Preconditioned Conjugate Gradient Research Articles

Low-complexity soft-output signal detector based on AI-SSOR preconditioned conjugate gradient method over massive MIMO correlated channel

Massive multiple-input multiple-output (MIMO) techniques with a large number of antenna elements at base station (BS) have been proved as an alternative to provide potential opportunity to increase the spectrum and energy efficiency. However, in the system, there generally exists a spatial correlation effect due to insufficient antenna elements spacing and/or the lack of rich scattering at BS. The minimum mean square error (MMSE) method performs signal detection at the expense of large-scale matrix inversion operation. Thus, the conjugate gradient (CG) method has received a lot of attentions to realize the MMSE detection efficiently. Unfortunately, this efficiency can be compromised due to the ill-conditioned equalization matrix of MMSE method over the correlated channel environments. Moreover, the hard output signal detection exhibits a sharply degradation in performance for higher-order quadrature amplitude modulation (QAM). Therefore, the modern communication systems use the soft-output information, i.e., log-likelihood ratio (LLR) along with the forward error-correcting code (FEC) to achieve satisfactory performance. The LLR computation along with a higher-order QAM remains challenging due to the exhaustive search of symbol in the modulation constellation. In this paper, a low-complexity soft-output signal detector based on approximate inverse symmetric successive over-relaxation preconditioned conjugate gradient (AI-SSOR-CG-SOD) method is proposed to realize MMSE method detection for uplink multiuser massive MIMO correlated channel. In the proposed method, a new preconditioner, an AI-SSOR, which is based on the Neumann series approximation of the inverse of the conventional SSOR preconditioner is firstly developed to handle ill-conditioned matrix, and then incorporated with CG method to improve the convergence rate and performance. According to the characteristic of the Gray-coding that adjacent symbols in the constellation set have only one different bit, the constellation set is divided multiple times based on the bits of the inphase and the quadrature components of the symbol, which reduces the complexity of the LLR computation of the transmitted bits by avoiding the exhaustive search process. Simulation results show that the AI-SSOR preconditioner is robust against spatial correlation effect, and the proposed detector converges at 3 iterations. Simulation results also show that the proposed detector achieves a better trade-off between the complexity and the performance compared to other existing detectors.

Fast solution of three‐dimensional elliptic equations with randomly generated jumping coefficients by using tensor‐structured preconditioners

AbstractIn this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in , . Both the two‐dimensional (2D) and three‐dimensional (3D) elliptic problems are considered for the jumping equation coefficients built as a checkerboard type configuration of bumps randomly distributed on a large , or lattice, respectively. The finite element method discretization procedure on a 3D uniform tensor grid is described in detail, and the Kronecker tensor product approach is proposed for fast generation of the stiffness matrix. We introduce tensor techniques for the construction of the low Kronecker rank spectrally equivalent preconditioner in a periodic setting to be used in the framework of the preconditioned conjugate gradient iteration. The discrete 3D periodic Laplacian pseudo‐inverse is first diagonalized in the Fourier basis, and then the diagonal matrix is reshaped into a fully populated third‐order tensor of size . The latter is approximated by a low‐rank canonical tensor by using the multigrid Tucker‐to‐canonical tensor transform. As an example, we apply the presented solver in numerical analysis of stochastic homogenization method where the 3D elliptic equation should be solved many hundred times, and where for every random sampling of the equation coefficient one has to construct the new stiffness matrix and the right‐hand side. The computational characteristics of the presented solver in terms of a lattice parameter and the grid‐size, , in both 2D and 3D cases are illustrated in numerical tests. Our solver can be used in various applications where the elliptic problem should be solved for a number of different coefficients for example, in many‐particle dynamics, protein docking problems or stochastic modeling.

Goal-oriented adaptive finite element methods with optimal computational complexity

We consider a linear symmetric and elliptic PDE and a linear goal functional. We design and analyze a goal-oriented adaptive finite element method, which steers the adaptive mesh-refinement as well as the approximate solution of the arising linear systems by means of a contractive iterative solver like the optimally preconditioned conjugate gradient method or geometric multigrid. We prove linear convergence of the proposed adaptive algorithm with optimal algebraic rates. Unlike prior work, we do not only consider rates with respect to the number of degrees of freedom but even prove optimal complexity, i.e., optimal convergence rates with respect to the total computational cost.

Efficient SfM for Large-Scale UAV Images Based on Graph-Indexed BoW and Parallel-Constructed BA Optimization

Structure from Motion (SfM) for large-scale UAV (Unmanned Aerial Vehicle) images has been widely used in the fields of photogrammetry and computer vision. Its efficiency, however, decreases dramatically as well as with the memory occupation rising steeply due to the explosion of data volume and the iterative BA (bundle adjustment) optimization. In this paper, an efficient SfM solution is proposed to solve the low-efficiency and high memory consumption problems. First, an algorithm is designed to find UAV image match pairs based on a graph-indexed bag-of-words (BoW) model (GIBoW), which treats visual words as vertices and link relations as edges to build a small-world graph structure. The small-world graph structure can be used to search the nearest-neighbor visual word for query features with extremely high efficiency. Reliable UAV image match pairs can effectively improve feature matching efficiency. Second, a central bundle adjustment with object point-wise parallel construction of the Schur complement (PSCBA) is proposed, which is designed as the combination of the LM (Levenberg–Marquardt) algorithm with the preconditioned conjugate gradients (PCG). The PSCBA method can dramatically reduce the memory consumption in both error and normal equations, as well as improve efficiency. Finally, by using four UAV datasets, the effectiveness of the proposed SfM solution is verified through comprehensive analysis and comparison. The experimental results show that compared with Colmap-Bow and Dbow2, the proposed graph index BoW retrieval algorithm improves the efficiency of image match pair selection with an acceleration ratio ranging from 3 to 7. Meanwhile, the parallel-constructed BA optimization algorithm can achieve accurate bundle adjustment results with an acceleration ratio by 2 to 7 times and reduce memory occupation by 2 to 3 times compared with the BA optimization using Ceres solver. For large-scale UAV images, the proposed method is an effective and reliable solution.

Acceleration of structural topology optimization using symmetric element-by-element strategy for unstructured meshes on GPU

PurposeStructural topology optimization is computationally expensive due to the involvement of high-resolution mesh and repetitive use of finite element analysis (FEA) for computing the structural response. Since FEA consumes most of the computational time in each optimization iteration, a novel GPU-based parallel strategy for FEA is presented and applied to the large-scale structural topology optimization of 3D continuum structures.Design/methodology/approachA matrix-free solver based on preconditioned conjugate gradient (PCG) method is proposed to minimize the computational time associated with solution of linear system of equations in FEA. The proposed solver uses an innovative strategy to utilize only symmetric half of elemental stiffness matrices for implementation of the element-by-element matrix-free solver on GPU.FindingsUsing solid isotropic material with penalization (SIMP) method, the proposed matrix-free solver is tested over three 3D structural optimization problems that are discretized using all hexahedral structured and unstructured meshes. Results show that the proposed strategy demonstrates 3.1× –3.3× speedup for the FEA solver stage and overall speedup of 2.9× –3.3× over the standard element-by-element strategy on the GPU. Moreover, the proposed strategy requires almost 1.8× less GPU memory than the standard element-by-element strategy.Originality/valueThe proposed GPU-based matrix-free element-by-element solver takes a more general approach to the symmetry concept than previous works. It stores only symmetric half of the elemental matrices in memory and performs matrix-free sparse matrix-vector multiplication (SpMV) without any inter-thread communication. A customized data storage format is also proposed to store and access only symmetric half of elemental stiffness matrices for coalesced read and write operations on GPU over the unstructured mesh.

A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems

Abstract In this paper, based on the idea of symmetric-triangular (ST) decomposition and matrix splitting, a product-type preconditioner and its inexact version are developed for generalized saddle point problems. These preconditioners can be combined appropriately with the efficient preconditioned conjugate gradient (PCG) method to solve the generalized saddle point problems, although neither these preconditioners nor the generalized saddle point matrices are symmetric and positive definite, which is the advantage of our proposed preconditioners. The proposed PCG method belongs to the group of nonstandard inner product CG methods, so the convergence theorem of the former is given by the use of that of the latter. The difference between the existing nonstandard inner product CG methods and our proposed method is studied. Theoretical analysis shows that the spectrum of the preconditioned matrix corresponding to the proposed inexact preconditioner is contained in a real, positive interval and the quasi-optimal parameter of the preconditioner is effective and easy to apply in practice. Numerical experiments are given to illustrate the effectiveness and robustness of the proposed inexact preconditioner and show the advantages of the preconditioner over the existing state-of-the-art preconditioners for saddle point problems.

Accelerating Certifiable Estimation with Preconditioned Eigensolvers

Convex (specifically semidefinite) relaxation provides a powerful approach to constructing robust machine perception systems, enabling the recovery of certifiably globally optimal solutions of challenging estimation problems in many practical settings. However, solving the large-scale semidefinite relaxations underpinning this approach remains a formidable computational challenge. A dominant cost in many state-of-the-art (Burer-Monteiro factorization-based) certifiable estimation methods is solution verification (testing the global optimality of a given candidate solution), which entails computing a minimum eigenpair of a certain symmetric certificate matrix. In this letter, we show how to significantly accelerate this verification step, and thereby the overall speed of certifiable estimation methods. First, we show that the certificate matrices arising in the Burer-Monteiro approach generically possess spectra that make the verification problem expensive to solve using standard iterative eigenvalue methods. We then show how to address this challenge using preconditioned eigensolvers; specifically, we design a specialized solution verification algorithm based upon the locally optimal block preconditioned conjugate gradient (LOBPCG) method together with a simple yet highly effective algebraic preconditioner. Experimental evaluation on a variety of simulated and real-world examples shows that our proposed verification scheme is very effective in practice, accelerating solution verification by up to 280x, and the overall Burer-Monteiro method by up to 16x, versus the standard Lanczos method when applied to relaxations derived from large-scale SLAM benchmarks.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

Signed Graph Metric Learning via Gershgorin Disc Perfect Alignment.

Given a convex and differentiable objective Q(M) for a real symmetric matrix M in the positive definite (PD) cone-used to compute Mahalanobis distances-we propose a fast general metric learning framework that is entirely projection-free. We first assume that M resides in a space S of generalized graph Laplacian matrices corresponding to balanced signed graphs. M ∈ S that is also PD is called a graph metric matrix. Unlike low-rank metric matrices common in the literature, S includes the important diagonal-only matrices as a special case. The key theorem to circumvent full eigen-decomposition and enable fast metric matrix optimization is Gershgorin disc perfect alignment (GDPA): given M ∈ S and diagonal matrix S, where Sii = 1/vi and v is the first eigenvector of M, we prove that Gershgorin disc left-ends of similarity transform B = SMS-1 are perfectly aligned at the smallest eigenvalue λmin. Using this theorem, we replace the PD cone constraint in the metric learning problem with tightest possible linear constraints per iteration, so that the alternating optimization of the diagonal / off-diagonal terms in M can be solved efficiently as linear programs via the Frank-Wolfe method. We update v using Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) with warm start as entries in M are optimized successively. Experiments show that our graph metric optimization is significantly faster than cone-projection schemes, and produces competitive binary classification performance.

New Interior-Point Approach for One- and Two-Class Linear Support Vector Machines Using Multiple Variable Splitting

Multiple variable splitting is a general technique for decomposing problems by using copies of variables and additional linking constraints that equate their values. The resulting large optimization problem can be solved with a specialized interior-point method that exploits the problem structure and computes the Newton direction with a combination of direct and iterative solvers (i.e. Cholesky factorizations and preconditioned conjugate gradients for linear systems related to, respectively, subproblems and new linking constraints). The present work applies this method to solving real-world binary classification and novelty (or outlier) detection problems by means of, respectively, two-class and one-class linear support vector machines (SVMs). Unlike previous interior-point approaches for SVMs, which were practical only with low-dimensional points, the new proposal can also deal with high-dimensional data. The new method is compared with state-of-the-art solvers for SVMs that are based on either interior-point algorithms (such as SVM-OOPS) or specific algorithms developed by the machine learning community (such as LIBSVM and LIBLINEAR). The computational results show that, for two-class SVMs, the new proposal is competitive not only against previous interior-point methods—and much more efficient than they are with high-dimensional data—but also against LIBSVM, whereas LIBLINEAR generally outperformed the proposal. For one-class SVMs, the new method consistently outperformed all other approaches, in terms of either solution time or solution quality.

Research on application of LU decomposition and CG method in nonlinear iteration of transmission line finite element method

PurposeThe purpose of this study is to develop a reliable finite element algorithm based on the transmission line method (TLM) to solve the nonlinear problem in electromagnetic field calculation.Design/methodology/approachIn this paper, the TLM has been researched and applied to solve nonlinear iteration in FEM. LU decomposition method and the Jacobi preconditioned conjugate gradient method have been investigated to solve the equations in transmission line finite element method (FEM-TLM). The algorithms have been developed in C++ language. The algorithm is applied to analyze the magnetic field of a long straight current-carrying wire and a single-phase transformer.FindingsFEM-TLM is more effective than traditional FEM in nonlinear iteration. The results of FEM-TLM have been compared and analyzed under different calculation scales. It is found that the LU decomposition method is more suitable for FEM-TLM because there is no need to repeatedly assemble the global coefficient matrix in the iterative solution process and it is not affected by the disturbance of the right-hand vector.Originality/valueAn effective algorithm is provided for solving nonlinear problems in the electromagnetic field, which can save a lot of computing costs. The efficiency of LU decomposition and CG method in FEM-TLM nonlinear iteration is investigated, which also makes a preliminary exploration for the research of FEM-TLM parallel algorithms.

Preconditioned Conjugate Gradient Acceleration on FPGA-Based Platforms

Reconfigurable computing can significantly improve the performance and energy efficiency of many applications. However, FPGA-based chips are evolving rapidly, increasing the difficulty of evaluating the impact of new capabilities such as HBM and high-speed links. In this paper, a real-world application was implemented on different FPGAs in order to better understand the new capabilities of modern FPGAs and how new FPGA technology improves performance and scalability. The aforementioned application was the preconditioned conjugate gradient (PCG) method that is utilized in underground analysis. The implementation was done on four different FPGAs, including an MPSoC, taking into account each platform’s characteristics. The results show that today’s FPGA-based chips offer eight times better performance on a memory-bound problem than 5-year-old FPGAs, as they incorporate HBM and can operate at higher clock frequencies.

Linear programming with a feasible direction interior point technique for smooth optimization

We propose an adaptation of the Feasible Direction Interior Points Algorithm (FDIPA) of J. Herskovits, for solving large-scale linear programs. At each step, the solution of two linear systems with the same coefficient matrix is determined. This step involves a significant computational effort. Reducing the solution time of linear systems is, therefore, a way to improve the performance of the method. The linear systems to be solved are associated with definite positive symmetric matrices. Therefore, we use Split Preconditioned Conjugate Gradient (SPCG) method to solve them, together with an Incomplete Cholesky preconditioner using Matlab’s ICHOL function. We also propose to use the first iteration of the conjugate gradient, and to presolve before applying the algorithm, in order to reduce the computational cost. Following, we then provide mathematica proof that show that the iterations approach Karush–Kuhn–Tucker points of the problem under reasonable assumptions. Finally, numerical evidence show that the method not only works in theory but is also competitive with more advanced methods.

A cascade of preconditioned conjugate gradient networks for accelerated magnetic resonance imaging

Background and objective: Recent unfolding based compressed sensing magnetic resonance imaging (CS-MRI) methods only reinterpret conventional CS-MRI optimization algorithms and, consequently, inherit the weaknesses of the alternating optimization strategy. In order to avoid the structural complexity of the alternating optimization strategy and achieve better reconstruction performance, we propose to directly optimize the ℓ1 regularized convex optimization problem using a deep learning approach. Method: In order to achieve direct optimization, a system of equations solving the ℓ1 regularized optimization problem is constructed from the optimality conditions of a novel primal-dual form proposed for the effective training of the sparsifying transform. The optimal solution is obtained by a cascade of unfolding networks of the preconditioned conjugate gradient (PCG) algorithm trained to minimize the mean element-wise absolute difference (ℓ1 loss) between the terminal output and ground truth image in an end-to-end manner. The performance of the proposed method was compared with that of U-Net, PD-Net, ISTA-Net+, and the recently proposed projection-based cascaded U-Net, using single-coil knee MR images of the fastMRI dataset. Results: In our experiment, the proposed network outperformed existing unfolding-based networks and the complex version of U-Net in several subsampling scenarios. In particular, when using the random Cartesian subsampling mask with 25 % sampling rate, the proposed model outperformed PD-Net by 0.76 dB, ISTA-Net+ by 0.43 dB, and U-Net by 1.21 dB on the positron density without suppression (PD) dataset in term of peak signal to noise ratio. In comparison with the projection-based cascade U-Net, the proposed algorithm achieved approximately the same performance when the sampling rate was 25% with only 1.62% number of network parameters at the cost of a longer reconstruction time (approximately twice). Conclusion: A cascade of unfolding networks of the PCG algorithm was proposed to directly optimize the ℓ1 regularized CS-MRI optimization problem. The proposed network achieved improved reconstruction performance compared with U-Net, PD-Net, and ISTA-Net+, and achieved approximately the same performance as the projection-based cascaded U-Net while using significantly fewer network parameters.

Robust BDDC algorithms for the Brinkman problem with HDG discretizations

The balancing domain decomposition by constraints methods (BDDC) have been applied to solve the saddle point problem arising from a hybridizable discontinuous Galerkin (HDG) discretization for the Brinkman equations. Edge/face average constraints are enforced across the subdomain interface for each velocity component to ensure that the BDDC preconditioned conjugate gradient (CG) iterations stay in a special subspace, where the reduced system from the original saddle point problem is positive definite. The condition number is proved to be uniformly bounded under both Stokes and Darcy dominant cases with a deluxe scaling for the parameters with jumps only across the subdomain interface. When the parameters are highly discontinuous with large jumps inside each subdomains, additional adaptively chosen primal constraints, obtained by solving local generalized eigenvalue problems, are introduced to control the condition numbers. Numerical experiments confirm the theory.

Research of the Algebraic Multigrid Method for Electron Optical Simulator.

At present, electron optical simulator (EOS) takes a long time to solve linear FEM systems. The algebraic multigrid preconditioned conjugate gradient (AMGPCG) method can improve the efficiency of solving systems. This paper is focused on the implementation of the AMGPCG method in EOS. The aggregation-based scheme, which uses two passes of a pairwise matching algorithm and the K-cyle scheme, is adopted in the aggregation-based algebraic multigrid method. Numerical experiments show the advantages and disadvantages of the AMG algorithm in peak memory and solving efficiency. The AMGPCG is more efficient than the iterative methods used in the past and only needs one coarsening when EOS computes the particle motion trajectory.

Seismic Imaging of Complex Velocity Structures by 2D Pseudo-Viscoelastic Time-Domain Full-Waveform Inversion

In the presented study, multi-parameter inversion in the presence of attenuation is used for the reconstruction of the P- and the S- wave velocities and the density models of a synthetic shallow subsurface structure that contains a dipping high-velocity layer near the surface with varying thicknesses. The problem of high-velocity layers also complicates selection of an appropriate initial velocity model. The forward problem is solved with the finite difference, and the inverse problem is solved with the preconditioned conjugate gradient. We used also the adjoint wavefield approach for computing the gradient of the misfit function without explicitly build the sensitivity matrix. The proposed method is capable of either minimizing the least-squares norm of the data misfit or use the Born approximation for estimating partial derivative wavefields. It depends on which characteristics of the recorded data—such as amplitude, phase, logarithm of the complex-valued data, envelope in the misfit, or the linearization procedure of the inverse problem—are used. It showed that by a pseudo-viscoelastic time-domain full-waveform inversion, structures below the high-velocity layer can be imaged. However, by inverting attenuation of P- and S- waves simultaneously with the velocities and mass density, better results would be obtained.

A GPU implementation of the PCG method for large-scale image-based finite element analysis in heterogeneous periodic media

Image-based physics simulations in heterogeneous media at a microscopic scale are a growing trend in various fields within and around Scientific Computing. We consider the scope of Numerical Homogenization, where micro-computed tomography is used to obtain digital models of physical samples, naturally leading to pixel and voxel-based solutions. In this context, the Finite Element Method (FEM) is commonly employed to solve the governing differential equations, via a system of algebraic equations. As image dimensions increase, the memory allocation due to the matrix associated with the FEM quickly becomes unfeasible, even in sparse format. Assembly-free strategies are adopted to reduce memory usage, with the caveat of increased computational cost. The Preconditioned Conjugate Gradient (PCG) method is widely employed to solve this sort of large-scale sparse linear systems , and is fitting to be adapted for assembly-free implementations. This work focuses on a massively parallel PCG solver applied to finite element analyses of heat conduction and linear elasticity on image-based models. Memory-efficiency is one of our main concerns, in an attempt to make feasible the employment of personal-use GPUs for large-scale simulations. The resulting solver is validated with an analytical benchmark, and by comparing the obtained results for a microtomographic model of a cast iron sample against experimental values found in the literature. Time and memory metrics are presented and discussed. It is shown that the developed program allows for homogenization studies of nearly 500 million degrees-of-freedom to be conducted in personal computers equipped with CUDA-enabled devices of 8 GB RAM, taking seconds or a few minutes per system solution with the PCG method. Up to 400× speed-up was observed in comparison to an analogous solver running in a 16-thread CPU. Our GPU implementation makes it possible to conduct, in a matter of minutes, homogenization studies that would take hours, or even days, in personal CPUs.

BDDC Preconditioners for Divergence Free Virtual Element Discretizations of the Stokes Equations

The virtual element method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.

Bernstein-Bézier Galerkin-Characteristics Finite Element Method for Convection-Diffusion Problems

A class of Bernstein-Bézier basis based high-order finite element methods is developed for the Galerkin-characteristics solution of convection-diffusion problems. The Galerkin-characteristics formulation is derived using a semi-Lagrangian discretization of the total derivative in the considered problems. The spatial discretization is performed using the finite element method on unstructured meshes. The Lagrangian interpretation in this approach greatly reduces the time truncation errors in the Eulerian methods. To achieve high-order accuracy in the Galerkin-characteristics solver, the semi-Lagrangian method requires high-order interpolating procedures. In the present work, this step is carried out using the Bernstein-Bézier basis functions to evaluate the solution at the departure points. Triangular Bernstein-Bézier patches are constructed in a simple and inherent manner over finite elements along the characteristics. An efficient preconditioned conjugate gradient solver is used for the linear systems of algebraic equations. Several numerical examples including advection-diffusion equations with known analytical solutions and the viscous Burgers problem are considered to illustrate the accuracy, robustness and performance of the proposed approach. The computed results support our expectations for a stable and highly accurate Bernstein-Bézier Galerkin-characteristics finite element method for convection-diffusion problems.