We show that the mass matrix derived from finite elements can be effectively used as a preconditioner for iteratively solving the linear system arising from finite-difference discretization of the Poisson equation, using the conjugate gradient method. We derive analytically the condition number of the preconditioned operator. Theoretical analysis shows that the ratio of the condition number of the Laplacian to the preconditioned operator is $8/3$ in one dimension, $9/2$ in two dimensions, and $2^9/3^4 \approx 6.3$ in three dimensions. From this it follows that the expected iteration count for achieving a fixed reduction of the norm of the residual is smaller than a half of the number of the iterations of unpreconditioned CG in 2D and 3D. The scheme is easy to implement, and numerical experiments show its efficiency.
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Neural networks have been used for channel decoding, channel detection, channel evaluation, and resource management in multi-input and multi-output (MIMO) wireless communication systems. In this paper, we consider the problem of finding a precoding matrix with high spectral efficiency (SE) using a variational autoencoder. There is an optimization procedure for finding optimal precoding matrices. Our goal is to build a less time-consuming algorithm with minimal loss of quality from the optimal one. As a solution to achieve this goal, we used two types of variational autoencoders to build precoding matrices: the classical variational autoencoder (VAE) and the conditional variational autoencoder (CVAE). Both methods can be used to study a wide range of optimal precoding matrices. The VAE and CVAE methods allow restoring the distribution of the predicted value by sampling random variables from the normal distribution at the intermediate stage of calculations. The construction of precoding matrices and their distribution for the SE objective function using VAE and CVAE methods is described in the literature for the first time.
We present a method for randomizing formulas for bilinear computation of matrix products. We consider the implications of such randomization when there are two sources of error: One due to the formula itself only being approximately correct, and one due to using floating point arithmetic. Our theoretical results and numerical experiments indicate that our method can improve performance when each of these error sources are present individually, as well as when they are present at the same time.
We propose a uniform block diagonal preconditioner for condensed H(div)-conforming HDG schemes for parameter-dependent saddle point problems including the generalized Stokes problem and the linear elasticity. An optimal preconditioner is obtained for the stiffness matrix on the global velocity/displacement space via the auxiliary space preconditioning (ASP) technique [49]. A robust preconditioner spectrally equivalent to the Schur complement on the element-wise constant pressure space is also constructed. Finally, numerical results of the generalized Stokes and the steady linear elasticity equations verify the robustness of our proposed preconditioner with respect to model parameters and mesh size.
The FEAST eigensolver is extended to the computation of the singular triplets of a large matrix $A$ with the singular values in a given interval. It is subspace iteration in nature applied to an approximate spectral projector associated with the cross-product matrix $A^TA$ and constructs approximate left and right singular subspaces corresponding to the desired singular values, onto which $A$ is projected to obtain approximations to the desired singular triplets. Approximate spectral projectors are constructed using the Chebyshev--Jackson series expansion other than contour integration and quadrature rules, and they are proven to be always symmetric positive semi-definite with the eigenvalues in $[0,1]$. Compact estimates are established for pointwise approximation errors of a specific step function that corresponds to the exact spectral projector, the accuracy of the approximate spectral projector, the number of desired singular triplets,the distance between the desired right singular subspace and the subspace generated each iteration, and the convergence of the FEAST SVDsolver. Practical selection strategies are proposed for the series degree and the subspace dimension. Numerical experiments illustrate that the FEAST SVDsolver is robust and efficient.
Inverse source problems arise often in real-world applications, such as localizing unknown groundwater contaminant sources. Being different from Tikhonov regularization, the quasi-boundary value method has been proposed and analyzed as an effective way for regularizing such inverse source problems, which was shown to achieve an optimal order convergence rate under suitable assumptions. However, fast direct or iterative solvers for the resulting all-at-once large-scale linear systems have been rarely studied in the literature. In this work, we first proposed and analyzed a modified quasi-boundary value method, and then developed a diagonalization-based parallel-in-time (PinT) direct solver, which can achieve a dramatic speedup in CPU times when compared with MATLAB's sparse direct solver. In particular, the time-discretization matrix $B$ is shown to be diagonalizable, and the condition number of its eigenvector matrix $V$ is proven to exhibit quadratic growth, which guarantees the roundoff errors due to diagonalization is well controlled. Several 1D and 2D examples are presented to demonstrate the very promising computational efficiency of our proposed method, where the CPU times in 2D cases can be speedup by three orders of magnitude.
Domain decomposition methods are among the most efficient for solving sparse linear systems of equations. Their effectiveness relies on a judiciously chosen coarse space. Originally introduced and theoretically proved to be efficient for self-adjoint operators, spectral coarse spaces have been proposed in the past few years for indefinite and non-self-adjoint operators. This paper presents a new spectral coarse space that can be constructed in a fully-algebraic way unlike most existing spectral coarse spaces. We present theoretical convergence result for Hermitian positive definite diagonally dominant matrices. Numerical experiments and comparisons against state-of-the-art preconditioners in the multigrid community show that the resulting two-level Schwarz preconditioner is efficient especially for non-self-adjoint operators. Furthermore, in this case, our proposed preconditioner outperforms state-of-the-art preconditioners.
We introduce a new hybridized discontinuous Galerkin method for the incompressible magnetohydrodynamics equations. If particular velocity, pressure, magnetic field, and magnetic pressure spaces are employed for both element and trace solution fields, we arrive at an energy stable method which returns pointwise divergence-free velocity fields and magnetic fields and properly balances linear momentum. We discretize in time using a second-order-in-time generalized-$\alpha$ method, and we present a block iterative method for solving the resulting nonlinear system of equations at each time step. We numerically examine the effectiveness of our method using a manufactured solution and observe our method yields optimal convergence rates in the $L_2$ norm for the velocity field, pressure field, magnetic field, and magnetic pressure field. We further find our method is pressure robust. We then apply our method to a selection of benchmark problems and numerically confirm our method is energy stable.
Optimal experimental design (OED) plays an important role in the problem of identifying uncertainty with limited experimental data. In many applications, we seek to minimize the uncertainty of a predicted quantity of interest (QoI) based on the solution of the inverse problem, rather than the inversion model parameter itself. In these scenarios, we develop an efficient method for goal-oriented optimal experimental design (GOOED) for large-scale Bayesian linear inverse problem that finds sensor locations to maximize the expected information gain (EIG) for a predicted QoI. By deriving a new formula to compute the EIG, exploiting low-rank structures of two appropriate operators, we are able to employ an online-offline decomposition scheme and a swapping greedy algorithm to maximize the EIG at a cost measured in model solutions that is independent of the problem dimensions. We provide detailed error analysis of the approximated EIG, and demonstrate the efficiency, accuracy, and both data- and parameter-dimension independence of the proposed algorithm for a contaminant transport inverse problem with infinite-dimensional parameter field.
Advanced finite-element discretizations and preconditioners for models of poroelasticity have attracted significant attention in recent years. The equations of poroelasticity offer significant challenges in both areas, due to the potentially strong coupling between unknowns in the system, saddle-point structure, and the need to account for wide ranges of parameter values, including limiting behavior such as incompressible elasticity. This paper was motivated by an attempt to develop monolithic multigrid preconditioners for the discretization developed in [48]; we show here why this is a difficult task and, as a result, we modify the discretization in [48] through the use of a reduced quadrature approximation, yielding a more "solver-friendly" discretization. Local Fourier analysis is used to optimize parameters in the resulting monolithic multigrid method, allowing a fair comparison between the performance and costs of methods based on Vanka and Braess-Sarazin relaxation. Numerical results are presented to validate the LFA predictions and demonstrate efficiency of the algorithms. Finally, a comparison to existing block-factorization preconditioners is also given.
The positive definiteness of discrete time-fractional derivatives is fundamental to the numerical stability (in the energy sense) for time-fractional phase-field models. A novel technique is proposed to estimate the minimum eigenvalue of discrete convolution kernels generated by the nonuniform L1, half-grid based L1 and time-averaged L1 formulas of the fractional Caputo's derivative. The main discrete tools are the discrete orthogonal convolution kernels and discrete complementary convolution kernels. Certain variational energy dissipation laws at discrete levels of the variable-step L1-type methods are then established for time-fractional Cahn-Hilliard model.They are shown to be asymptotically compatible, in the fractional order limit $\alpha\rightarrow1$, with the associated energy dissipation law for the classical Cahn-Hilliard equation. Numerical examples together with an adaptive time-stepping procedure are provided to demonstrate the effectiveness of the proposed methods.
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