The objective of this paper is to extend, in the context of multicore architectures, the concepts of algorithms-by-tiles [Buttari et al., 2007] for Cholesky, LU, QR factorizations to the family of twosided factorizations. In particular, the bidiagonal reduction of a general, dense matrix is very often used as a pre-processing step for calculating the singular value decomposition. Furthermore, i...

This paper discusses weighted tensor Golub–Kahan-type bidiagonalization processes using the t-product. product was introduced in M.E. Kilmer and C.D. Martin (2011). A few steps of a process with least squares norm are carried out to reduce large-scale linear discrete ill-posed problem small size. The weights determined by symmetric positive definite (SPD) tensors. Tikhonov regularization is app...

This paper discusses several transform-based methods for solving linear discrete ill-posed problems third order tensor equations based on a tensor-tensor product defined by an invertible transform. Linear products were first introduced in Kernfeld et al. (2015) [16]. These are applied to derive Tikhonov regularization Golub-Kahan-type bidiagonalization and Arnoldi-type processes. GMRES-type sol...

The model-based image reconstruction techniques for photoacoustic (PA) tomography require an explicit regularization. An error estimate (?2) minimization-based approach was proposed and developed for the determination of a regularization parameter for PA imaging. The regularization was used within Lanczos bidiagonalization framework, which provides the advantage of dimensionality reduction for ...

lsmr (least squares minimal residual) is an iterative method for the solution of the linear system of equations and leastsquares problems. this paper presents a block version of the lsmr algorithm for solving linear systems with multiple right-hand sides. the new algorithm is based on the block bidiagonalization and derived by minimizing the frobenius norm of the resid ual matrix of normal equa...

A challenging class of problems arising in many GPU applications, called batched problems, involves linear algebra operations on many small-sized matrices. We designed batched BLAS (Basic Linear Algebra Subroutines) routines, and in particular the Level-2 BLAS GEMV and the Level-3 BLAS GEMM routines, to solve them. We proposed device functions and big-tile settings in our batched BLAS design. W...

We revisit unitary representation of centrally extended psu(2|2) excitation superalgebra. We find most generally that ‘pseudo-momentum’, not lattice momentum, diagonalizes spin chain Hamiltonian and leads to generalized dynamic spin chain. All known results point to lattice momentum diagonalization for N = 4 super Yang-Mills theory. Having different interacting structure, we ask if N = 6 superc...

Communication – the movement of data between levels of memory hierarchy or between processors over a network – is the most expensive operation in terms of both time and energy at all scales of computing. Achieving scalable performance in terms of time and energy thus requires a dramatic shift in the field of algorithmic design. Solvers for sparse linear algebra problems, ubiquitous throughout s...