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In geometry processing, numerical optimization methods often involve solving sparse linear systems of equations. These linear systems have a structure that strongly resembles to adjacency graphs of the underlying mesh. We observe how…
Recent advancements in quantum computing and quantum-inspired algorithms have sparked renewed interest in binary optimization. These hardware and software innovations promise to revolutionize solution times for complex problems. In this…
We investigate the use of piecewise linear systems, whose coefficient matrix is a piecewise constant function of the solution itself. Such systems arise, for example, from the numerical solution of linear complementarity problems and in the…
A matrix approach to continuous iteration is proposed for general formal series. It leads, in particular, to an order{to{order iteration of the exponential function, and consequently to an algorithmic approach to tetration. Lower{order…
Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community. This paper focuses on iterative methods, which use linear system solvers, like conjugate gradients, alternating…
In this paper, several Kaczmarz-type numerical methods for solving the matrix equation $AX=B$ and $XA=C$ are proposed, where the coefficient matrix $A$ may be full rank or rank deficient. These methods are iterative methods without matrix…
New solution method for the systems of linear equations in commutative integral domains is proposed. Its complexity is the same that the complexity of the matrix multiplication.
The present work has been designed for students in secondary school and their teachers in mathematics. We will show how with the help of our knowledge of number systems we can solve problems from other fields of mathematics for example in…
This paper deals with investigating numerical methods for solving coupled system of nonlinear parabolic problems. We utilize block monotone iterative methods based on Jacobi and Gauss--Seidel methods to solve difference schemes which…
In this work we discuss the possibility to reduce the computational complexity of modal methods, i.e. methods based on eigenmodes expansion, from the third power to the second power of the number of eigenmodes. The proposed approach is…
The motivation of this work is to illustrate the efficiency of some often overlooked alternatives to deal with optimization problems in systems and control. In particular, we will consider a problem for which an iterative linear matrix…
Several recent works have developed a new, probabilistic interpretation for numerical algorithms solving linear systems in which the solution is inferred in a Bayesian framework, either directly or by inferring the unknown action of the…
The Kaczmarz algorithm is an iterative method that solves linear systems of equations. It stands out among iterative algorithms when dealing with large systems for two reasons. First, at each iteration, the Kaczmarz algorithm uses a single…
We use a unified framework to summarize sixteen randomized iterative methods including Kaczmarz method, coordinate descent method, etc. Some new iterative schemes are given as well. Some relationships with \textsc{mg} and \textsc{ddm} are…
This research introduces an innovative mathematical learning approach that integrates generative AI to cultivate a structured learning rather than quick solution. Our method combines chatbot capabilities and generative AI to offer…
An iterative method is derived for image reconstruction. Among other attributes, this method allows constraints unrelated to the radiation measurements to be incorporated into the reconstructed image. A comparison is made with the widely…
A new method, the Dynamical Systems Method (DSM), justified recently, is applied to solving ill-conditioned linear algebraic system (ICLAS). The DSM gives a new approach to solving a wide class of ill-posed problems. In this paper a new…
In this paper, we introduce a new iterative method which we call one step back approach: the main idea is to anticipate the consequence of the iterative computation per coordinate and to optimize on the choice of the sequence of the…
Since its introduction by Gauss, Matrix Algebra has facilitated understanding of scientific problems, hiding distracting details and finding more elegant and efficient ways of computational solving. Today's largest problems, which often…
Substitutions play a crucial role in a wide range of contexts, from analyzing the dynamics of social opinions and conducting mathematical computations to engaging in game-theoretical analysis. For many situations, considering one-step…