Systematic Literature Review.

The medicinal plant L. was traditionally used in the Siddha and Ayurvedic systems of medicine in India. The Indian center of origin holds a vast variability in its seed color. The objective of this study was to assess the total monomeric anthocyanin, flavonol, as well as the antioxidative potential, protein content and ash content among the accessions. A total of 99 accessions conserved in the Indian National Genebank were used in this study. The methods used for the estimation of total monomeric anthocyanin, flavonol, as well as the antioxidative potential, protein content and ash content were the pH differential method, Oomah method, Ferric Reducing Antioxidant Potential, Dumas method and gravimetric method, respectively. The completely black colored accession was recorded with highest total monomeric anthocyanin (51.95 mg/100 g of cyanidin 3-glucoside equivalent) and flavonol content (66.41 mg/g of quercetin equivalent). Red + black colored accessions have recorded the maximum value with respect to antioxidants (14.18 mg/g of gallic acid equivalent). The highest amount of protein content was found in a completely white colored accession (20.67%) and the maximum ash content was recorded in red + black colored accession (4.01%). The promising accessions identified can be used by pharmaceutical companies in drug development and in curing degenerative diseases.

Integrating inorganic and polymerized organic functionalities to create composite materials presents an efficient strategy for the discovery and fabrication of multifunctional materials. The characteristics of these composites go beyond a simple sum of individual component properties; they are profoundly influenced by the spatial arrangement of these components and the resulting homo-/hetero-interactions. In this work, we develop a facile and highly adaptable approach for crafting nanostructured polymer-inorganic composites, leveraging hierarchically assembling mixed-graft block copolymers (mGBCPs) as templates. These mGBCPs, composed of diverse polymeric side chains that are covalently tethered with a defined sequence to a linear backbone polymer, self-assemble into ordered hierarchical structures with independently tuned nano- and mesoscale lattice features. Through the coassembly of mGBCPs with diversely sized inorganic fillers such as metal ions (ca. 0.1 nm), metal oxide clusters (0.5-2 nm), and metallic nanoparticles (>2 nm), we create three-dimensional filler arrays with controlled interfiller separation and arrangement. Multiple types of inorganic fillers are simultaneously integrated into the mGBCP matrix by introducing orthogonal interactions between distinct fillers and mGBCP side chains. This results in nanocomposites where each type of filler is selectively segregated into specific nanodomains with matrix-defined orientations. The developed coassembly strategy offers a versatile and scalable pathway for hierarchically structured nanocomposites, unlocking new possibilities for advanced materials in the fields of optoelectronics, sensing, and catalysis.

An interval union is a finite set of closed and disjoint intervals. In this paper we introduce the interval union Gauss-Seidel procedure to rigorously enclose the solution set of linear systems with uncertainties given by intervals or interval unions. We also present the interval union midpoint and Gauss-Jordan preconditioners. The Gauss-Jordan preconditioner is used in a mixed strategy to improve the quality and efficiency of the algorithm. Numerical experiments on interval linear systems generated at random show the capabilities of our approach.

We present a number of new contributions to the topic of constructing efficient higher-order splitting methods for the numerical integration of evolution equations. Particular schemes are constructed via setup and solution of polynomial systems for the splitting coefficients. To this end we use and modify a recent approach for generating these systems for a large class of splittings. In particular, various types of pairs of schemes intended for use in adaptive integrators are constructed.

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle. Our approach permits to recast in a unified framework a number of integrators previously studied in the literature, and presents a general methodology to derive new structure-preserving numerical schemes. The resulting integrators are symplectic; they preserve integrals of motion related to Lie group symmetries; and they include stochastic symplectic Runge-Kutta methods as a special case. Several new low-stage stochastic symplectic methods of mean-square order 1.0 derived using this approach are presented and tested numerically to demonstrate their superior long-time numerical stability and energy behavior compared to nonsymplectic methods.

Using perfectly matched layers for the computation of resonances in open systems typically produces artificial or spurious resonances. We analyze the dependency of these artificial resonances with respect to the discretization parameters and the complex scaling function. In particular, we study the differences between a standard frequency independent complex scaling and a frequency dependent one. While the standard scaling leads to a linear eigenvalue problem, the frequency dependent scaling leads to a polynomial one. Our studies show that the location of artificial resonances is more convenient for the frequency dependent scaling than for a standard scaling. Moreover, the artificial resonances of a frequency dependent scaling are less sensitive to the discretization parameters. Hence, the use of a frequency dependent scaling simplifies the choice of the corresponding discretization parameters.

An a posteriori estimate for the error of a standard Krylov approximation to the matrix exponential is derived. The estimate is based on the defect (residual) of the Krylov approximation and is proven to constitute a rigorous upper bound on the error, in contrast to existing asymptotical approximations. It can be computed economically in the underlying Krylov space. In view of time-stepping applications, assuming that the given matrix is scaled by a time step, it is shown that the bound is asymptotically correct (with an order related to the dimension of the Krylov space) for the time step tending to zero. This means that the deviation of the error estimate from the true error tends to zero faster than the error itself. Furthermore, this result is extended to Krylov approximations of -functions and to improved versions of such approximations. The accuracy of the derived bounds is demonstrated by examples and compared with different variants known from the literature, which are also investigated more closely. Alternative error bounds are tested on examples, in particular a version based on the concept of effective order. For the case where the matrix exponential is used in time integration algorithms, a step size selection strategy is proposed and illustrated by experiments.

In neuroscience, the distribution of a decision time is modelled by means of a one-dimensional Fokker-Planck equation with time-dependent boundaries and space-time-dependent drift. Efficient approximation of the solution to this equation is required, e.g., for model evaluation and parameter fitting. However, the prescribed boundary conditions lead to a strong singularity and thus to slow convergence of numerical approximations. In this article we demonstrate that the solution can be related to the solution of a parabolic PDE on a rectangular space-time domain with homogeneous initial and boundary conditions by transformation and subtraction of a known function. We verify that the solution of the new PDE is indeed more regular than the solution of the original PDE and proceed to discretize the new PDE using a space-time minimal residual method. We also demonstrate that the solution depends analytically on the parameters determining the boundaries as well as the drift. This justifies the use of a sparse tensor product interpolation method to approximate the PDE solution for various parameter ranges. The predicted convergence rates of the minimal residual method and that of the interpolation method are supported by numerical simulations.

In this work, the Parareal algorithm is applied to evolution problems that admit good low-rank approximations and for which the dynamical low-rank approximation (DLRA) can be used as time stepper. Many discrete integrators for DLRA have recently been proposed, based on splitting the projected vector field or by applying projected Runge-Kutta methods. The cost and accuracy of these methods are mostly governed by the rank chosen for the approximation. These properties are used in a new method, called low-rank Parareal, in order to obtain a time-parallel DLRA solver for evolution problems. The algorithm is analyzed on affine linear problems and the results are illustrated numerically.

In algorithms for solving optimization problems constrained to a smooth manifold, retractions are a well-established tool to ensure that the iterates stay on the manifold. More recently, it has been demonstrated that retractions are a useful concept for other computational tasks on manifold as well, including interpolation tasks. In this work, we consider the application of retractions to the numerical integration of differential equations on fixed-rank matrix manifolds. This is closely related to dynamical low-rank approximation (DLRA) techniques. In fact, any retraction leads to a numerical integrator and, vice versa, certain DLRA techniques bear a direct relation with retractions. As an example for the latter, we introduce a new retraction, called KLS retraction, that is derived from the so-called unconventional integrator for DLRA. We also illustrate how retractions can be used to recover known DLRA techniques and to design new ones. In particular, this work introduces two novel numerical integration schemes that apply to differential equations on general manifolds: the accelerated forward Euler (AFE) method and the Projected Ralston-Hermite (PRH) method. Both methods build on retractions by using them as a tool for approximating curves on manifolds. The two methods are proven to have local truncation error of order three. Numerical experiments on classical DLRA examples highlight the advantages and shortcomings of these new methods.

The numerical solution of the generalized eigenvalue problem for a singular matrix pencil is challenging due to the discontinuity of its eigenvalues. Classically, such problems are addressed by first extracting the regular part through the staircase form and then applying a standard solver, such as the QZ algorithm, to that regular part. Recently, several novel approaches have been proposed to transform the singular pencil into a regular pencil by relatively simple randomized modifications. In this work, we analyze three such methods by Hochstenbach, Mehl, and Plestenjak that modify, project, or augment the pencil using random matrices. All three methods rely on the normal rank and do not alter the finite eigenvalues of the original pencil. We show that the eigenvalue condition numbers of the transformed pencils are unlikely to be much larger than the -weak eigenvalue condition numbers, introduced by Lotz and Noferini, of the original pencil. This not only indicates favorable numerical stability but also reconfirms that these condition numbers are a reliable criterion for detecting simple finite eigenvalues. We also provide evidence that, from a numerical stability perspective, the use of complex instead of real random matrices is preferable even for real singular matrix pencils and real eigenvalues. As a side result, we provide sharp left tail bounds for a product of two independent random variables distributed with the generalized beta distribution of the first kind or Kumaraswamy distribution.

A preconditioner is proposed for Laplace exterior boundary value problems on multi-screens. To achieve this, the quotient-space boundary element method and operator preconditioning are combined. For a fairly general subclass of multi-screens, it is shown that this approach paves the way for block diagonal Calderón preconditioners which achieve a spectral condition number that grows only logarithmically with decreasing mesh size, just as in the case of simple screens. Since the resulting scheme contains many more degrees of freedom than strictly required, strategies are presented to remove almost all redundancy without significant loss of effectiveness of the preconditioner. The performance of this method is verified by providing representative numerical results. Further numerical experiments suggest that these results can be extended to a much wider class of multi-screens that cover essentially all geometries encountered in practice, leading to a significantly reduced simulation cost.

In this paper sharp lower error bounds for numerical methods for jump-diffusion stochastic differential equations (SDEs) with discontinuous drift are proven. The approximation of jump-diffusion SDEs with non-adaptive as well as jump-adapted approximation schemes is studied and lower error bounds of order 3/4 for both classes of approximation schemes are provided. This yields optimality of the transformation-based jump-adapted quasi-Milstein scheme.

This paper presents a novel multi-scale method for convection-dominated diffusion problems in the regime of large Péclet numbers. The method involves applying the solution operator to piecewise constant right-hand sides on an arbitrary coarse mesh, which defines a finite-dimensional coarse ansatz space with favorable approximation properties. For some relevant error measures, including the -norm, the Galerkin projection onto this generalized finite element space even yields -independent error bounds, being the singular perturbation parameter. By constructing an approximate local basis, the approach becomes a novel multi-scale method in the spirit of the Super-Localized Orthogonal Decomposition (SLOD). The error caused by basis localization can be estimated in an a posteriori way. In contrast to existing multi-scale methods, numerical experiments indicate -robust convergence without pre-asymptotic effects even in the under-resolved regime of large mesh Péclet numbers.