183 research outputs found
From density-matrix renormalization group to matrix product states
In this paper we give an introduction to the numerical density matrix
renormalization group (DMRG) algorithm, from the perspective of the more
general matrix product state (MPS) formulation. We cover in detail the
differences between the original DMRG formulation and the MPS approach,
demonstrating the additional flexibility that arises from constructing both the
wavefunction and the Hamiltonian in MPS form. We also show how to make use of
global symmetries, for both the Abelian and non-Abelian cases.Comment: Numerous small changes and clarifications, added a figur
The principle of indirect elimination
The principle of indirect elimination states that an algorithm for solving
discretized differential equations can be used to identify its own
bad-converging modes. When the number of bad-converging modes of the algorithm
is not too large, the modes thus identified can be used to strongly improve the
convergence. The method presented here is applicable to any standard algorithm
like Conjugate Gradient, relaxation or multigrid. An example from theoretical
physics, the Dirac equation in the presence of almost-zero modes arising from
instantons, is studied. Using the principle, bad-converging modes are removed
efficiently. Applied locally, the principle is one of the main ingredients of
the Iteratively Smooting Unigrid algorithm.Comment: 16 pages, LaTeX-style espart (elsevier preprint style). Three
.eps-figures are now added with the figure command
A GPU-based hyperbolic SVD algorithm
A one-sided Jacobi hyperbolic singular value decomposition (HSVD) algorithm,
using a massively parallel graphics processing unit (GPU), is developed. The
algorithm also serves as the final stage of solving a symmetric indefinite
eigenvalue problem. Numerical testing demonstrates the gains in speed and
accuracy over sequential and MPI-parallelized variants of similar Jacobi-type
HSVD algorithms. Finally, possibilities of hybrid CPU--GPU parallelism are
discussed.Comment: Accepted for publication in BIT Numerical Mathematic
A hierarchically blocked Jacobi SVD algorithm for single and multiple graphics processing units
We present a hierarchically blocked one-sided Jacobi algorithm for the
singular value decomposition (SVD), targeting both single and multiple graphics
processing units (GPUs). The blocking structure reflects the levels of GPU's
memory hierarchy. The algorithm may outperform MAGMA's dgesvd, while retaining
high relative accuracy. To this end, we developed a family of parallel pivot
strategies on GPU's shared address space, but applicable also to inter-GPU
communication. Unlike common hybrid approaches, our algorithm in a single GPU
setting needs a CPU for the controlling purposes only, while utilizing GPU's
resources to the fullest extent permitted by the hardware. When required by the
problem size, the algorithm, in principle, scales to an arbitrary number of GPU
nodes. The scalability is demonstrated by more than twofold speedup for
sufficiently large matrices on a Tesla S2050 system with four GPUs vs. a single
Fermi card.Comment: Accepted for publication in SIAM Journal on Scientific Computin
A robust, open-source implementation of the locally optimal block preconditioned conjugate gradient for large eigenvalue problems in quantum chemistry
We present two open-source implementations of the locally optimal block preconditioned conjugate gradient (lobpcg) algorithm to find a few eigenvalues and eigenvectors of large, possibly sparse matrices. We then test lobpcg for various quantum chemistry problems, encompassing medium to large, dense to sparse, well-behaved to ill-conditioned ones, where the standard method typically used is Davidson’s diagonalization. Numerical tests show that while Davidson’s method remains the best choice for most applications in quantum chemistry, LOBPCG represents a competitive alternative, especially when memory is an issue, and can even outperform Davidson for ill-conditioned, non-diagonally dominant problems
Non-normal Recurrent Neural Network (nnRNN): learning long time dependencies while improving expressivity with transient dynamics
A recent strategy to circumvent the exploding and vanishing gradient problem
in RNNs, and to allow the stable propagation of signals over long time scales,
is to constrain recurrent connectivity matrices to be orthogonal or unitary.
This ensures eigenvalues with unit norm and thus stable dynamics and training.
However this comes at the cost of reduced expressivity due to the limited
variety of orthogonal transformations. We propose a novel connectivity
structure based on the Schur decomposition and a splitting of the Schur form
into normal and non-normal parts. This allows to parametrize matrices with
unit-norm eigenspectra without orthogonality constraints on eigenbases. The
resulting architecture ensures access to a larger space of spectrally
constrained matrices, of which orthogonal matrices are a subset. This crucial
difference retains the stability advantages and training speed of orthogonal
RNNs while enhancing expressivity, especially on tasks that require
computations over ongoing input sequences
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