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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We describe a complete set of algorithms for ab initio molecular simulations based on numerically tabulated atom-centered orbitals (NAOs) to capture a wide range of molecular and materials properties from quantum-mechanical first principles. The full algorithmic framework described here is embodied in the Fritz Haber Institute “ab initio molecular simulations” (FHI-aims) computer program package. Its comprehensive description should be relevant to any other first-principles implementation based on NAOs. The focus here is on density-functional theory (DFT) in the local and semilocal (generalized gradient) approximations, but an extension to hybrid functionals, Hartree–Fock theory, and MP2 GW electron self-energies for total energies and excited states is possible within the same underlying algorithms. An all-electron full-potential treatment that is both computationally efficient and accurate is achieved for periodic and cluster geometries on equal footing, including relaxation and ab initio molecular dynamics. We demonstrate the construction of transferable, hierarchical basis sets, allowing the calculation to range from qualitative tight-binding like accuracy to meV-level total energy convergence with the basis set. Since all basis functions are strictly localized, the otherwise computationally dominant grid-based operations scale as O(N) with system size N. Together with a scalar-relativistic treatment, the basis sets provide access to all elements from light to heavy. Both low-communication parallelization of all real-space grid based algorithms and a ScaLapack-based, customized handling of the linear algebra for all matrix operations are possible, guaranteeing efficient scaling (CPU time and memory) up to massively parallel computer systems with thousands of CPUs. In this paper we propose a scheme for choosing basis functions for quantum dynamics calculations. Direct product bases are frequently used. The number of direct product functions required to converge a spectrum, compute a rate constant, etc., is so large that direct product calculations are impossible for molecules or reacting systems with more than four atoms. It is common to extract a smaller working basis from a huge direct product basis by removing some of the product functions. We advocate a build and prune strategy of this type. The one-dimensional (1D) functions from which we build the direct product basis are chosen to satisfy two conditions: (1) they nearly diagonalize the full Hamiltonian matrix; (2) they minimize off-diagonal matrix elements that couple basis functions with diagonal elements close to those of the energy levels we wish to compute. By imposing these conditions we increase the number of product functions that can be removed from the multidimensional basis without degrading the acc... We describe a new method to truncate and recouple basis functions in general variational calculations based on a direct‐product representation of multidimensional wave functions. The method is presented for molecular vibrations; however, the procedure is quite general and can be used in any basis set expansion method. The direct‐product Hamiltonian matrix H is decomposed into a block diagonal matrix H0 plus a remainder H1 . A new subset of basis functions is obtained by diagonalizing H0 . This subset of basis functions is shown to be eigenfunctions of a Hamiltonian in a reduced dimensionality space, ‘‘dressed’’ by the remaining degrees of freedom. These dressed eigenfunctions are then augmented by the component of the original direct‐product basis in which H0 is diagonal. The new basis is recoupled using an energy selection criterion, yielding a substantial reduction in the size of the final full Hamiltonian matrix. The method also suggests a generalization of the vibrational self‐consistent field method,... 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. We propose an iterative method for computing vibrational spectra that significantly reduces the memory cost of calculations. It uses a direct product primitive basis, but does not require storing vectors with as many components as there are product basis functions. Wavefunctions are represented in a basis each of whose functions is a sum of products (SOP) and the factorizable structure of the Hamiltonian is exploited. If the factors of the SOP basis functions are properly chosen, wavefunctions are linear combinations of a small number of SOP basis functions. The SOP basis functions are generated using a shifted block power method. The factors are refined with a rank reduction algorithm to cap the number of terms in a SOP basis function. The ideas are tested on a 20-D model Hamiltonian and a realistic CH3CN (12 dimensional) potential. For the 20-D problem, to use a standard direct product iterative approach one would need to store vectors with about 1020 components and would hence require about 8 × 1011 GB... Precise calculation of molecular electronic wavefunctions by methods such as coupled-cluster requires the computation of tensor contractions, the cost of which has polynomial computational scaling with respect to the system and basis set sizes. Each contraction may be executed via matrix multiplication on a properly ordered and structured tensor. However, data transpositions are often needed to reorder the tensors for each contraction. Writing and optimizing distributed-memory kernels for each transposition and contraction is tedious since the number of contractions scales combinatorially with the number of tensor indices. We present a distributed-memory numerical library (Cyclops Tensor Framework (CTF)) that automatically manages tensor blocking and redistribution to perform any user-specified contractions. CTF serves as the distributed-memory contraction engine in Aquarius, a new program designed for high-accuracy and massively-parallel quantum chemical computations. Aquarius implements a range of coupled-cluster and related methods such as CCSD and CCSDT by writing the equations on top of a C++ templated domain-specific language. This DSL calls CTF directly to manage the data and perform the contractions. Our CCSD and CCSDT implementations achieve high parallel scalability on the BlueGene Q and Cray XC30 supercomputer architectures showing that accurate electronic structure calculations can be effectively carried out on top of general distributed-memory tensor primitives. We introduce Cyclops Tensor Framework (CTF), a distributed-memory library for tensor contractions.CTF is able to perform tensor decomposition, redistribution, and contraction at runtime.CTF enables the expression of massively-parallel coupled-cluster methods via a concise tensor contraction interface.The quantum chemistry software suite Aquarius employs CTF to execute two coupled-cluster methods: CCSD and CCSDT.The Aquarius CCSD and CCSDT codes scale well on BlueGene Q and Cray XC30, comparing favorably to NWChem. We present Spectral Registration with Multilayer Resampling (SRMR) as a 6 Degrees Of Freedom (DOF) registration method for noisy 3D data with partial overlap. The algorithm is based on decoupling 3D rotation from 3D translation by a corresponding resampling process of the spectral magnitude of a 3D Fast Fourier Transform (FFT) calculation on discretized 3D range data. The registration of all 6DOF is then subsequently carried out with spectral registrations using Phase Only Matched Filtering (POMF). There are two main aspects for the fast and robust registration of Euler angles from spherical information in SRMR. First of all, there is the permanent use of phase matching. Second, based on the FFT on a discrete Cartesian grid, not only one spherical layer but also a complete stack of layers are processed in one step. Experiments are presented with challenging datasets with respect to interference and overlap. The results include the fast and robust registration of artificially transformed data for ground-truth comparison, scans from the Stanford Bunny dataset, high end 3D laser range finder (LRF) scans of a city center, and range data from a low-cost actuated LRF in a disaster response scenario. 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. Methods based on l1-relaxation, such as basis pursuit and the Lasso, are very popular for sparse regression in high dimensions. The conditions for success of these methods are now well-understood: (1) exact recovery in the noiseless setting is possible if and only if the design matrix X satisfies the restricted nullspace property, and (2) the squared l2-error of a Lasso estimate decays at the minimax optimal rate k log p n, where k is the sparsity of the p-dimensional regression problem with additive Gaussian noise, whenever the design satisfies a restricted eigenvalue condition. The key issue is thus to determine when the design matrix X satisfies these desirable properties. Thus far, there have been numerous results showing that the restricted isometry property, which implies both the restricted nullspace and eigenvalue conditions, is satisfied when all entries of X are independent and identically distributed (i.i.d.), or the rows are unitary. This paper proves directly that the restricted nullspace and eigenvalue conditions hold with high probability for quite general classes of Gaussian matrices for which the predictors may be highly dependent, and hence restricted isometry conditions can be violated with high probability. In this way, our results extend the attractive theoretical guarantees on l1-relaxations to a much broader class of problems than the case of completely independent or unitary designs. The self-consistent solution of a Schrodinger-like equation for the density matrix is a critical and computationally demanding step in quantum-based models of interatomic bonding. This step was tackled historically via the diagonalization of the Hamiltonian. We have investigated the performance and accuracy of the second-order spectral projection (SP2) algorithm for the computation of the density matrix via a recursive expansion of the Fermi operator in a series of generalized matrix–matrix multiplications. We demonstrate that owing to its simplicity, the SP2 algorithm [Niklasson, A. M. N. Phys. Rev. B2002, 66, 155115] is exceptionally well suited to implementation on graphics processing units (GPUs). The performance in double and single precision arithmetic of a hybrid GPU central processing unit (CPU) and full GPU implementation of the SP2 algorithm exceed those of a CPU-only implementation of the SP2 algorithm and traditional matrix diagonalization when the dimensions of the matrices exceed about 2000 ... Basis sets are some of the most important input data for computational models in the chemistry, materials, biology, and other science domains that utilize computational quantum mechanics methods. Providing a shared, Web-accessible environment where researchers can not only download basis sets in their required format but browse the data, contribute new basis sets, and ultimately curate and manage the data as a community will facilitate growth of this resource and encourage sharing both data and knowledge. We describe the Basis Set Exchange (BSE), a Web portal that provides advanced browsing and download capabilities, facilities for contributing basis set data, and an environment that incorporates tools to foster development and interaction of communities. The BSE leverages and enables continued development of the basis set library originally assembled at the Environmental Molecular Sciences Laboratory. The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar. We review some recently published methods to represent atomic neighbourhood environments, and analyse their relative merits in terms of their faithfulness and suitability for fitting potential energy surfaces. The crucial properties that such representations (sometimes called descriptors) must have are differentiability with respect to moving the atoms, and invariance to the basic symmetries of physics: rotation, reflection, translation, and permutation of atoms of the same species. We demonstrate that certain widely used descriptors that initially look quite different are specific cases of a general approach, in which a finite set of basis functions with increasing angular wave numbers are used to expand the atomic neighbourhood density function. Using the example system of small clusters, we quantitatively show that this expansion needs to be carried to higher and higher wave numbers as the number of neighbours increases in order to obtain a faithful representation, and that variants of the descriptors converge at very different rates. We also propose an altogether new approach, called Smooth Overlap of Atomic Positions (SOAP), that sidesteps these difficulties by directly defining the similarity between any two neighbourhood environments, and show that it is still closely connected to the invariant descriptors. We test the performance of the various representations by fitting models to the potential energy surface of small silicon clusters and the bulk crystal. A simple and practical method for variationally optimizing numerical atomic orbitals used in density functional calculations is presented based on the force theorem. The derived equation provides the same procedure for the optimization of atomic orbitals as that for the geometry optimization. The optimized orbitals well reproduce convergent results calculated by a larger number of unoptimized orbitals. In addition, we demonstrate that the optimized orbitals significantly reduce the computational effort in the geometry optimization, while keeping a high degree of accuracy. Abstract Nonlinear dimensionality reduction embeddings computed from datasets do not provide a mechanism to compute the inverse map. In this paper, we address the problem of computing a stable inverse map to such a general bi-Lipschitz map. Our approach relies on radial basis functions (RBFs) to interpolate the inverse map everywhere on the low-dimensional image of the forward map. We demonstrate that the scale-free cubic RBF kernel performs better than the Gaussian kernel: it does not suffer from ill-conditioning, and does not require the choice of a scale. The proposed construction is shown to be similar to the Nystrom extension of the eigenvectors of the symmetric normalized graph Laplacian matrix. Based on this observation, we provide a new interpretation of the Nystrom extension with suggestions for improvement. We introduce a fermionic formula associated with any quantum affine algebra U q (X N (r) . Guided by the interplay between corner transfer matrix and the Bethe ansatz in solvable lattice models, we study several aspects related to representation theory, most crucially, the crystal basis theory. They include one-dimensional sums over both finite and semi-infinite paths, spinon character formulae, Lepowsky—Primc type conjectural formula for vacuum string functions, dilogarithm identities, Q-systems and their solution by characters of various classical subalgebras and so forth. The results expand [HKOTY1] including the twisted cases and more details on inhomogeneous paths consisting of non-perfect crystals. As a most intriguing example, certain inhomogeneous one-dimensional sums conjecturally give rise to branching functions of an integrable G 2 (1) -module related to the embedding G 2 (1) ↪ B 3 (1) ↪ D 4 1 . This work investigates the problem of 6-Degrees-Of-Freedom (6-DOF) object tracking from RGB-D images, where the object is rigid and a 3D model of the object is known. As in many previous works, we utilize a Particle Filter (PF) framework. In order to have a fast tracker, the key aspect is to design a clever proposal distribution which works reliably even with a small number of particles. To achieve this we build on a recently developed state-of-the-art system for single image 6D pose estimation of known 3D objects, using the concept of so-called 3D object coordinates. The idea is to train a random forest that regresses the 3D object coordinates from the RGB-D image. Our key technical contribution is a two-way procedure to integrate the random forest predictions in the proposal distribution generation. This has many practical advantages, in particular better generalization ability with respect to occlusions, changes in lighting and fast-moving objects. We demonstrate experimentally that we exceed state-of-the-art on a given, public dataset. To raise the bar in terms of fast-moving objects and object occlusions, we also create a new dataset, which will be made publicly available. We use polyharmonic Radial Basis Functions (RBFs) to reconstruct smooth, manifold surfaces from point-cloud data and to repair incomplete meshes. An object's surface is defined implicitly as the zero set of an RBF fitted to the given surface data. Fast methods for fitting and evaluating RBFs allow us to model large data sets, consisting of millions of surface points, by a single RBF — previously an impossible task. A greedy algorithm in the fitting process reduces the number of RBF centers required to represent a surface and results in significant compression and further computational advantages. The energy-minimisation characterisation of polyharmonic splines result in a “smoothest” interpolant. This scale-independent characterisation is well-suited to reconstructing surfaces from non-uniformly sampled data. Holes are smoothly filled and surfaces smoothly extrapolated. We use a non-interpolating approximation when the data is noisy. The functional representation is in effect a solid model, which means that gradients and surface normals can be determined analytically. This helps generate uniform meshes and we show that the RBF representation has advantages for mesh simplification and remeshing applications. Results are presented for real-world rangefinder data. The use of gradient techniques for the development of energy-optimized basis sets has been investigated. The region where the energy surface is approximately quadratic with a positive definite Hessian is found to be very small for large basis sets. However, scaled Newton-Raphson methods prove quite effective even when the starting point is outside this region. The analytic calculation of the Hessian is found to be most efficient in terms of computing time.