Finite Projection Research Articles

Vector bundles whose restriction to a linear section is Ulrich

An Ulrich sheaf on an n-dimensional projective variety \(X \subseteq \mathbb {P}^{N}\) is an initialized ACM sheaf which has the maximum possible number of global sections. Using a construction based on the representation theory of Roby–Clifford algebras, we prove that every normal ACM variety admits a reflexive sheaf whose restriction to a general 1-dimensional linear section is Ulrich; we call such sheaves \(\delta \)-Ulrich. In the case \(n=2,\) where \(\delta \)-Ulrich sheaves satisfy the property that their direct image under a general finite linear projection to \(\mathbb {P}^2\) is a semistable instanton bundle on \(\mathbb {P}^{2}\), we show that some high Veronese embedding of X admits a \(\delta \)-Ulrich sheaf with a global section.

DISCREPANCY SETS FOR COMBINED LEAST SQUARES PROJECTION AND TIKHONOV REGULARIZATION

To solve a linear ill-posed problem, a combination of the finite dimensional least squares projection method and the Tikhonov regularization is considered. The dimension of the projection is treated as the second parameter of regularization. A two-parameter discrepancy principle defines a discrepancy set for any data error bound. The aim of the paper is to describe this set and to indicate its subset such that for regularization parameters from this subset the related regularized solution has the same order of accuracy as the Tikhonov regularization with the standard discrepancy principle but without any discretization.

Superconvergence of semidiscrete finite element methods for bilinear parabolic optimal control problems

In this paper, a semidiscrete finite element method for solving bilinear parabolic optimal control problems is considered. Firstly, we present a finite element approximation of the model problem. Secondly, we bring in some important intermediate variables and their error estimates. Thirdly, we derive a priori error estimates of the approximation scheme. Finally, we obtain the superconvergence between the semidiscrete finite element solutions and projections of the exact solutions. A numerical example is presented to verify our theoretical results.

Fluctuations for analytic test functions in the Single Ring Theorem

We consider a non-Hermitian random matrix $A$ whose distribution is invariant under the left and right actions of the unitary group. The so-called Single Ring Theorem, proved by Guionnet, Krishnapur and Zeitouni, states that the empirical eigenvalue distribution of $A$ converges to a limit measure supported by a ring $S$. In this text, we establish the convergence in distribution of random variables of the type $Tr (f(A)M)$ where $f$ is analytic on $S$ and the Frobenius norm of $M$ has order $\sqrt{N}$. As corollaries, we obtain central limit theorems for linear spectral statistics of $A$ (for analytic test functions) and for finite rank projections of $f(A)$ (like matrix entries). As an application, we locate outliers in multiplicative perturbations of $A$.

Implementation of variable parameters in the Krylov-based finite state projection for solving the chemical master equation

The finite state projection (FSP) algorithm is a reduction method for solving the chemical master equation (CME). The Krylov-FSP improved on the original FSP by using an embedded scheme where the action of the matrix exponential is evaluated by the Krylov subspace method of Expokit for greater efficiency. There are parameters that impact the method, such as the stepsize that must be controlled to ensure the accuracy of the computed matrix exponentials, or to ensure the accuracy of the FSP. Other parameters include the dimension of the Krylov basis, or even the extent of reachability when expanding the FSP. In this work, we incorporate adaptive strategies to automatically vary these parameters. Numerical experiments comparing the resulting variants are reported, showing how certain choices perform better than others.

Is Finnish topic prominent?

Finnish finite clause exhibits topic prominence in the sense that the preverbal subject position is occupied by the topic (for example, by the direct object topic), not necessarily by the grammatical subject. Three currently unexplained facts concerning the Finnish free word order phenomenon and topicalization are noted in this paper: subject-verb agreement interacts with word order; the preverbal “topic” position is not reserved exclusively for topics; and noun phrase (DP) arguments are also able to dislocate to the right edge of a (potentially very long) finite clause. A generalized morphosyntactic agreement mechanism that requires the presence of nominal phi-features inside the highest finite projection of a clause is posited to explain the link between agreement and word order. The problem with topicality is accounted for by assuming that the topic-focus mechanism operates outside of narrow syntax. Free word order and non-configurationality are argued to result from argument adjunction, not from movement. Finally, it is concluded that the Finnish EPP is connected neither to morphosyntax nor to discourse.

Finite state projection based bounds to compare chemical master equation models using single-cell data.

Emerging techniques now allow for precise quantification of distributions of biological molecules in single cells. These rapidly advancing experimental methods have created a need for more rigorous and efficient modeling tools. Here, we derive new bounds on the likelihood that observations of single-cell, single-molecule responses come from a discrete stochastic model, posed in the form of the chemical master equation. These strict upper and lower bounds are based on a finite state projection approach, and they converge monotonically to the exact likelihood value. These bounds allow one to discriminate rigorously between models and with a minimum level of computational effort. In practice, these bounds can be incorporated into stochastic model identification and parameter inference routines, which improve the accuracy and efficiency of endeavors to analyze and predict single-cell behavior. We demonstrate the applicability of our approach using simulated data for three example models as well as for experimental measurements of a time-varying stochastic transcriptional response in yeast.

Finite volume element method and its stability analysis for analyzing the behavior of sub-diffusion problems

In this paper, we analyze the spatially semi-discrete piecewise linear finite volume element method for the time fractional sub-diffusion problem in two dimensions, and give an approximate solution of this problem. At first, we introduce bilinear finite volume element method with interpolated coefficients and derive some error estimates between exact solution and numerical solution in both finite element and finite volume element methods. Furthermore, we use the standard finite element Ritz projection and also the elliptic projection defined by the bilinear form associated with the variational formulation of the finite volume element method. Finally, some numerical examples are included to illustrate the effectiveness of the new technique.

Maximum-norm stability of the finite element Ritz projection under mixed boundary conditions

As a model of the second order elliptic equation with non-trivial boundary conditions, we consider the Laplace equation with mixed Dirichlet and Neumann boundary conditions on convex polygonal domains. Our goal is to establish that finite element discrete harmonic functions with mixed Dirichlet and Neumann boundary conditions satisfy a weak (Agmon---Miranda) discrete maximum principle, and then prove the stability of the Ritz projection with mixed boundary conditions in $$L^\infty $$L? norm. Such results have a number of applications, but are not available in the literature. Our proof of the maximum-norm stability of the Ritz projection is based on converting the mixed boundary value problem to a pure Neumann problem, which is of independent interest.

A parallel Large Eddy Simulation in a finite element projection method for all flow regimes

ABSTRACTA new Finite Element Method (FEM) system has been developed for engine and combustion modeling, KIVA-hpFE. This new FEM uses a projection method for the solution of the momentum equations and runs on parallel computer systems using the Message Passing Interface (MPI). The modeled fluid is multispecies and employs either Reynolds Averaged Navier–Stokes (RANS) (k-ϖ) or a dynamic Large Eddy Simulation (LES) model for the solution of turbulent reactive flow. This FEM-based code provides an excellent platform for developing better in-cylinder fuel and species evolution. A dynamic LES method applicable through transitional to fully turbulent flow is described in this paper. LES uses a fine grid resolution to provide the accurate subgrid-scale (SGS) modeling of turbulence and to capture the majority of the turbulent energy in the resolved regions (the grid scale). To this end, a parallel method is required, particularly for engineering solutions of 3D domains, and is described along with the linear algebraic processes.

Understanding the finite state projection and related methods for solving the chemical master equation

The finite state projection (FSP) method has enabled us to solve the chemical master equation of some biological models that were considered out of reach not long ago. Since the original FSP method, much effort has gone into transforming it into an adaptive time-stepping algorithm as well as studying its accuracy. Some of the improvements include the multiple time interval FSP, the sliding windows, and most notably the Krylov-FSP approach. Our goal in this tutorial is to give the reader an overview of the current methods that build on the FSP.

Computational study of p53 regulation via the chemical master equation

A stochastic model of cellular p53 regulation was established in Leenders, and Tuszynski (2013 Front. Oncol. 3 1–16) to study the interactions of p53 with MDM2 proteins, where the stochastic analysis was done using a Monte Carlo approach. We revisit that model here using an alternative scheme, which is to directly solve the chemical master equation (CME) by an adaptive Krylov-based finite state projection method that combines the stochastic simulation algorithm with other computational strategies, namely Krylov approximation techniques to the matrix exponential, divide and conquer, and aggregation. We report numerical results that demonstrate the extend of tackling the CME with this combination of tools.

Hölder regularity of the densities for the Navier–Stokes equations with noise

We prove that the densities of the finite dimensional projections of weak solutions of the Navier–Stokes equations driven by Gaussian noise are bounded and Holder continuous, thus improving the results of Debussche and Romito [8]. The proof is based on analytical estimates on a conditioned Fokker–Planck equation solved by the density, that has a “non–local” term that takes into account the influence of the rest of the infinite dimensional dynamics over the finite subspace under observation.

Multiscale modelling of evolving foams

We present a set of multi-scale interlinked algorithms to model the dynamics of evolving foams. These algorithms couple the key effects of macroscopic bubble rearrangement, thin film drainage, and membrane rupture. For each of the mechanisms, we construct consistent and accurate algorithms, and couple them together to work across the wide range of space and time scales that occur in foam dynamics. These algorithms include second order finite difference projection methods for computing incompressible fluid flow on the macroscale, second order finite element methods to solve thin film drainage equations in the lamellae and Plateau borders, multiphase Voronoi Implicit Interface Methods to track interconnected membrane boundaries and capture topological changes, and Lagrangian particle methods for conservative liquid redistribution during rearrangement and rupture. We derive a full set of numerical approximations that are coupled via interface jump conditions and flux boundary conditions, and show convergence for the individual mechanisms. We demonstrate our approach by computing a variety of foam dynamics, including coupled evolution of three-dimensional bubble clusters attached to an anchored membrane and collapse of a foam cluster.

FISH: Finite Projective Plane Based Interference Cancellation for Slotted ALOHA

Due to the large number of light weight, low power devices in the future Internet, an efficient distributed multiple access protocol that can support a small fixed number of machines with continuous traffic along with a large variable number of devices with sporadic traffic is needed. The traffic in the future Internet has fundamentally different requirements: first, the accessing devices are of low power, which limit the transmission range; secondly, the delay is very stringent; thirdly, the number of active devices is time varying. In this paper, we propose a finite projective plane (FPP) based multiple access protocol (FISH) to meet such requirements. A low power node can transmit the replicas of a data packet in multiple time slots to achieve the same range of transmission as a high power node at one time. The initialization time slot of a frame allows for the estimation of the number of active nodes and the adjustment of the size of the frame to maximize the throughput. The interference cancellation and conflict resolution scheme of the FISH protocol is based on the theory of FPP. By using the property that the union of the sets of the FPP sharing one common point is a complete set, we are able to extract each of the individual user signals. One distinguishing feature of the FISH protocol is the amplification of the signal-to-noise ratio (SNR), which can either extend the transmission range or improve the bit error rate without sacrificing the throughput. In the performance evaluation, we demonstrate that the throughput is identical to the Slotted Aloha, while the SNR is magnified m times, where m is the order of the FPP. The proposed FISH protocol is of great use in the machine-to-machine communications as well as traditional cellular and satellite systems.

Data assimilation algorithm for 3D Bénard convection in porous media employing only temperature measurements

In this paper we propose a continuous data assimilation (downscaling) algorithm for the Bénard convection in porous media using only discrete spatial-mesh measurements of the temperature. In this algorithm, we incorporate the observables as a feedback (nudging) term in the evolution equation of the temperature. We show that under an appropriate choice of the nudging parameter and the size of the mesh, and under the assumption that the observed data is error free, the solution of the proposed algorithm converges at an exponential rate, asymptotically in time, to the unique exact unknown reference solution of the original system, associated with the observed (finite dimensional projection of) temperature data. Moreover, we note that in the case where the observational measurements are not error free, one can estimate the error between the solution of the algorithm and the exact reference solution of the system in terms of the error in the measurements.

CERENA: ChEmical REaction Network Analyzer--A Toolbox for the Simulation and Analysis of Stochastic Chemical Kinetics.

Gene expression, signal transduction and many other cellular processes are subject to stochastic fluctuations. The analysis of these stochastic chemical kinetics is important for understanding cell-to-cell variability and its functional implications, but it is also challenging. A multitude of exact and approximate descriptions of stochastic chemical kinetics have been developed, however, tools to automatically generate the descriptions and compare their accuracy and computational efficiency are missing. In this manuscript we introduced CERENA, a toolbox for the analysis of stochastic chemical kinetics using Approximations of the Chemical Master Equation solution statistics. CERENA implements stochastic simulation algorithms and the finite state projection for microscopic descriptions of processes, the system size expansion and moment equations for meso- and macroscopic descriptions, as well as the novel conditional moment equations for a hybrid description. This unique collection of descriptions in a single toolbox facilitates the selection of appropriate modeling approaches. Unlike other software packages, the implementation of CERENA is completely general and allows, e.g., for time-dependent propensities and non-mass action kinetics. By providing SBML import, symbolic model generation and simulation using MEX-files, CERENA is user-friendly and computationally efficient. The availability of forward and adjoint sensitivity analyses allows for further studies such as parameter estimation and uncertainty analysis. The MATLAB code implementing CERENA is freely available from http://cerenadevelopers.github.io/CERENA/.

A Galerkin-based formulation of the probability density evolution method for general stochastic finite element systems

The present paper proposes a Galerkin finite element projection scheme for the solution of the partial differential equations (pde's) involved in the probability density evolution method, for the linear and nonlinear static analysis of stochastic systems. According to the principle of preservation of probability, the probability density evolution of a stochastic system is expressed by its corresponding Fokker---Planck (FP) stochastic partial differential equation. Direct integration of the FP equation is feasible only for simple systems with a small number of degrees of freedom, due to analytical and/or numerical intractability. However, rewriting the FP equation conditioned to the random event description, a generalized density evolution equation (GDEE) can be obtained, which can be reduced to a one dimensional pde. Two Galerkin finite element method schemes are proposed for the numerical solution of the resulting pde's, namely a time-marching discontinuous Galerkin scheme and the StreamlineUpwind/Petrov Galerkin (SUPG) scheme. In addition, a reformulation of the classical GDEE is proposed, which implements the principle of probability preservation in space instead of time, making this approach suitable for the stochastic analysis of finite element systems. The advantages of the FE Galerkin methods and in particular the SUPG over finite difference schemes, like the modified Lax---Wendroff, which is the most frequently used method for the solution of the GDEE, are illustrated with numerical examples and explored further.

Isometries of Grassmann spaces

Botelho, Jamison, and Molnár have recently described the general form of surjective isometries of Grassmann spaces on complex Hilbert spaces under certain dimensionality assumptions. In this paper we provide a new approach to this problem which enables us first, to give a shorter proof and second, to remove dimensionality constraints completely. In one of the low dimensional cases, which was not covered by Botelho, Jamison, and Molnár, an exceptional possibility occurs. As a byproduct, we are able to handle the real case as well. Furthermore, in finite dimensions we remove the surjectivity assumption. A variety of tools is used in order to achieve our goal, such as topological, geometrical and linear algebra techniques. The famous two projections theorem for two finite rank projections will be re-proven using linear algebraic methods. A theorem of Györy and the second author on orthogonality preservers on Grassmann spaces will be strengthened as well. This latter result will be obtained by using Chow's fundamental theorem of geometry of Grassmannians.

Abridged Continuous Data Assimilation for the 2D Navier–Stokes Equations Utilizing Measurements of Only One Component of the Velocity Field

We introduce a continuous data assimilation (downscaling) algorithm for the two-dimensional Navier-Stokes equations employing coarse mesh measurements of only one component of the velocity field. This algorithm can be implemented with a variety of finitely many observables: low Fourier modes, nodal values, finite volume averages, or finite elements. We provide conditions on the spatial resolution of the observed data, under the assumption that the observed data is free of noise, which are sufficient to show that the solution of the algorithm approaches, at an exponential rate asymptotically in time, to the unique exact unknown reference solution, of the 2D Navier-Stokes equations, associated with the observed (finite dimensional projection of) velocity.