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An essential feature of the subdiffusion equations with the $\alpha$-order time fractional derivative is the weak singularity at the initial time. The weak regularity of the solution is usually characterized by a regularity parameter $\sigma\in (0,1)\cup(1,2)$. Under this general regularity assumption, we here obtain the pointwise-in-time error estimate of the widely used L1 scheme for nonlinear subdiffusion equations. To the end, we present a refined discrete fractional-type Gr\"onwall inequality and a rigorous analysis for the truncation errors. Numerical experiments are provided to demonstrate the effectiveness of our theoretical analysis.

Topology optimization problems often support multiple local minima due to a lack of convexity. Typically, gradient-based techniques combined with continuation in model parameters are used to promote convergence to more optimal solutions; however, these methods can fail even in the simplest cases. In this paper, we present an algorithm to perform a systematic exploratory search for the solutions of the optimization problem via second-order methods without a good initial guess. The algorithm combines the techniques of deflation, barrier methods and primal-dual active set solvers in a novel way. We demonstrate this approach on several numerical examples, observe mesh-independence in certain cases and show that multiple distinct local minima can be recovered.

We consider sequential prediction with expert advice when data are generated from distributions varying arbitrarily within an unknown constraint set. We quantify relaxations of the classical i.i.d. assumption in terms of these constraint sets, with i.i.d. sequences at one extreme and adversarial mechanisms at the other. The Hedge algorithm, long known to be minimax optimal in the adversarial regime, was recently shown to be minimax optimal for i.i.d. data. We show that Hedge with deterministic learning rates is suboptimal between these extremes, and present a new algorithm that adaptively achieves the minimax optimal rate of regret with respect to our relaxations of the i.i.d. assumption, and does so without knowledge of the underlying constraint set. We analyze our algorithm using the follow-the-regularized-leader framework, and prove it corresponds to Hedge with an adaptive learning rate that implicitly scales as the square root of the entropy of the current predictive distribution, rather than the entropy of the initial predictive distribution.

Several novel mixed-integer linear and bilinear formulations are proposed for the optimum communication spanning tree problem. They implement the distance-based approach: graph distances are directly modeled by continuous, integral, or binary variables, and interconnection between distance variables is established using the recursive Bellman-type conditions or using matrix equations from algebraic graph theory. These non-linear relations are used either directly giving rise to the bilinear formulations, or, through the big-M reformulation, resulting in the linear programs. A branch-and-bound framework of Gurobi 9.0 optimization software is employed to compare performance of the novel formulations on the example of an optimum requirement spanning tree problem with additional vertex degree constraints. Several real-world requirements matrices from transportation industry are used to generate a number of examples of different size, and computational experiments show the superiority of the two novel linear distance-based formulations over the the traditional multicommodity flow model.

In this work, we consider the numerical recovery of a spatially dependent diffusion coefficient in a subdiffusion model from distributed observations. The subdiffusion model involves a Caputo fractional derivative of order $\alpha\in(0,1)$ in time. The numerical estimation is based on the regularized output least-squares formulation, with an $H^1(\Omega)$ penalty. We prove the well-posedness of the continuous formulation, e.g., existence and stability. Next, we develop a fully discrete scheme based on the Galerkin finite element method in space and backward Euler convolution quadrature in time. We prove the subsequential convergence of the sequence of discrete solutions to a solution of the continuous problem as the discretization parameters (mesh size and time step size) tend to zero. Further, under an additional regularity condition on the exact coefficient, we derive convergence rates in a weighted $L^2(\Omega)$ norm for the discrete approximations to the exact coefficient {in the one- and two-dimensional cases}. The analysis relies heavily on suitable nonstandard nonsmooth data error estimates for the direct problem. We provide illustrative numerical results to support the theoretical study.

We construct Two-Point Flux Approximation (TPFA) finite volume schemes to solve the quadratic optimal transport problem in its dynamic form, namely the problem originally introduced by Benamou and Brenier. We show numerically that these type of discretizations are prone to form instabilities in their more natural implementation, and we propose a variation based on nested meshes in order to overcome these issues. Despite the lack of strict convexity of the problem, we also derive quantitative estimates on the convergence of the method, at least for the discrete potential and the discrete cost. Finally, we introduce a strategy based on the barrier method to solve the discrete optimization problem.

We explore an algorithm for approximating roots of integers, discuss its motivation and derivation, and analyze its convergence rates with varying parameters and inputs. We also perform comparisons with established methods for approximating square roots and other rational powers.

Multidimensional packing problems generalize the classical packing problems such as Bin Packing, Multiprocessor Scheduling by allowing the jobs to be $d$-dimensional vectors. While the approximability of the scalar problems is well understood, there has been a significant gap between the approximation algorithms and the hardness results for the multidimensional variants. In this paper, we close this gap by giving almost tight hardness results for these problems. 1. We show that Vector Bin Packing has no polynomial time $\Omega( \log d)$ factor asymptotic approximation algorithm when $d$ is a large constant, assuming $\textsf{P}\neq \textsf{NP}$. This matches the $\ln d + O(1)$ factor approximation algorithms (Chekuri, Khanna SICOMP 2004, Bansal, Caprara, Sviridenko SICOMP 2009, Bansal, Eli\'{a}s, Khan SODA 2016) upto constants. 2. We show that Vector Scheduling has no polynomial time algorithm with an approximation ratio of $\Omega\left( (\log d)^{1-\epsilon}\right)$ when $d$ is part of the input, assuming $\textsf{NP}\nsubseteq \textsf{ZPTIME}\left( n^{(\log n)^{O(1)}}\right)$. This almost matches the $O\left( \frac{\log d}{\log \log d}\right)$ factor algorithms(Harris, Srinivasan JACM 2019, Im, Kell, Kulkarni, Panigrahi SICOMP 2019). We also show that the problem is NP-hard to approximate within $(\log \log d)^{\omega(1)}$. 3. We show that Vector Bin Covering is NP-hard to approximate within $\Omega\left( \frac{\log d}{\log \log d}\right)$ when $d$ is part of the input, almost matching the $O(\log d)$ factor algorithm (Alon et al., Algorithmica 1998). Previously, no hardness results that grow with $d$ were known for Vector Scheduling and Vector Bin Covering when $d$ is part of the input and for Vector Bin Packing when $d$ is a fixed constant.

This study examines the time complexities of the unbalanced optimal transport problems from an algorithmic perspective for the first time. We reveal which problems in unbalanced optimal transport can/cannot be solved efficiently. Specifically, we prove that the Kantorovich Rubinstein distance and optimal partial transport in the Euclidean metric cannot be computed in strongly subquadratic time under the strong exponential time hypothesis. Then, we propose an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric. The proposed algorithm processes a tree with one million nodes in less than one second. Our analysis forms a foundation for the theoretical study of unbalanced optimal transport algorithms and opens the door to the applications of unbalanced optimal transport to million-scale datasets.

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.