Matrix-free Approach Research Articles

Detection of Ru potential metallodrug in human urine by MALDI-TOF mass spectrometry: Validation and options to enhance the sensitivity

We studied the possibility of detection of [Ru(η5-C5H5)(PPh3)2Cl] (abbreviated by RuCp) complex as a model system for Ru-based metallodrugs in human urine by using matrix-assisted laser desorption/ionization time-of-flight mass spectrometry (MALDI-TOF MS) without previous purification or removal of inorganic salts. Inorganic salts might prevent the detection of RuCp by MALDI-TOF MS, most likely through the increased number and intensity of background/organic matrix signals. This problem might be overcome by the acquisition of matrix-free spectra and the addition of nanoparticles, such as carbon dots, to the urine solution. Our results suggest that RuCp is easily detectable by MALDI-TOF MS in all acquisition conditions, with the CHCA matrix being the best for acquisition in phosphate-containing solutions, whereas in urine, DHB and matrix-free approach demonstrated the highest sensitivity, precision, and reproducibility. The sensitivity of matrix-free MALDI detection of RuCp could be increased by the addition of carbon dots to the urine. Based on theoretical calculations for all matrix/analyte combinations, the model for the interaction of RuCp with carbon dots was established, and higher sensitivity explained.

Time-domain elastic Gauss–Newton full-waveform inversion: a matrix-free approach

SUMMARYWe present a time-domain matrix-free elastic Gauss–Newton full-waveform inversion (FWI) algorithm. Our algorithm consists of a Gauss–Newton update with a search direction calculated via elastic least-squares reverse time migration (LSRTM). The conjugate gradient least-squares (CGLS) method solves the LSRTM problem with forward and adjoint operators derived via the elastic Born approximation. The Hessian of the Gauss–Newton method is never explicitly formed or saved in memory. In other words, the CGLS algorithm solves for the Gauss–Newton direction via the application of implicit-form forward and adjoint operators which are equivalent to elastic Born modelling and elastic reverse time migration, respectively. We provide numerical examples to test the proposed algorithm where we invert for P- and S-wave velocities simultaneously. The proposed algorithm performs positively on mid-size problems where we report solutions of slight improvement than those computed using the conventional non-linear conjugate gradient method. In spite of the aforementioned limited gain, the theory developed in this paper contributes to a better understanding of time-domain elastic Gauss–Newton FWI.

Recent validation study of a Method of Characteristics based Neutron Transport Code

We have reported the development of a computer code based on Method of Characteristics (MOC) to solve neutron transport equation for mainly lattice (i.e. assembly) level calculations with reflective and periodic boundary conditions and to some extent, core level calculation with vacuum boundary condition. Performance of the code has also been demonstrated with flat and linear source assumptions. In a subsequent work, we have implemented Krylov subspace iteration method for accelerating the MOC solution and in that context, introduced the concept of a matrix-free approach for using the code in solving large problems in effective manner. The acceleration technique is then used in combination with both kind of source assumptions, mentioned above, to solve a number of benchmark problems and demonstrate the capability and validity of the code. In continuation of the study, few more benchmark problems are selected from the literature to verify the performance of the code and their results are reported in the present paper.

A matrix-free Chimera approach based on Dirichlet–Dirichlet coupling for domain composition purposes

A novel approach using an algebraic-based Chimera type domain composition method in the context of the finite element method for non-matching overlapping unstructured grids is proposed in this work. The scheme is based on both the transfer of information across each grid interface via Dirichlet boundary conditions and a high-order interpolation algorithm to obtain one global solution of the system. The solution can be obtained iteratively, with a convergence rate that is similar to that obtained with an analogous conformal mesh, and the matrix–vector operator can be computed with completely decoupled operations on both meshes. Furthermore, the scheme can be set as a linear operator that can be fed to a matrix-free efficient iterative solver, such as the BiConjugate Gradient Stabilized method. Several numerical examples using non-matching unstructured grids that are partially and completely overlapped with different element sizes are presented, assessing the precision and convergence rate of the method.

A Jacobian-free approximate Newton–Krylov startup strategy for RANS simulations

The favorable convergence rates of Newton–Krylov-based solution algorithms have increased their popularity for computational fluid dynamics applications. Unfortunately, these methods perform poorly during the initial stages of convergence, particularly for three-dimensional Reynolds-averaged Navier–Stokes simulations. Addressing this problem requires the use of a globalization method such as pseudo-transient continuation, along with an approximate Newton–Krylov startup stage. This class of methods marches the solution in pseudo-time with a matrix-based approximate Jacobian that has a lower bandwidth and better conditioning properties compared with the exact Jacobian. However, this matrix-based approach also has shortcomings, including the large cost of computing and storing the approximate Jacobian and its preconditioner, along with the need for an updated Jacobian for every nonlinear iteration. To rectify these shortcomings, we use approximate residual formulations in a Jacobian-free approximate Newton–Krylov algorithm. With the approximate Jacobian, we compute the vector products by using the approximate residual computations in a matrix-free manner while forming a preconditioner based on the matrix-based approximate Jacobian. This approach keeps the approximate Jacobian up to date and mitigates the cost of forming a matrix-based Jacobian and its preconditioner at each iteration by lagging the preconditioner between nonlinear iterations. We use varying levels of approximations with the matrix-free approach and thereby demonstrate the trade-off between rate of convergence and the cost of each nonlinear iteration. The proposed implementation uses only the exact and approximate residual formulations and can therefore be generalized with minimal additional implementation effort to a range of solvers and discretizations. The code is available under an open-source license.

A biradical-tagged phospholipid as a polarizing agent for solid-state MAS Dynamic Nuclear Polarization NMR of membrane proteins

A novel Dynamic Nuclear Polarization (DNP) NMR polarizing agent ToSMTSL-PTE representing a phospholipid with a biradical TOTAPOL tethered to the polar head group has been synthesized, characterized, and employed to enhance solid-state Nuclear Magnetic Resonance (SSNMR) signal of a lipid-reconstituted integral membrane protein proteorhodopsin (PR). A matrix-free PR formulation for DNP improved the absolute sensitivity of NMR signal by a factor of ca. 4 compared to a conventional preparation with TOTAPOL dispersed in a glassy glycerol/water matrix. DNP enhancements measured at 400 MHz/263 GHz and 600 MHz/395 GHz showed a strong field dependence but remained moderate at both fields, and comparable to those obtained for PR covalently modified with ToSMTSL. Additional continuous wave (CW) X-band electron paramagnetic resonance (EPR) experiments with ToSMTSL-PTE in solutions and in lipid bilayers revealed that an unfavorable conformational change of the linker connecting mononitroxides could be one of the reasons for moderate DNP enhancements. Further, differential scanning calorimetry (DSC) and CW EPR experiments indicated an inhomogeneous distribution and/or a possibility of a partial aggregation of ToSMTSL-PTE in DMPC:DMPA bilayers when the concentration of the polarizing agent was increased to 20 mol% to maximize the DNP enhancement. Thus, conformational changes and an inhomogeneous distribution of the lipid-based biradicals in lipid bilayers emerged as important factors to consider for further development of this matrix-free approach for DNP of membrane proteins.

The Eigenvalues Slicing Library (EVSL): Algorithms, Implementation, and Software

This paper describes a software package called EVSL (for eigenvalues slicing library) for solving large sparse real symmetric standard and generalized eigenvalue problems. As its name indicates, the package exploits spectrum slicing, a strategy that consists of dividing the spectrum into a number of subintervals and extracting eigenpairs from each subinterval independently. In order to enable such a strategy, the methods in EVSL utilize a quick calculation of the spectral density of a given matrix (or matrix pair). What distinguishes EVSL from other available packages is that EVSL relies entirely on filtering techniques. Both polynomial and rational classes of filtering are implemented and are coupled with Krylov subspace methods as well as subspace iteration. On the implementations, the package offers interfaces for various scenarios including matrix-free approaches, whereby user-specific functions can be supplied to perform matrix-vector operations or solve linear systems. The paper describes the algorithms in EVSL, provides details on their implementations, and discusses performance issues for the various methods and on various computing platforms.

Large-scale simulation of mantle convection based on a new matrix-free approach

In this paper, we report on a two-scale approach for efficient matrix-free finite element simulations. It is an extended version of our previous conference publication [1]. The proposed method is based on surrogate element matrices constructed by low order polynomial approximations. It is applied to a Stokes-type PDE system with variable viscosity as is a key component in mantle convection models. We set the ground for a rigorous performance analysis inspired by the concept of parallel textbook multigrid efficiency and study the weak scaling behavior on SuperMUC, a peta-scale supercomputer system. For a complex geodynamical model, we achieve, on up to 47 250 compute cores, a parallel efficiency of 93% for application of the discrete operator and 83% for a complete Uzawa V-cycle including the coarse grid solve. Our largest simulation uses a trillion (O(1012)) degrees of freedom for a global mesh resolution of 1.5 km. Applicability of our new approach for geodynamical problems is demonstrated by investigating dynamic topography for classical benchmark settings as well as for high-resolution models with lateral viscosity variations.

Use of the Extended Fujita method for representing the molecular weight and molecular weight distributions of native and processed oat beta-glucans

Beta 1–3, 1–4 glucans (“beta-glucans”) are one of the key components of the cell wall of cereals, complementing the main structural component cellulose. Beta-glucans are also an important source of soluble fibre in foods containing oats with claims of other beneficial nutritional properties such as plasma cholesterol lowering in humans. Key to the function of beta-glucans is their molecular weight and because of their high polydispersity - molecular weight distribution. Analytical ultracentrifugation provides a matrix-free approach (not requiring separation columns or media) to polymer molecular weight distribution determination. The sedimentation coefficient distribution is converted to a molecular weight distribution via a power law relation using an established procedure known as the Extended Fujita approach. We establish and apply the power law relation and Extended Fujita method for the first time to a series of native and processed oat beta-glucans. The application of this approach to beta-glucans from other sources is considered.

3D global hydrodynamic stability analysis of a diffusion flame

This work investigates the three-dimensional global hydrodynamic stability of a diffusion flame. The low-Mach-number Navier–Stokes (LMN-NS) equations for reacting flows are solved together with a transport equation for the mixture fraction. A source term is added to the energy conservation equation to model the chemical heat release as a function of the Damköhler (Da) number and of the reaction rate, computed according to an Arrhenius law. The global stability analysis has been performed by a matrix-free time-stepper approach applied to the LMN-NS equations, using an Arnoldi method to compute the most unstable modes. Increasing the value of Da, direct numerical simulations show a transition from an oscillating unstable regime towards a stable one. In the unstable regime, stability analyses show two different flame behaviors: a highly unstable weak-flame and a typical diffusion flame. In the latter case, two different families of modes have been identified: the low-frequency most unstable one related to the premixing zone of the flame and a high-frequency stable branch representative of the Kelvin–Helmholtz instability of the diffusive rear region of the flame. The present three-dimensional stability analysis has been able to compute, for the first time, the eigenmodes responsible for the cellular structure of the flame.

A Computational Architecture for Coupling Heterogeneous Numerical Models and Computing Coupled Derivatives

One of the challenges in computational modeling is coupling models to solve multidisciplinary problems. Flow-based computational frameworks alleviate part of the challenge through a modular approach, where data flows from component to component. However, existing flow-based frameworks are inefficient when coupled derivatives are needed for optimization. To address this, we develop the modular analysis and unified derivatives (MAUD) architecture. MAUD formulates the multidisciplinary model as a nonlinear system of equations, which leads to a linear equation that unifies all methods for computing derivatives. This enables flow-based frameworks that use the MAUD architecture to provide a common interface for the chain rule, adjoint method, coupled adjoint method, and hybrid methods; MAUD automatically uses the appropriate method for the problem. A hierarchical, matrix-free approach enables modern solution techniques such as Newton--Krylov solvers to be used within this monolithic formulation without computational overhead. Two demonstration problems are solved using a Python implementation of MAUD: a nanosatellite optimization with more than 2 million unknowns and 25,000 design variables, and an aircraft optimization involving over 6,000 design variables and 23,000 constraints. MAUD is now implemented in the open source framework OpenMDAO, which has been used to solve aircraft, satellite, wind turbine, and turbofan engine design problems.

A differential variational approach for handling fluid–solid interaction problems via smoothed particle hydrodynamics

The fluid–solid interaction (FSI) problem is customarily solved by starting with the fluid dynamics component. One chooses an established computational fluid dynamics (CFD) method and subsequently embeds the solid phase dynamics within the CFD solution leading to either a monolithic or a staggered/co-simulated solution. The approach discussed here takes the opposite tack. We start with a differential variational framework to handle the solid phase; i.e., the multi-body dynamics problem in the presence of contact, friction, and bilateral kinematic constraints. The dynamics of the fluid phase, which is captured via smoothed particle hydrodynamics (SPH), is subsequently embedded into this framework in which the incompressibility attribute of the flow is enforced via kinematic constraint equations that involve SPH particles. The resulting monolithic FSI solution methodology relies on a half-implicit symplectic time integration method that uses a matrix-free iterative approach to solve a cone constrained quadratic optimization problem at each time step. This problem yields the contact forces, friction forces, boundary condition Lagrange multipliers, fluid–solid coupling terms, and bilateral constraint Lagrange multipliers. The solution of the optimization problem represents the computationally taxing component of the method. Large integration time steps, tight enforcement of incompressibility, a unified approach for handling the fluid and solid phases, and linear scaling are listed as the attractive attributes of the proposed method. The numerical experiments reported include three validation studies (incompressibility, dam break, and sloshing), a scaling analysis, and a tracked vehicle fording simulation.

Posing Multibody Dynamics With Friction and Contact as a Differential Complementarity Problem

This technical brief revisits the method outlined in Tasora and Anitescu 2011 [“A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 200(5–8), pp. 439–453], which was introduced to solve the rigid multibody dynamics problem in the presence of friction and contact. The discretized equations of motion obtained here are identical to the ones in Tasora and Anitescu 2011 [“A Matrix-Free Cone Complementarity Approach for Solving Large-Scale, Nonsmooth, Rigid Body Dynamics,” Comput. Methods Appl. Mech. Eng., 200(5–8), pp. 439–453]; what is different is the process of obtaining these equations. Instead of using maximum dissipation conditions as the basis for the Coulomb friction model, the approach detailed uses complementarity conditions that combine with contact unilateral constraints to augment the classical index-3 differential algebraic equations of multibody dynamics. The resulting set of differential, algebraic, and complementarity equations is relaxed after time discretization to a cone complementarity problem (CCP) whose solution represents the first-order optimality condition of a quadratic program with conic constraints. The method discussed herein has proven reliable in handling large frictional contact problems. Recently, it has been used with promising results in fluid–solid interaction applications. Alas, this solution is not perfect, and it is hoped that the detailed account provided herein will serve as a starting point for future improvements.

A laid-back trip through the Hennigian Forests.

BackgroundThis paper is a comment on the idea of matrix-free Cladistics. Demonstration of this idea’s efficiency is a major goal of the study. Within the proposed framework, the ordinary (phenetic) matrix is necessary only as “source” of Hennigian trees, not as a primary subject of the analysis. Switching from the matrix-based thinking to the matrix-free Cladistic approach clearly reveals that optimizations of the character-state changes are related not to the real processes, but to the form of the data representation.MethodsWe focused our study on the binary data. We wrote the simple ruby-based script FORESTER version 1.0 that helps represent a binary matrix as an array of the rooted trees (as a “Hennigian forest”). The binary representations of the genomic (DNA) data have been made by script 1001. The Average Consensus method as well as the standard Maximum Parsimony (MP) approach has been used to analyze the data.Principle findingsThe binary matrix may be easily re-written as a set of rooted trees (maximal relationships). The latter might be analyzed by the Average Consensus method. Paradoxically, this method, if applied to the Hennigian forests, in principle can help to identify clades despite the absence of the direct evidence from the primary data. Our approach may handle the clock- or non clock-like matrices, as well as the hypothetical, molecular or morphological data.DiscussionOur proposal clearly differs from the numerous phenetic alignment-free techniques of the construction of the phylogenetic trees. Dealing with the relations, not with the actual “data” also distinguishes our approach from all optimization-based methods, if the optimization is defined as a way to reconstruct the sequences of the character-state changes on a tree, either the standard alignment-based techniques or the “direct” alignment-free procedure. We are not viewing our recent framework as an alternative to the three-taxon statement analysis (3TA), but there are two major differences between our recent proposal and the 3TA, as originally designed and implemented: (1) the 3TA deals with the three-taxon statements or minimal relationships. According to the logic of 3TA, the set of the minimal trees must be established as a binary matrix and used as an input for the parsimony program. In this paper, we operate directly with maximal relationships written just as trees, not as binary matrices, while also using the Average Consensus method instead of the MP analysis. The solely ‘reversal’-based groups can always be found by our method without the separate scoring of the putative reversals before analyses.

A matrix-free approach to efficient affine-linear image registration on CPU and GPU

This paper presents a generic approach to highly efficient image registration in two and three dimensions. Both monomodal and multimodal registration problems are considered. We focus on the important class of affine-linear transformations in a derivative-based optimization framework. Our main contribution is an explicit formulation of the objective function gradient and Hessian approximation that allows for very efficient, parallel derivative calculation with virtually no memory requirements. The flexible parallelism of our concept allows for direct implementation on various hardware platforms. Derivative calculations are fully matrix free and operate directly on the input data, thereby reducing the auxiliary space requirements from $${\mathcal {O}}(n)$$O(n) to $${\mathcal {O}}(1)$$O(1). The proposed approach is implemented on multicore CPU and GPU. Our GPU code outperforms a conventional matrix-based CPU implementation by more than two orders of magnitude, thus enabling usage in real-time scenarios. The computational properties of our approach are extensively evaluated, thereby demonstrating the performance gain for a variety of real-life medical applications.

A matrix-free approach for solving systems of nonlinear equations

The computation of the Jacobian, its inverse and $n\times n$ matrix storage requirement in each iteration limits the application of Newton-like method for solving systems of nonlinear equations, because computing the Jacobian entails derivative and to compute its inverse, it must be non-singular. We present an enhancement of a three-step Newton-like method by J. R. Sharma and P. Gupta (2014) where the Jacobian inverse is approximated into a diagonal matrix. Numerical results proved that the proposed method in terms of CPU time and number of iterations compared to the Sharma and Gupta's method is very encouraging.

A matrix-free augmented lagrangian algorithm with application to large-scale structural design optimization

In many large engineering design problems, it is not computationally feasible or realistic to store Jacobians or Hessians explicitly. Matrix-free implementations of standard optimization methods—implementations that do not explicitly form Jacobians and Hessians, and possibly use quasi-Newton approximations—circumvent those restrictions, but such implementations are virtually non-existent. We develop a matrix-free augmented-Lagrangian algorithm for nonconvex problems with both equality and inequality constraints. Our implementation is developed in the Python language, is available as an open-source package, and allows for approximating Hessian and Jacobian information.We show that our approach solves problems from the CUTEr and COPS test sets in a comparable number of iterations to state-of-the-art solvers. We report numerical results on a structural design problem that is typical in aircraft wing design optimization. The matrix-free approach makes solving problems with thousands of design variables and constraints tractable, even when function and gradient evaluations are costly.

Matrix-free Brownian dynamics simulation technique for semidilute polymeric solutions.

Evaluating the concentration dependence of static and dynamic properties of macromolecules in semidilute polymer solutions requires accurate calculation of long-range hydrodynamic interactions (HI) and short range excluded volume (EV) forces. In conventional Brownian dynamics simulations (BDS), computation of HI necessitates construction of a dense diffusion tensor commonly performed via Ewald summation. Krylov subspace techniques allow efficient decomposition of this tensor [computational cost scales as O(N^{2}), where N is the total number of beads in bead-spring representation of macromolecules in a simulation box] and computation of Brownian displacements in the box. In this paper, a matrix-free approach for calculation of HI is implemented which leads to O(NlogN) scaling of computational expense. The fidelity of the algorithm is demonstrated by evaluating the asymptotic value of center-of-mass diffusivity of polymer molecules at very low concentrations and their radius of gyration scaling as a function of number of beads for dilute and semidilute solutions (with concentrations up to 5 times the overlap concentration). In turn, a favorable comparison between our results and the blob theory is shown.

Matrix-free DNP-enhanced NMR spectroscopy of liposomes using a lipid-anchored biradical.

Magic-angle spinning dynamic nuclear polarization (MAS-DNP) has been proven to be a powerful technique to enhance the sensitivity of solid-state NMR (SSNMR) in a wide range of systems. Here, we show that DNP can be used to polarize lipids using a lipid-anchored polarizing agent. More specifically, we introduce a C16-functionalized biradical, which allows localization of the polarizing agents in the lipid bilayer and DNP experiments to be performed in the absence of excess cryo-protectant molecules (glycerol, dimethyl sulfoxide, etc.). This constitutes another original example of the matrix-free DNP approach that we recently introduced.

Implicit scheme for meshless compressible Euler solver

In this paper, an implicit scheme is presented for a meshless compressible Euler solver based on the Least Square Kinetic Upwind Method (LSKUM). The Jameson and Yoon's split flux Jacobians formulation is very popular in finite volume methodology, which leads to a scalar diagonal dominant matrix for an efficient implicit procedure (Jameson & Yoon, 1987). However, this approach leads to a block diagonal matrix when applied to the LSKUM meshless method. The above split flux Jacobian formulation, along with a matrix-free approach, has been adopted to obtain a diagonally dominant, robust and cheap implicit time integration scheme. The efficacy of the scheme is demonstrated by computing 2D flow past a NACA 0012 airfoil under subsonic, transonic and supersonic flow conditions. The results obtained are compared with available experiments and other reliable computational fluid dynamics (CFD) results. The present implicit formulation shows good convergence acceleration over the RK4 explicit procedure. Further, the accuracy and robustness of the scheme in 3D is demonstrated by computing the flow past an ONERA M6 wing and a clipped delta wing with aileron deflection. The computed results show good agreement with wind tunnel experiments and other CFD computations.