We analyze the convergence of compressive sensing based sampling techniques for the efficient evaluation of functionals of solutions for a class of high-dimensional, affine-parametric, linear operator equations which depend on possibly infinitely many parameters. The proposed algorithms are based on so-called “non-intrusive” sampling of the high-dimensional parameter space, reminiscent of Monte...

In this work we have studied the QCD phase structure and critical dynamics related to 3-$d$ $O(4)$ $Z(2)$ symmetry universality classes in two-flavor quark-meson low energy effective theory within functional renormalization group approach. We employed expansion of Chebyshev polynomials solve flow equation for order-parameter potential. The chiral transition line limit, end points explicit break...

The singularity of cylindrical or spherical coordinate systems at the origin imposes certain regularity conditions on the spectral expansion of any infinitely differentiable function. There are two efficient choices of a set of radial basis functions suitable for discretising the solution of a partial differential equation posed in either such geometry. One choice is methods based on standard C...

This paper presents a new technique for studying the stability properties of dynamic systems modeled by delay-differential equations (DDEs) with time-periodic parameters. By employing a shifted Chebyshev polynomial approximation in each time interval with length equal to the delay and parametric excitation period, the dynamic system can be reduced to a set of linear difference equations for the...

A linear stability analysis is applied to a system consisting of a horizontal fluid layer overlying a layer of porous medium saturated with the same fluid, with uniform heating from below in the presence of a vertical magnetic field. The flow in porous medium is assumed to be governed by Darcy’s law. The Beavers-Joseph condition is applied at the interface between the two layers. Numerical solu...

We consider elliptic stochastic partial differential equations (SPDEs) with random coefficients and solve them by expanding the solution using generalized polynomial chaos (gPC). Under some mild conditions on the coefficients, the solution is ‘‘sparse’’ in the random space, i.e., only a small number of gPC basis makes considerable contribution to the solution. To exploit this sparsity, we emplo...

A common way of computing the roots of a polynomial is to find the eigenvalues of a linearization, such as the companion (when the polynomial is expressed in the monomial basis), colleague (Chebyshev basis) or comrade matrix (general orthogonal polynomial basis). For the monomial case, many studies exist on the stability of linearization-based rootfinding algorithms. By contrast, little seems t...

We consider dimension reduction techniques for the Liouville–von Neumann equation for the evaluation of the expectation values in a mixed quantum system. We describe several existing methods that have appeared in the literature, showing the failure of them when the system is scaled up. We introduce a new method termed direct expectation values via Chebyshev based on evaluation of a trace formul...

Quantum control plays a key role in quantum technology, in particular for steering quantum systems. As problem size grows exponentially with the system size, it is necessary to deal with fast numerical algorithms and implementations. We improved an existing code for quantum control concerning two linear algebra tasks: The computation of the matrix exponential and efficient parallelisation of pr...

In this work, the definition of first and second types multiplicative Chebyshev differential equations is given solutions these are investigated with help power series. Also, properties type polynomials proved. Finally, studies supported numerical examples.