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We present a numerical method for solving the Poisson equation on a nested grid. The nested grid consists of uniform grids having different grid spacing and is designed to cover the space closer to the center with a finer grid. Thus our…
In the Noisy Intermediate Scale Quantum (NISQ) era, finding implementations of quantum algorithms that minimize the number of expensive and error prone multi-qubit gates is vital to ensure computations produce meaningful outputs. Unitary…
First-order methods for minimization and saddle point (min-max) problems are widely used for solving large-scale problems, in particular arising in machine learning. The majority of works obtain favorable complexity guarantees of such…
The stochastic gradient descent (SGD) method is a widely used approach for solving stochastic optimization problems, but its convergence is typically slow. Existing variance reduction techniques, such as SAGA, improve convergence by…
This paper develops a unified distributed method for solving two classes of constrained networked optimization problems, i.e., optimal consensus problem and resource allocation problem with non-identical set constraints. We first transform…
We consider the task of computing solutions of linear systems that only differ by a shift with the identity matrix as well as linear systems with several different right hand sides. In the past Krylov subspace methods have been developed…
We establish the pointwise convergence of the iterative Lloyd algorithm, also known as $k$-means algorithm, when the quadratic quantization error of the starting grid (with size $N\ge 2$) is lower than the minimal quantization error with…
In this chapter, we investigate recently proposed nonlinear conjugate gradient (NCG) methods for shape optimization problems. We briefly introduce the methods as well as the corresponding theoretical background and investigate their…
This paper surveys an important class of methods that combine iterative projection methods and variational regularization methods for large-scale inverse problems. Iterative methods such as Krylov subspace methods are invaluable in the…
There exist two major problems in application of the conventional block BiCGSTAB method to the O(a)-improved Wilson-Dirac equation with multiple right-hand-sides: One is the deviation between the true and the recursive residuals. The other…
We propose in this paper a multilevel correction method to solve optimal control problems constrained by elliptic equations with the finite element method. In this scheme, solving optimization problem on the finest finite element space is…
We deal with the numerical solution of linear partial differential equations (PDEs) with focus on the goal-oriented error estimates including algebraic errors arising by an inaccurate solution of the corresponding algebraic systems. The…
Gradient Descent (GD) is a ubiquitous algorithm for finding the optimal solution to an optimization problem. For reduced computational complexity, the optimal solution $\mathrm{x^*}$ of the optimization problem must be attained in a minimum…
It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint…
Enlarged Krylov subspace methods and their s-step versions were introduced [7] in the aim of reducing communication when solving systems of linear equations Ax = b. These enlarged CG methods consist of enlarging the Krylov subspace by a…
In this paper, we propose two algorithms for solving convex optimization problems with linear ascending constraints. When the objective function is separable, we propose a dual method which terminates in a finite number of iterations. In…
The notion of well-separated sets is crucial in fast multipole methods as the main idea is to approximate the interaction between such sets via cluster expansions. We revisit the one-parameter multipole acceptance criterion in a general…
Due to the limited number of bits in floating-point or fixed-point arithmetic, rounding is a necessary step in many computations. Although rounding methods can be tailored for different applications, round-off errors are generally…
We investigate the application of Krylov space methods to the solution of shifted linear systems of the form (A+\sigma) x - b = 0 for several values of \sigma simultaneously, using only as many matrix-vector operations as the solution of a…
The vast majority of real world classification problems are imbalanced, meaning there are far fewer data from the class of interest (the positive class) than from other classes. We propose two machine learning algorithms to handle highly…