<div>The design of advanced engineering systems generally results in large-scale numerical problems, which require efficient computational electromagnetic (CEM) solutions. Among existing CEM methods, iterative methods have been a popular choice since conventional direct solutions are computationally expensive. The optimal complexity of an iterative solver is <i>O(NN_itN_rhs)</i> with <i>N</i> being matrix size, <i>N_it</i>the number of iterations and <i>N_rhs</i> the number of right hand sides. How to invert or factorize a dense matrix or a sparse matrix of size <i>N</i> in <i>O(N)</i> (optimal) complexity with explicitly controlled accuracy has been a challenging research problem. For solving a dense matrix of size <i>N</i>, the computational complexity of a conventional direct solution is <i>O(N³)</i>; for solving a general sparse matrix arising from a 3-D EM analysis, the best computational complexity of a conventional direct solution is <i>O(N²)</i>. Recently, an <i>H²</i>-matrix based mathematical framework has been developed to obtain fast dense matrix algebra. However, existing linear-complexity <i>H²</i>-based matrix-matrix multiplication and matrix inversion lack an explicit accuracy control. If the accuracy is to be controlled, the inverse as well as the matrix-matrix multiplication algorithm must be completely changed, as the original formatted framework does not offer a mechanism to control the accuracy without increasing complexity.</div><div> </div><div>In this work, we develop a series of new accuracy controlled fast <i>H²</i> arithmetic, including matrix-matrix multiplication (MMP) without formatted multiplications, minimal-rank MMP, new accuracy controlled <i>H²</i> factorization and inversion, new accuracy controlled <i>H²</i> factorization and inversion with concurrent change of cluster bases, <i>H²</i>-based direct sparse solver and new <i>HSS</i> recursive inverse with directly controlled accuracy. For constant-rank <i>H²</i>-matrices, the proposed accuracy directly controlled <i>H²</i> arithmetic has a strict <i>O(N)</i> complexity in both time and memory. For rank that linearly grows with the electrical size, the complexity of the proposed <i>H²</i> arithmetic is <i>O(NlogN)</i> in factorization and inversion time, and <i>O(N)</i> in solution time and memory for solving volume IEs. Applications to large-scale interconnect extraction as well as large-scale scattering analysis, and comparisons with state-of-the-art solvers have demonstrated the clear advantages of the proposed new <i>H²</i> arithmetic and resulting fast direct solutions with explicitly controlled accuracy. In addition to electromagnetic analysis, the new <i>H²</i> arithmetic developed in this work can also be applied to other disciplines, where fast and large-scale numerical solutions are being pursued. </div>

<p>Among computational electromagnetic methods, Integral Equation (IE) solvers have a great capability in solving open-region problems such as scattering and radiation, due to no truncation boundary condition required. Volume Integral Equation (VIE) solvers are especially capable of handling arbitrarily shaped geometries and inhomogeneous materials. However, the numerical system resulting from a VIE analysis is a dense system, having $N^2$ nonzero elements for a problem of $ N $ unknowns. The dense numerical system in conjunction with the large number of unknowns resulting from a volume discretization prevents a practical use of the VIE for solving large-scale problems.</p> <p>In this work, two fast algorithms of $ O(N \log N) $ complexity to generate an rank-minimized $ H^2 $-representation for electrically large VIEs are developed. The algorithms systematically compress the off-diagonal admissible blocks of full VIE matrix into low-rank forms of total storage of $O(N)$. Both algorithms are based on nested cross approximation, which are purely algebraic. The first one is a two-stage algorithm. The second one is optimized to only use one-stage, and has a significant speedup. Numerical experiments on electrically large examples with over 33 million unknowns demonstrate the efficiency and accuracy of the proposed algorithms. </p> <p>Important applications of VIEs in thermal and quantum engineering have also been explored in this work. Creating spin(circularly)-polarized infrared thermal radiation source without an external magnetic field is important in science and engineering. Here we study two materials, magnetic Weyl semimetals and manganese-bismuth(MnBi), which both have permittivity tensors of large gyrotropy, and can emit circularly-polarized thermal radiations without an external magnetic field. We also design symmetry-broken metasurfaces, which show strong circularly-polarized radiations in simulations and experiments. In spin qubit quantum computing systems, metallic gates and antennas are necessary for quantum gate operations. But their existence greatly enhances evanescent wave Johnson noise (EWJN), which induces the decay of spin qubits and limits the quantum gate operation fidelity. Here we first use VIE to accurately simulate realistic quantum gate designs and quantify the influence on gate fidelity due to this noise.</p>