructors face is how varied the backgrounds, abilities, and interests of students are. In order to simultaneously instill confidence in those with weaker preparations and still challenge those able to go faster, an instructor must be prepared to give problems of different difficulty levels. Using Dirichlet's Theorem as a case study, we create and discuss a family of problems in number theory that highlight the relative strengths and weaknesses of different ways to approach a question and show how to invite students to extend the problems and explore research-level mathematics.

math.HO↗

arXiv2024

Existence of Solutions for Fractional Optimal Control Problems with Superlinear-subcritical Controls

This paper gives an existence result for solutions to an elliptic optimal control problem based on a general fractional kernel, where the admissible controls come from a class satisfying both a growth bound and a superlinear-subcritical condition. Each admissible control is known to produce a nontrivial corresponding state by applying the Mountain Pass Theorem to fractional equations. The main theoretical contribution is the construction of a suitable set of admissible controls on which the the standard existence theory for control problems with linear and semi-linear state constraints can be adapted. Extra care is taken to explain what new difficulties arise for these types of control problems, to justify the limitations of this theory. For completeness, the corresponding local elliptic control problem is also studied.

math.OC↗

arXiv2024

Walking to Infinity Along Some Number Theory sequences

An interesting open conjecture asks whether it is possible to walk to infinity along primes, where each term in the sequence has one digit more than the previous. We present different greedy models for prime walks to predict the long-time behavior of the trajectories of orbits, one of which has similar behavior to the actual backtracking one. Furthermore, we study the same conjecture for square-free numbers, which is motivated by the fact that they have a strictly positive density, as opposed to primes. We introduce stochastic models and analyze the walks' expected length and frequency of digits added. Lastly, we prove that it is impossible to walk to infinity in other important number-theoretical sequences or on primes in different bases.

math.NT↗

arXiv2023

On the Optimal Control of a Linear Peridynamics Model

We study a non-local optimal control problem involving a linear, bond-based peridynamics model. In addition to existence and uniqueness of solutions to our problem, we investigate their behavior as the horizon parameter $δ$, which controls the degree of nonlocality, approaches zero. We then study a finite element-based discretization of this problem, its convergence, and the so-called asymptotic compatibility as the discretization parameter $h$ and the horizon parameter $δ$ tend to zero simultaneously.

math.OC↗

arXiv2022

On the Enumeration and Asymptotic Analysis of Fibonacci Compositions

We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where summands have a maximum possible value. Then, we consider asymptotic properties of Fibonacci compositions, comparing them to compositions whose terms come from positive linear recurrence sequences. Finally, we consider analogues of these results where we do not allow the use of a certain number of consecutive Fibonacci numbers starting from $F_2 = 1$.

math.NT↗

arXiv2021

On the Combinatorics of Placing Balls into Ordered Bins

In this paper, we use techniques of enumerative combinatorics to study the following problem: we count the number of ways to split $n$ balls into nonempty, ordered bins so that the most crowded bin has exactly $k$ balls. We find closed forms for three of the different cases that can arise: $k > \frac{n}{2}$, $k = \frac{n}{2}$, and when there exists $j < k$ such that $n = 2k + j$. As an immediate result of our proofs, we find a closed form for the number of positive integer solutions to $x_1 + x_2 + \dots + x_{\ell} = n$ with the attained maximum of $\{x_1, x_2, \dots, x_{\ell}\}$ being equal to $k$, when $n$ and $k$ have one of the aforementioned algebraic relationships to each other. The problem is generalized to find a formula that enumerates the total number of ways without specific conditions on $n, \ell, k$. Subsequently, various additional identities and estimates related to this enumeration are proven and interpreted.

math.CO↗

arXiv2020

Zeckendorf's Theorem Using Indices in an Arithmetic Progression

Zeckendorf's Theorem states that any positive integer can be uniquely decomposed into a sum of distinct, non-adjacent Fibonacci numbers. There are many generalizations, including results on existence of decompositions using only even indexed Fibonacci numbers. We extend these further and prove that similar results hold when only using indices in a given arithmetic progression. As part of our proofs, we generate a range of new recurrences for the Fibonacci numbers that are of interest in their own right.

math.NT↗

arXiv2020

Central Limit Theorems for Compound Paths on the 2-Dimensional Lattice

Zeckendorf proved that every integer can be written uniquely as a sum of non-consecutive Fibonacci numbers $\{F_n\}$, and later researchers showed that the distribution of the number of summands needed for such decompositions of integers in $[F_n, F_{n+1})$ converges to a Gaussian as $n\to\infty$. Decomposition problems have been studied extensively for a variety of different sequences and notions of a legal decompositions; for the Fibonacci numbers, a legal decomposition is one for which each summand is used at most once and no two consecutive summands may be chosen. Recently, Chen et al. [CCGJMSY] generalized earlier work to $d$-dimensional lattices of positive integers; there, a legal decomposition is a path such that every point chosen had each component strictly less than the component of the previous chosen point in the path. They were able to prove Gaussianity results despite the lack of uniqueness of the decompositions; however, their results should hold in the more general case where some components are identical. The strictly decreasing assumption was needed in that work to obtain simple, closed form combinatorial expressions, which could then be well approximated and led to the limiting behavior. In this work we remove that assumption through inclusion-exclusion arguments. These lead to more involved combinatorial sums; using generating functions and recurrence relations we obtain tractable forms in $2$ dimensions and prove Gaussianity again; a more involved analysis should work in higher dimensions.

math.NT↗

arXiv2019

Recasting the Proof of Parseval's Identity

We generalize aspects of Fourier Analysis from intervals on $\mathbb{R}$ to bounded and measurable subsets of $\mathbb{R}^n$. In doing so, we obtain a few interesting results. The first is a new proof of the famous Integral Cauchy-Schwarz Inequality. The second is a restatement of Parseval's Identity that doubles as a representation of integrating bounded and measurable functions over bounded and measurable subsets of $\mathbb{R}^n$. Finally, we apply these first two results to develop some sufficient criteria for additional integral inequalities that are elementary in nature.

math.CA↗

arXiv2018

Gaussian Behavior in Zeckendorf Decompositions From Lattices

Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of lattice points such that each point is included at most once. Once a point is chosen, all future points must have strictly smaller coordinates, and the pairwise sum of the values of the points chosen equals $n$. We prove that the distribution of the number of summands in these lattice decompositions converges to a Gaussian distribution in any number of dimensions. As an immediate corollary we obtain a new proof for the asymptotic number of certain lattice paths.

math.NT↗