Spectral projectors of second order differential operators play an important role in quantum physics and other scientific and engineering applications. In order to resolve local features and to obtain converged results, typically the number of degrees of freedom needed is much larger than the rank of the spectral projector. This leads to significant cost in terms of both computation and storage. In this paper, we develop a method to construct a basis set that is adaptive to the given differential operator. The basis set is systematically improvable, and the local features of the projector is built into the basis set. As a result the required number of degrees of freedom is only a small constant times the rank of the projector. The construction of the basis set uses a randomized procedure and only requires applying the differential operator to a small number of vectors on the global domain, while each basis function itself is supported on strictly local domains and is discontinuous across the global domain...
In the context of quantum physics, many tailored basis set have been designed to reduce the number of DOFs to represent spectral projectors (or density matrices in physics terminology). Notable examples include the Gaussian basis set and the atomic orbital basis set @cite_54 @cite_19 @cite_43 . Such basis sets are developed based on physical intuition, and can provide relatively accurate solution with much reduced number of degrees of freedom compared to more conventional basis sets. However, expert knowledge is often required to systematically converge the solution. These basis sets have also been used to enrich'' conventional basis sets to achieve a balance between the small number of DOFs and the systematic convergence property @cite_13 @cite_22 . However, the number of DOFs in the mixed basis representation is often much larger than those using Gaussian orbital or atomic orbital basis sets alone.
@cite_22 @cite_54 @cite_19 @cite_43 @cite_13
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Various contracted Gaussian basis sets for atoms up to Kr are presented which have been determined by optimizing atomic self‐consistent field ground state energies with respect to all basis set parameters, i.e., orbital exponents and contraction coefficients. We have developed and implemented a selfconsistent density functional method using standard norm-conserving pseudopotentials and a flexible, numerical linear combination of atomic orbitals basis set, which includes multiple-zeta and polarization orbitals. Exchange and correlation are treated with the local spin density or generalized gradient approximations. The basis functions and the electron density are projected on a real-space grid, in order to calculate the Hartree and exchange-correlation potentials and matrix elements, with a number of operations that scales linearly with the size of the system. We use a modified energy functional, whose minimization produces orthogonal wavefunctions and the same energy and density as the Kohn-Sham energy functional, without the need for an explicit orthogonalization. Additionally, using localized Wannier-like electron wavefunctions allows the computation time and memory required to minimize the energy to also scale linearly with the size of the system. Forces and stresses are also calculated efficiently and accurately, thus allowing structural relaxation and molecular dynamics simulations. In studies of the electronic structure of solids, the augmented plane wave (APW) method is the basis for the solution of the Kohn–Sham equations of density functional theory (DFT). The different versions and developing steps are discussed in terms of linearization, full potential, local orbitals, mixed basis sets, relativistic effects and computational aspects, as employed in the WIEN2k code. We prove that low-rank matrices can be recovered efficiently from a small number of measurements that are sampled from orbits of a certain matrix group. As a special case, our theory makes statements about the phase retrieval problem. Here, the task is to recover a vector given only the amplitudes of its inner product with a small number of vectors from an orbit. Variants of the group in question have appeared under different names in many areas of mathematics. In coding theory and quantum information, it is the complex Clifford group; in time-frequency analysis the oscillator group; and in mathematical physics the metaplectic group. It affords one particularly small and highly structured orbit that includes and generalizes the discrete Fourier basis: While the Fourier vectors have coefficients of constant modulus and phases that depend linearly on their index, the vectors in said orbit have phases with a quadratic dependence. In quantum information, the orbit is used extensively and is known as the set of stabilizer states. We argue that due to their rich geometric structure and their near-optimal recovery properties, stabilizer states form an ideal model for structured measurements for phase retrieval. Our results hold for @math measurements, where the oversampling factor k varies between @math and @math depending on the orbit. The reconstruction is stable towards both additive noise and deviations from the assumption of low rank. If the matrices of interest are in addition positive semidefinite, reconstruction may be performed by a simple constrained least squares regression. Our proof methods could be adapted to cover orbits of other groups.