Numerical methods of the ADER family, in particular finite-element ADER-DG and finite-volume ADER-WENO methods, are among the most accurate numerical methods for solving quasilinear hyperbolic PDE systems. The internal structure of ADER-DG and ADER-WENO numerical methods contains a large number of basic linear algebra operations related to matrix multiplications. The main interface of software libraries for matrix multiplications for high-performance computing is BLAS. An effective method for integration the standard functions of the BLAS interface into the implementation of these numerical methods is presented. The calculated matrices are small matrices; and this allows to use effectively JIT technologies. The proposed approach immediately operates on AoS, which allows to efficiently calculate flux, source and non-conservative terms without transposition. The obtained computational costs demonstrated that the effective implementation, based on the use of the JIT functions of the BLAS, outperformed both the implementation based on the general BLAS functions and the vanilla implementations by several orders of magnitude. The complexity of developing an implementation based on the proposed approach does not exceed the complexity of developing a vanilla implementation. Performance analysis using roofline partly explains the observed features of the decreasing of computational costs.

Raising the order of the multipole expansion is a feasible approach for improving the accuracy of the treecode algorithm. However, a uniform order for the expansion would result in the inefficiency of the implementation, especially when the kernel function is singular. In this paper, a $p$-adaptive treecode algorithm is designed to resolve the efficiency issue for problems defined on a general domain. Such a $p$-adaptive method is realized through i). conducting a systematical error analysis for the treecode algorithm, ii). designing a strategy for a non-uniform distribution of the order of multipole expansion towards a given error tolerance, and iii). employing a hierarchy geometry tree structure for coding the algorithm. The proposed $p$-adaptive treecode algorithm is validated by a number of numerical experiments, from which the desired performance is observed successfully, i.e., the computational complexity is reduced dramatically compared with the uniform order case, making our algorithm a competitive one for bottleneck problems such as the demagnetizing field calculation in computational micromagnetics.

The quantum thermal average plays a central role in describing the thermodynamic properties of a quantum system. Path integral molecular dynamics (PIMD) is a prevailing approach for computing quantum thermal averages by approximating the quantum partition function as a classical isomorphism on an augmented space, enabling efficient classical sampling, but the theoretical knowledge of the ergodicity of the sampling is lacking. Parallel to the standard PIMD with $N$ ring polymer beads, we also study the Matsubara mode PIMD, where the ring polymer is replaced by a continuous loop composed of $N$ Matsubara modes. Utilizing the generalized $\Gamma$ calculus, we prove that both the Matsubara mode PIMD and the standard PIMD have uniform-in-$N$ ergodicity, i.e., the convergence rate towards the invariant distribution does not depend on the number of modes or beads $N$.

The principal goal of 1-bit compressive sampling is to decode $n$-dimensional signals with a sparsity level of $s$ from $m$ binary measurements. This task presents significant challenges due to nonlinearity, noise, and sign flips. In this paper, we propose the use of the cardinality-constrained least squares decoder as an optimal solution. We establish that, with high probability, the proposed decoder achieves a minimax estimation error, up to a constant $c$, as long as $m ≥ \mathcal{O}(s{\rm log} \ n)$. In terms of computational efficiency, we employ a generalized Newton algorithm (GNA) to solve the cardinality-constrained minimization problem. At each iteration, this approach incurs the cost of solving a least squares problem with a small size. Through rigorous analysis, we demonstrate that, with high probability, the $ℓ_∞$ norm of the estimation error between the output of GNA and the underlying target diminishes to $\mathcal{O}( \sqrt{\frac{{\rm log} \ n}{m}})$ after at most $\mathcal{O}({\rm log} \ s)$ iterations. Furthermore, provided that the target signal is detectable, we can recover the underlying support with high probability within $\mathcal{O}({\rm log} \ s)$ steps. To showcase the robustness of our proposed decoder and the efficiency of the GNA algorithm, we present extensive numerical simulations and comparisons with state-of-the-art methods.

In this work, we develop a multidomain hybrid discontinuous Galerkin (DG) method and finite difference(FD) method for solving two-dimensional compressible Navier-Stokes equations on the hybrid meshes. The direct discontinuous Galerkin (DDG) method and central difference(CD) scheme are utilized to discretize the viscous fluxes respectively. This approach combines the flexibility for the complex geometries of the DG method on the unstructured meshes, and the computational efficiency of the FD method on Cartesian grids. At the artificial interfaces between the DG subdomain and FD subdomain, the square ghost cells are generated and the weighted essentially non-oscillatory (WENO) interpolation is employed to reconstruct the degrees of freedom of these ghost cells. To ensure the accuracy in smooth regions and the correct position of the shock wave, the troubled cell indicator is adopted to determine the nonconservative or conservative coupling modes. The construction process of the numerical fluxes at the artificial interfaces is described specifically and the corresponding WENO interpolation coefficients are given in detail. Numerous numerical results demonstrate that the multidomain hybrid method achieves high-order accuracy in smooth regions, robustness in shock simulations, flexibility in handling complex geometries, and significant computational cost savings compared to the traditional DG method on the hybrid meshes.

This paper proposes an Adaptive Basis-inspired Deep Neural Network (ABI-DNN) for solving partial differential equations with localized phenomena such as sharp gradients and singularities. Like the adaptive finite element method, ABI-DNN incorporates an iteration of "solve, estimate, mark, enhancement", which automatically identifies challenging regions and adds new neurons to enhance its capability. A key challenge is to force new neurons to focus on identified regions with limited understanding of their roles in approximation. To address this, we draw inspiration from the finite element basis function and construct the novel Basis-inspired Block (BI-block), to help understand the contribution of each block. With the help of the BI-block and the famous Kolmogorov Superposition Theorem, we first develop a novel fixed network architecture named the Basis-inspired Deep Neural Network (BI-DNN), and then integrate it into the aforementioned adaptive framework to propose the ABI-DNN. Extensive numerical experiments demonstrate that both BI-DNN and ABI-DNN can effectively capture the challenging singularities in target functions. Compared to PINN, BI-DNN attains significantly lower relative errors with a similar number of trainable parameters. When a specified tolerance is set, ABI-DNN can adaptively learn an appropriate architecture that achieves an error comparable to that of BI-DNN with the same structure.

A decoupled finite element algorithm is developed for simulating the vortex dynamics on an elastic superconductor which couples the time-dependent Ginzburg-Landau equation with the complex-valued superconducting order parameter and the vector-valued magnetic potential, and the elasticity equation. We present an iterative algorithm for the decoupled system arising from the time and spatial discretization using a combination of preconditioner, algebraic multigrid method (AMG) and preconditioned conjugate gradient method (PCG). The iterative algorithm allows us to perform large-scale three-dimensional simulations of mesoscale pattern formation during superconducting phase transitions with arbitrary elastic boundary conditions. The performance and efficiency of the algorithm are numerically verified by several benchmark problems, exhibiting up to two orders of magnitude improvement depending on the scale of discrete system compared to the exact solver.

In this paper, we develop a provable third order bound-preserving (BP) nodal discontinuous Galerkin (DG) method for compressible miscible displacements. We consider the problem with a multi-component fluid mixture and physically the volumetric concentration of each component, $c_j(j=1,···,N)$, is between 0 and 1. The main idea is to apply a positivity-preserving (PP) method to all $c′_js,$ while enforce $∑_jc_j=1$ simultaneously. First, we treat the time derivative of the pressure as a source and choose suitable “consistent” numerical fluxes in the pressure and concentration equations to construct a nodal interior penalty DG (IPDG) method to enforce $∑_jc_j =1.$ For PP, we represent the cell average of $c_j$ as a weighted summation of Gaussian quadrature point values, and transform which to some other specially chosen point values. We prove that by taking appropriate parameters in the nodal IPDG method and a suitable time stability condition, the cell average can be kept positive, which further implies that the cell averages of all components are between 0 and 1. Finally, we apply a polynomial scaling limiter to obtain physically relevant numerical approximations without sacrificing accuracy. Numerical experiments are given to demonstrate desired accuracy, BP and good performances of our proposed approach.

Based on overlapping domain decomposition, two parallel Schwarz finite element algorithms are presented for solving a two-domain parabolic problem. Our algorithms are based on two ideas. One is the observation that the influence of the non-homogeneous boundary data diminishes exponentially towards the interior of the subdomains. Another is to apply proper partitions of unity to distribute the residuals of the discretized systems reasonably into many subdomains so as to avoid repeated correction in the overlapping parts. The resulting algorithms are fully parallel in each subdomain. At each time step, only one iteration step is required to reach the optimal order of accuracy. The convergence rate depending on the mesh parameters are analyzed. Finally, numerical results are reported, which verify the theoretical analysis.