Matrix-free Approach Research Articles

A matrix free, partitioned solution of fluid–structure interaction problems using finite volume and finite element methods

A fully-coupled partitioned finite volume–finite volume and hybrid finite volume–finite element fluid–structure interaction scheme is presented. The fluid domain is modelled as a viscous incompressible isothermal region governed by the Navier–Stokes equations and discretised using an edge-based hybrid-unstructured vertex-centred finite volume methodology. The structure, consisting of a homogeneous isotropic elastic solid undergoing large, non-linear deformations, is discretised using either an elemental/nodal-strain finite volume approach or isoparametric Q8 finite elements and is solved using a matrix-free dual-timestepping approach. Coupling is on the solver sub-iteration level leading to a tighter coupling than if the subdomains are converged separately. The solver is parallelised for distributed-memory systems using METIS for domain-decomposition and MPI for inter-domain communication. The developed technology is evaluated by application to benchmark problems for strongly-coupled fluid–structure interaction systems. It is demonstrated that the scheme results in full coupling between the fluid and solid domains, whilst furnishing accurate solutions.

Iterated preconditioned LSQR method for inverse problems on unstructured grids

This article presents a method for solving large-scale linear inverse imaging problems regularized with a nonlinear, edge-preserving penalty term such as total variation or the Perona–Malik technique. Our method is aimed at problems defined on unstructured meshes, where such regularizers naturally arise in unfactorized form as a stiffness matrix of an anisotropic diffusion operator and factorization is prohibitively expensive. In the proposed scheme, the nonlinearity is handled with lagged diffusivity fixed point iteration, which involves solving a large-scale linear least squares problem in each iteration. Because the convergence of Krylov methods for problems with discontinuities is notoriously slow, we propose to accelerate it by means of priorconditioning (Bayesian preconditioning). priorconditioning is a technique that, through transformation to the standard form, embeds the information contained in the prior (Bayesian interpretation of a regularizer) directly into the forward operator and thence into the solution space. We derive a factorization-free preconditioned LSQR algorithm (MLSQR), allowing implicit application of the preconditioner through efficient schemes such as multigrid. The resulting method is also matrix-free i.e. the forward map can be defined through its action on a vector. We illustrate the performance of the method on two numerical examples. Simple 1D-deblurring problem serves to visualize the discussion throughout the paper. The effectiveness of the proposed numerical scheme is demonstrated on a three-dimensional problem in fluorescence diffuse optical tomography with total variation regularization derived algebraic multigrid preconditioner, which is the type of large scale, unstructured mesh problem, requiring matrix-free and factorization-free approaches that motivated the work here.

A matrix-free approach to build band preconditioners for large-scale bound-constrained optimization

We propose a procedure for building symmetric positive definite band preconditioners for large-scale symmetric, possibly indefinite, linear systems, when the coefficient matrix is not explicitly available, but matrix–vector products involving it can be computed. We focus on linear systems arising in Newton-type iterations within matrix-free versions of projected methods for bound-constrained nonlinear optimization. In this case, the structure and the size of the matrix may significantly change in subsequent iterations, and preconditioner updating algorithms that exploit information from previous steps cannot be easily applied. Our procedure is based on a recursive approach that incrementally improves the quality of the preconditioner, while requiring a modest number of matrix–vector products. A strategy for dynamically choosing the bandwidth of the preconditioners is also presented. Numerical results are provided, showing the performance of our preconditioning technique within a trust-region Newton method for bound-constrained optimization.

Probing of Metabolites in Finely Powdered Plant Material by Direct Laser Desorption Ionization Mass Spectrometry

Natural products continue to serve as an important source of novel drugs since the beginning of human history. High-throughput techniques, such as MALDI-MS, can be techniques of choice for the rapid screening of natural products in plant materials. We present here a fast and reproducible matrix-free approach for the direct detection of UV active metabolites in plant materials without any prior sample preparation. The plant material is mechanically ground to a fine powder and then sieved through different mesh sizes. The collected plant material is dispersed using 1 μL solvent on a target plate is directly exposed to Nd:YAG 335 nm laser. The strategy was optimized for the analysis of plant metabolites after study of the different factors affecting the reproducibility and effectiveness of the analysis, including particle sizes effects, types of solvents used to disperse the sample, and the part of the plant analyzed. Moreover, several plant species, known for different classes of metabolites, were screened to establish the generality of the approach. The developed approach was validated by the characterization of withaferin A and nicotine in the leaves of Withania somnifera and Nicotiana tabacum, respectively, through comparison of its MS/MS data with the standard compound. Scanning electron microscopy (SEM) and atomic force microscopy (AFM) techniques were used for the tissue imaging purposes. This approach can be used to directly probe small molecules in plant materials as well as in herbal and pharmaceutical formulations for fingerprinting development.

Finite Element Algorithms and Data Structures on Graphical Processing Units

The finite element method (FEM) is one of the most commonly used techniques for the solution of partial differential equations on unstructured meshes. This paper discusses both the assembly and the solution phases of the FEM with special attention to the balance of computation and data movement. We present a GPU assembly algorithm that scales to arbitrary degree polynomials used as basis functions, at the expense of redundant computations. We show how the storage of the stiffness matrix affects the performance of both the assembly and the solution. We investigate two approaches: global assembly into the CSR and ELLPACK matrix formats and matrix-free algorithms, and show the trade-off between the amount of indexing data and stiffness data. We discuss the performance of different approaches in light of the implicit caches on Fermi GPUs and show a speedup over a two-socket 12-core CPU of up to 10 times in the assembly and up to 6 times in the solution phase. We present our sparse matrix-vector multiplication algorithms that are part of a conjugate gradient iteration and show that a matrix-free approach may be up to two times faster than global assembly approaches and up to 4 times faster than NVIDIA's cuSPARSE library, depending on the preconditioner used.

Six-Helix Bundle and Triangle DNA Origami Insulator-Based Dielectrophoresis

Self-assembled DNA nanostructures have large potential for nanoelectronic circuitry, targeted drug delivery, and intelligent sensing. Their applications require suitable methods for manipulation and nanoscale assembly as well as adequate concentration, purification, and separation methods. Insulator-based dielectrophoresis (iDEP) provides an efficient and matrix-free approach for manipulation of micro- and nanometer-sized objects. In order to exploit iDEP for DNA nanoassemblies, a detailed understanding of the underlying polarization and dielectrophoretic migration is essential. Here, we explore the dielectrophoretic behavior of six-helix bundle and triangle DNA origamis with identical sequence but large topological difference and reveal a characteristic frequency range of iDEP trapping. Moreover, the confinement of triangle origami in the iDEP trap required larger applied electric fields. To elucidate the observed DEP migration and trapping, we discuss polarizability models for the two species according to their structural difference complemented by numerical simulations, revealing a contribution of the electrophoretic transport of the DNA origami species in the iDEP trapping regions. The numerical model showed reasonable agreement with experiments at lower frequency. However, the extension of the iDEP trapping regions observed experimentally deviated considerably at higher frequencies. Our study demonstrates for the first time that DNA origami species can be successfully trapped and manipulated by iDEP and reveals distinctive iDEP behavior of the two DNA origamis. The experimentally observed trapping regimes will facilitate future exploration of DNA origami manipulation and assembly at the nano- and microscale as well as other applications of these nanoassemblies with iDEP.

Sensitivity and Accuracy of Organic Matrix-Assisted Laser Desorption and Ionisation Mass Spectrometry of FeCl3 is Higher Than in in Matrix-Free Approach

We compare the quality and reliability of laser desorption and ionisation mass spectra of FeCl3 acquired without the assistance of the matrix with the spectra acquired with different organic matrix molecules. Generally, inorganic salts tend to form clusters upon laser irradiation, the signals of which can be easily distinguished from ions arising from the matrix. In the presence of a matrix, cluster ions are, however, mostly suppressed. We have compared the number of analyte signals, accuracy of determination of isotope composition of the analyte and the sensitivity of FeCl3 detection between different approaches. The results obtained imply that the sensitivity of mass spectrometric analysis of FeCl3 is somewhat higher when matrices are applied than in the matrix-free approach. Among all matrices tested in this work, F20TPP seems to be the most promising for further applications as a matrix for mass spectrometry of inorganic salts.

Desorption/Ionization Efficiency of Common Amino Acids in Surface-Assisted Laser Desorption/Ionization Mass Spectrometry (SALDI-MS) with Nanostructured Platinum

Surface-assisted laser desorption/ionization mass spectrometry (SALDI-MS) using inorganic nanoparticles has been reported as an organic matrix-free approach. However, the correlation of desorption/ionization (DI) efficiency with analyte chemical structures in SALDI-MS is not clear. In this study, we investigated the DI efficiency of 20 common amino acids and several peptides in SALDI-MS with Pt nanoparticles with thin projections on the surface (termed with Pt nanoflowers, Pt Nfs) on silicon substrates. The fluorocarbon-based hydrophobic perfluorodecyltrichlorosilane (FDTS)-Pt Nf substrates enabled the simultaneous analysis of all 20 common amino acids in negative-ion mode, whereas MALDI-MS was able to detect only two amino acids, proline and glutamic acid, from the same mixture in negative-ion mode. The SALDI-MS produced high ion yields for arginine and proline in positive-ion mode as well as for glutamic acid and aspartic acid in negative-ion mode. A linear correlation was found between the ion yield an...

A Matrix-free Approach for Solving the Parametric Gaussian Process Maximum Likelihood Problem

Gaussian processes are the cornerstone of statistical analysis in many application areas. Nevertheless, most of the applications are limited by their need to use the Cholesky factorization in the computation of the likelihood. In this work, we present a matrix-free approach for computing the solution of the maximum likelihood problem involving Gaussian processes. The approach is based on a stochastic programming reformulation followed by sample average approximation applied to either the maximization problem or its optimality conditions. We provide statistical estimates of the approximate solution. The method is illustrated on several examples where the data is provided on a regular or irregular grid. In the latter case, the action of a covariance matrix on a vector is computed by means of fast multipole methods. For each of the examples presented, we demonstrate that the approach scales linearly with an increase in the number of sites.

A partly matrix-free solver for the gyrokinetic field equation in three-dimensional geometry

In the case of adiabatic electrons the gyrokinetic field equation for the electrostatic potential includes an averaging operator acting on flux surfaces. For realistic three-dimensional configurations, as e.g. in stellarator devices, the discretisation of this integro-differential equation leads to very large nearly dense matrices (full matrix approach) which typically cannot be stored in computer memory explicitly. A low memory consuming partly matrix-free approach, based on a preconditioned iterative matrix solver, has been developed where the Helmholtz part of the field equation is used in a matrix formulation while the averaging term is treated matrix-free. For matrices which could still be stored in memory explicitly, it is demonstrated that this approach is also much faster than the full matrix approach.

Laser desorption and ionization time-of-flightversus matrix-assisted laser desorption and ionization time-of-flight mass spectrometry of and metal complexes.

Matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) has been recently established as a powerful, "soft" ionization technique for the analysis of both transition metal complexes, which are used as metallo-drugs in the therapy of various types of tumors, and biomolecules. Since some metal complexes absorb light in the UV range, it should be possible to analyse them without additional matrices, i.e. using LDI-TOF MS. In this study, the matrix-free approach was tested for the analysis of [PtCl2(dach)] (dichloride(1,2-diamincyclohexane) platinum(ii)), [RuCl2(en)2]Cl (dichloridobis(ethylenediamine) ruthenium(iii) chloride) and [RuCl2(bipy)2]Cl (bis(bipyridine)dichloridoruthenium(iii) chloride) and the detection limit for these compounds was determined. In summary, the LDI-TOF mass spectra of [PtCl2(dach)] and [RuCl2(en)2]Cl are rather simple, whereas in the presence of 2,5-DHB as a matrix, additional peaks are generated. On the other hand, the standard MALDI-TOF mass spectrum of [RuCl2(bipy)2]Cl exhibits only one peak arising from the complex, in contrast to six peaks detectable in the LDI-TOF mass spectrum. The detection limit in the MALDI-TOF MS analysis of [PtCl2(dach)] and [RuCl2(bipy)2]Cl complexes was lower than that determined in LDI-TOF MS. Taking all into account, in this paper, we have demonstrated some advantages and drawbacks of the matrix-free LDI-TOF mass spectrometric analysis of transition metal complexes.

A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics

This paper proposes an iterative method that can simulate mechanical systems featuring a large number of contacts and joints between rigid bodies. The numerical method behaves as a contractive mapping that converges to the solution of a cone complementarity problem by means of iterated fixed-point steps with separable projections onto convex manifolds. Since computational speed and robustness are important issues when dealing with a large number of frictional contacts, we have performed special algorithmic optimizations in order to translate the numerical scheme into a matrix-free algorithm with O( n) space complexity and easy implementation. A modified version, that can run on parallel computers is discussed. A multithreaded version of the method has been used to simulate systems with more than a million contacts with friction.

Effect of urea surface modification and photocatalytic cleaning on surface‐assisted laser desorption ionization mass spectrometry with amorphous TiO2 nanoparticles

We have investigated the effect of urea surface modification and the photocatalytic cleaning on surface-assisted laser desorption ionization mass spectrometry (SALDI-MS) with amorphous TiO2 nanoparticles for the reduction of the background noise and the improvement of the sensitivity. In the use of nanoparticles of high surface area, chemical background signals arising from ambient environments and organic contaminants can frequently be serious problems below 500 Da, possibly reducing the advantages of the matrix-free approach. In this study, removal of contaminants and enhanced SALDI efficiency were easily achieved with UV irradiation via the photocatalyst effect of TiO2 before SALDI-MS measurements. The surface cleaning achieved by the UV photocatalytic procedure reduced the background noise and increased the peak intensities of peptides. In addition, we found that urea surface modification of TiO2 nanoparticles increased the performance of the TiO2-SALDI-MS. (1) The urea-surface modification of TiO2 made it possible to produce proton-adduct forms without citrate buffer, resulting in low background noises below 500 Da, in contrast to the essential use of a citrate buffer in the bare TiO2-SALDI-MS. (2) The detection sensitivity of angiotensin I increased to 0.3 fmol with the urea-surface modification, as compared to the use of bare TiO2 nanoparticles (6 fmol). The urea-TiO2 could ionize proteins of more than 20,000 Da such as trypsinogen (600 fmol). (3) The urea modification of TiO2 had the advantage of selective detection of phosphopeptides without sample clean up, or prefractionation in tryptic digest products of bovine hemoglobin.

Surface cleaning of germanium nanodot ionization substrate for surface‐assisted laser desorption/ionization mass spectrometry

In surface-assisted laser desorption/ionization mass spectrometry (SALDI-MS), a chemical background signal, arising from organic contaminants such as plasticizers, is frequently observed mainly under m/z ca. 600, which impairs the advantages of the matrix-free approach. Silver salts, which are used for the cationization of aromatic compounds, are also difficult to remove completely after the measurements. In this study, surface cleaning techniques used in semiconductor processing were used to clean our developed silicon-based SALDI substrate on which self-assembled germanium nanodots (GeNDs) had been deposited (termed a GeND chip). An immersion cleaning method using acetone with sonication, and a sulfuric-peroxide mixture (SPM) cleaning method using a mixture of H(2)SO(4)/H(2)O(2)/deionized water, were examined for their effectiveness in removing organic compounds and residual silver salts. Removal of both types of contaminants was successfully performed by SPM cleaning. The limit of detection for glutathione was improved from ca. 5 pmol without cleaning to ca. 50 fmol after the SPM cleaning. Since GeND chips can tolerate acidic cleaning and sonication due to their chemical inertness and rigid nanodot structures, they appear to be an ideal reusable SALDI substrate.

The use of desorption/ionization on porous silicon mass spectrometry for the detection of negative ions for fatty acids

The study of low molecular weight compounds by matrix-assisted laser desorption/ionization (MALDI) is difficult because of the presence of ions originating from the matrix in the low-m/z range. In order to resolve these problems, new matrix-free approaches were developed based on laser desorption/ionization from the surface of various materials such as graphite and porous silicon. Our work involves the use of 'desorption ionization on porous silicon mass spectrometry' (DIOS-MS) in the negative ion mode to study fatty acid compounds. The potential of the DIOS-MS technique is shown and an insight into the ionization mechanism provided.

A matrix-free spectral rotation approach to the computation of electromagnetic fields generated by a surface current distribution

A major step in the numerical solution of electromagnetic scattering problems involves the computation of the electric field generated by a surface current distribution. In this paper, we show that the desired field can be calculated over a generic plane by using a matrix-free approach based on a spectral domain technique combined with an analytic continuation procedure.

Krylov sub-space methods for K-eigenvalue problem in 3-D neutron transport

The K-eigenvalue problem in nuclear reactor physics is often formulated in the framework of Neutron Transport Theory. The fundamental mode solution of this problem is usually obtained by the power iteration method. Here, we are concerned with the use of a Krylov sub-space method, called ORTHOMIN(1), to obtain a more efficient solution of the K-eigenvalue problem. A Matrix-free approach is proposed which can be easily implemented by using a transport code which can perform fixed source calculations. The power iteration and ORTHOMIN(1) schemes are compared for two realistic 3-D multi-group cases with isotropic scattering: an LWR benchmark and a heavy water reactor problem. In both the schemes, within-group iterations over self-scattering source are required as intermediate procedures. These iterations are also accelerated using another Krylov method called conjugate gradient method. The overall work is based on the use of Sn-method and finite-differencing for discretisation of transport equation.

On two matrix-free continuation approaches for the determination of the bifurcation diagram of the von Kármán system

In this paper we present a combination of the asymptotic numerical continuation procedure with a preconditioned GMRES-solver as applied to the fourth-order non linear von Karman problem. Using a mixed equal order finite element discretization, we show how our “matrix free”approach allows for an efficient determination of the non-trivial branch of the bifurcation diagram. We also show that, using a steplength estimation of Gervais and Sadiky [GER 04], one can limit himself to a third order prediction without loosing too much in the number of continuation steps.

A Boundary Condition--Capturing Multigrid Approach to Irregular Boundary Problems

We propose a geometric multigrid method for solving linear systems arising from irregular boundary problems involving multiple interfaces in two and three dimensions. In this method, we adopt a matrix-free approach; i.e., we do not form the fine grid matrix explicitly and we never form nor store the coarse grid matrices, as many other robust multigrid methods do. The main idea is to construct an accurate interpolation which captures the correct boundary conditions at the interfaces via a level set function. Numerical results are given to compare our multigrid method with black box and algebraic multigrid methods.

Desorption–ionization on silicon mass spectrometry: an application in forensics

Desorption–ionization on silicon (DIOS) is a new, matrix-free laser desorption mass spectrometry approach that allows for the direct identification of low molecular weight compounds in the presence of potentially interfering compounds. The porous silicon surfaces provide a scaffold for trapping analyte molecules, and are readily adaptable to commercial time-of-flight instruments. As an example of its utility in forensic cases, DIOS mass spectrometry was used to distinguish between similar synthetic polymers and identify specific polymers from complex biological media. Despite the absence of matrix, specific low molecular weight polymers were rapidly identified without any fragmentation. This method was applied to the rapid identification of ethoxylate polymers during a criminal investigation.