Related papers: Developing robust incomplete Cholesky factorizatio…
Techniques of matrix completion aim to impute a large portion of missing entries in a data matrix through a small portion of observed ones. In practice including collaborative filtering, prior information and special structures are usually…
We consider the problem of writing an arbitrary symmetric matrix as the difference of two positive semidefinite matrices. We start with simple ideas such as eigenvalue decomposition. Then, we develop a simple adaptation of the Cholesky that…
In this work, we study a variant of nonnegative matrix factorization where we wish to find a symmetric factorization of a given input matrix into a sparse, Boolean matrix. Formally speaking, given $\mathbf{M}\in\mathbb{Z}^{m\times m}$, we…
Covariance estimation for high-dimensional datasets is a fundamental problem in modern day statistics with numerous applications. In these high dimensional datasets, the number of variables p is typically larger than the sample size n. A…
Nonnegative Matrix Factorization consists in (approximately) factorizing a nonnegative data matrix by the product of two low-rank nonnegative matrices. It has been successfully applied as a data analysis technique in numerous domains, e.g.,…
This work studies the strong duality of non-convex matrix factorization problems: we show that under certain dual conditions, these problems and its dual have the same optimum. This has been well understood for convex optimization, but…
Preconditioning techniques are crucial for enhancing the efficiency of solving large-scale linear equation systems that arise from partial differential equation (PDE) discretization. These techniques, such as Incomplete Cholesky…
Optimization problems with constraints in the form of a partial differential equation arise frequently in the process of engineering design. The discretization of PDE-constrained optimization problems results in large-scale linear systems…
The description of weakly bound electronic states is especially difficult with atomic orbital basis sets. The diffuse atomic basis functions that are necessary to describe the extended electronic state generate significant linear…
Many application problems that lead to solving linear systems make use of preconditioned Krylov subspace solvers to compute their solution. Among the most popular preconditioning approaches are incomplete factorization methods either as…
The algorithms in the current sequential numerical linear algebra libraries (e.g. LAPACK) do not parallelize well on multicore architectures. A new family of algorithms, the tile algorithms, has recently been introduced. Previous research…
Five new algorithms were proposed in order to optimize well conditioning of structural matrices. Along with decreasing the size and duration of analyses, minimizing analytical errors is a critical factor in the optimal computer analysis of…
This note proposes an efficient preconditioner for solving linear and semi-linear parabolic equations. With the Crank-Nicholson time stepping method, the algebraic system of equations at each time step is solved with the conjugate gradient…
This survey describes probabilistic algorithms for linear algebra computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problem instances. The paper…
Some new rigorous perturbation bounds for the generalized Cholesky factorization with normwise or componentwise perturbations in the given matrix are obtained, where the componentwise perturbation has the form of backward rounding error for…
A finite element methodology for large classes of variational boundary value problems is defined which involves discretizing two linear operators: (1) the differential operator defining the spatial boundary value problem; and (2) a Riesz…
The (Isogeometric) Finite Cell Method - in which a domain is immersed in a structured background mesh - suffers from conditioning problems when cells with small volume fractions occur. In this contribution, we establish a rigorous scaling…
Krylov subspace methods are linear solvers based on matrix-vector multiplications and vector operations. While easily parallelizable, they are sensitive to rounding errors and may experience convergence issues. ILU(0), an incomplete LU…
In this paper we present a novel algorithm developed for computing the QR factorisation of extremely ill-conditioned tall-and-skinny matrices on distributed memory systems. The algorithm is based on the communication-avoiding CholeskyQR2…
In the context of isogeometric analysis, we consider two discretization approaches that make the resulting stiffness matrix nonsymmetric even if the differential operator is self-adjoint. These are the collocation method and the…