Convergence results for some piecewise linear solvers

Let A be a real $$n\times n$$ n × n matrix and $$z,b\in \mathbb R^n$$ z , b ∈ R n . The piecewise linear equation system $$z-A\vert z\vert = b$$ z - A | z | = b is called an absolute value equation. In this note we consider two solvers for uniquely solvable instances of the latter problem, one direct, one semi-iterative. We slightly extend the existing correctness, resp. convergence, results for the latter algorithms and provide numerical tests.

The Old and the New: Can Physics-Informed Deep-Learning Replace Traditional Linear Solvers?

Physics-Informed Neural Networks (PINN) are neural networks encoding the problem governing equations, such as Partial Differential Equations (PDE), as a part of the neural network. PINNs have emerged as a new essential tool to solve various challenging problems, including computing linear systems arising from PDEs, a task for which several traditional methods exist. In this work, we focus first on evaluating the potential of PINNs as linear solvers in the case of the Poisson equation, an omnipresent equation in scientific computing. We characterize PINN linear solvers in terms of accuracy and performance under different network configurations (depth, activation functions, input data set distribution). We highlight the critical role of transfer learning. Our results show that low-frequency components of the solution converge quickly as an effect of the F-principle. In contrast, an accurate solution of the high frequencies requires an exceedingly long time. To address this limitation, we propose integrating PINNs into traditional linear solvers. We show that this integration leads to the development of new solvers whose performance is on par with other high-performance solvers, such as PETSc conjugate gradient linear solvers, in terms of performance and accuracy. Overall, while the accuracy and computational performance are still a limiting factor for the direct use of PINN linear solvers, hybrid strategies combining old traditional linear solver approaches with new emerging deep-learning techniques are among the most promising methods for developing a new class of linear solvers.

Quantum computation of Groebner basis

In this essay, we examine the feasibility of quantum computation of Groebner basis which is a fundamental tool of algebraic geometry. The classical method for computing Groebner basis is based on Buchberger's algorithm, and our question is how to adopt quantum algorithm there. A Quantum algorithm for finding the maximum is usable for detecting head terms of polynomials, which are required for the computation of S-polynomials. The reduction of S-polynomials with respect to a Groebner basis could be done by the quantum version of Gauss-Jordan elimination of echelon which represents polynomials. However, the frequent occurrence of zero-reductions of polynomials is an obstacle to the effective application of quantum algorithms. This is because zero-reductions of polynomials occur in non-full-rank echelons, for which quantum linear systems algorithms (through the inversion of matrices) are inadequate, as ever-known quantum linear solvers (such as Harrow-Hassidim-Lloyd) require the clandestine computations of the inverses of eigenvalues. Hence, for the quantum computation of the Groebner basis, the schemes to suppress the zero-reductions are necessary. To this end, the F5 algorithm or its variant (F5C) would be the most promising, as these algorithms have countermeasures against the occurrence of zero-reductions and can construct full-rank echelons whenever the inputs are regular sequences. Between these two algorithms, the F5C is the better match for algorithms involving the inversion of matrices.

Machine-Learned Preconditioners for Linear Solvers in Geophysical Fluid Flows

<p>Semi-implicit grid-point models for the atmosphere and the ocean require linear solvers that are working efficiently on modern supercomputers. The huge advantage of the semi-implicit time-stepping approach is that it enables large model time-steps. This however comes at the cost of having to solve a computationally demanding linear problem each model time-step to obtain an update to the model’s pressure/fluid-thickness field. In this study, we investigate whether machine learning approaches can be used to increase the efficiency of the linear solver.</p><p>Our machine learning approach aims at replacing a key component of the linear solver—the preconditioner. In the preconditioner an approximate matrix inversion is performed whose quality largely defines the linear solver’s performance. Embedding the machine-learning method within the framework of a linear solver circumvents potential robustness issues that machine learning approaches are often criticized for, as the linear solver ensures that a sufficient, pre-set level of accuracy is reached. The approach does not require prior availability of a conventional preconditioner and is highly flexible regarding complexity and machine learning design choices.</p><p>Several machine learning methods of different complexity from simple linear regression to deep feed-forward neural networks are used to learn the optimal preconditioner for a shallow-water model with semi-implicit time-stepping. The shallow-water model is specifically designed to be conceptually similar to more complex atmosphere models. The machine-learning preconditioner is competitive with a conventional preconditioner and provides good results even if it is used outside of the dynamical range of the training dataset.</p>

Evaluation of Solving Methods for the Fundamental Matrix Computation

Solving the fundamental matrix is a key step in many image calibration and 3D reconstruction systems. The goal of this paper is to study the performance of non-linear solvers for estimating the fundamental matrix in projector-camera calibration. To prevent measurements errors from distorting our understanding, synthetic data are created from ground-truth camera and projector parameters and then used for the assessment of four nonlinear solving strategies.