Showing papers on "Basis (linear algebra) published in 2016"

Testing the manifold hypothesis

Abstract: The hypothesis that high dimensional data tend to lie in the vicinity of a low dimensional manifold is the basis of manifold learning. The goal of this paper is to develop an algorithm (with accompanying complexity guarantees) for fitting a manifold to an unknown probability distribution supported in a separable Hilbert space, only using i.i.d samples from that distribution. More precisely, our setting is the following. Suppose that data are drawn independently at random from a probability distribution $P$ supported on the unit ball of a separable Hilbert space $H$. Let $G(d, V, \tau)$ be the set of submanifolds of the unit ball of $H$ whose volume is at most $V$ and reach (which is the supremum of all $r$ such that any point at a distance less than $r$ has a unique nearest point on the manifold) is at least $\tau$. Let $L(M, P)$ denote mean-squared distance of a random point from the probability distribution $P$ to $M$. We obtain an algorithm that tests the manifold hypothesis in the following sense. The algorithm takes i.i.d random samples from $P$ as input, and determines which of the following two is true (at least one must be): (a) There exists $M \in G(d, CV, \frac{\tau}{C})$ such that $L(M, P) \leq C \epsilon.$ (b) There exists no $M \in G(d, V/C, C\tau)$ such that $L(M, P) \leq \frac{\epsilon}{C}.$ The answer is correct with probability at least $1-\delta$.

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SparseMaps—A systematic infrastructure for reduced-scaling electronic structure methods. IV. Linear-scaling second-order explicitly correlated energy with pair natural orbitals

Abstract: We present a formulation of the explicitly correlated second-order Moller-Plesset (MP2-F12) energy in which all nontrivial post-mean-field steps are formulated with linear computational complexity in system size. The two key ideas are the use of pair-natural orbitals for compact representation of wave function amplitudes and the use of domain approximation to impose the block sparsity. This development utilizes the concepts for sparse representation of tensors described in the context of the domain based local pair-natural orbital-MP2 (DLPNO-MP2) method by us recently [Pinski et al., J. Chem. Phys. 143, 034108 (2015)]. Novel developments reported here include the use of domains not only for the projected atomic orbitals, but also for the complementary auxiliary basis set (CABS) used to approximate the three- and four-electron integrals of the F12 theory, and a simplification of the standard B intermediate of the F12 theory that avoids computation of four-index two-electron integrals that involve two CABS indices. For quasi-1-dimensional systems (n-alkanes), the ON DLPNO-MP2-F12 method becomes less expensive than the conventional ON5 MP2-F12 for n between 10 and 15, for double- and triple-zeta basis sets; for the largest alkane, C200H402, in def2-TZVP basis, the observed computational complexity is N∼1.6, largely due to the cubic cost of computing the mean-field operators. The method reproduces the canonical MP2-F12 energy with high precision: 99.9% of the canonical correlation energy is recovered with the default truncation parameters. Although its cost is significantly higher than that of DLPNO-MP2 method, the cost increase is compensated by the great reduction of the basis set error due to explicit correlation.

Analytical State-Average Complete-Active-Space Self-Consistent Field Nonadiabatic Coupling Vectors: Implementation with Density-Fitted Two-Electron Integrals and Application to Conical Intersections.

Abstract: Analytical state-average complete-active-space self-consistent field derivative (nonadiabatic) coupling vectors are implemented. Existing formulations are modified such that the implementation is compatible with Cholesky-based density fitting of two-electron integrals, which results in efficient calculations especially with large basis sets. Using analytical nonadiabatic coupling vectors, the optimization of conical intersections is implemented within the projected constrained optimization method. The standard description and characterization of conical intersections is reviewed and clarified, and a practical and unambiguous system for their classification and interpretation is put forward. These new tools are subsequently tested and benchmarked for 19 different conical intersections. The accuracy of the derivative coupling vectors is validated, and the information that can be drawn from the proposed characterization is discussed, demonstrating its usefulness.

Review of Discontinuous Galerkin Finite Element Methods for Partial Differential Equations on Complicated Domains

Abstract: The numerical approximation of partial differential equations (PDEs) posed on complicated geometries, which include a large number of small geometrical features or microstructures, represents a challenging computational problem. Indeed, the use of standard mesh generators, employing simplices or tensor product elements, for example, naturally leads to very fine finite element meshes, and hence the computational effort required to numerically approximate the underlying PDE problem may be prohibitively expensive. As an alternative approach, in this article we present a review of composite/agglomerated discontinuous Galerkin finite element methods (DGFEMs) which employ general polytopic elements. Here, the elements are typically constructed as the union of standard element shapes; in this way, the minimal dimension of the underlying composite finite element space is independent of the number of geometrical features. In particular, we provide an overview of hp-version inverse estimates and approximation results for general polytopic elements, which are sharp with respect to element facet degeneration. On the basis of these results, a priori error bounds for the hp-DGFEM approximation of both second-order elliptic and first-order hyperbolic PDEs will be derived. Finally, we present numerical experiments which highlight the practical application of DGFEMs on meshes consisting of general polytopic elements.

Effortless quasi-interpolation in hierarchical spaces

Abstract: We present a general and simple procedure to construct quasi-interpolants in hierarchical spaces. Such spaces are composed of a hierarchy of nested spaces and provide a flexible framework for local refinement. The proposed hierarchical quasi-interpolants are described in terms of the so-called truncated hierarchical basis. Assuming a quasi-interpolant is selected for each space associated with a particular level in the hierarchy, the hierarchical quasi-interpolants are obtained without any additional manipulation. The main properties (like polynomial reproduction) of the quasi-interpolants selected at each level are locally preserved in the hierarchical construction. We show how to construct hierarchical local projectors, and the local approximation order of the underling hierarchical space is also investigated. The presentation is detailed for the truncated hierarchical B-spline basis, and we discuss its extension to a more general framework.

Transforming differential equations of multi-loop Feynman integrals into canonical form

Abstract: The method of differential equations has been proven to be a powerful tool for the computation of multi-loop Feynman integrals appearing in quantum field theory. It has been observed that in many instances a canonical basis can be chosen, which drastically simplifies the solution of the differential equation. In this paper, an algorithm is presented that computes the transformation to a canonical basis, starting from some basis that is, for instance, obtained by the usual integration-by-parts reduction techniques. The algorithm requires the existence of a rational transformation to a canonical basis, but is otherwise completely agnostic about the differential equation. In particular, it is applicable to problems involving multiple scales and allows for a rational dependence on the dimensional regulator. It is demonstrated that the algorithm is suitable for current multi-loop calculations by presenting its successful application to a number of non-trivial examples.

Local complete active space second-order perturbation theory using pair natural orbitals (PNO-CASPT2)

Abstract: We present a CASPT2 method which exploits local approximations to achieve linear scaling of the computational effort with the molecular size, provided the active space is small and local. The inactive orbitals are localized, and the virtual space for each electron pair is spanned by a domain of pair-natural orbitals (PNOs). The configuration space is internally contracted, and the PNOs are defined for uniquely defined orthogonal pairs. Distant pair energies are obtained by multipole approximations, so that the number of configurations that are explicitly treated in the CASPT2 scales linearly with molecular size (assuming a constant active space). The PNOs are generated using approximate amplitudes obtained in a pair-specific semi-canonical basis of projected atomic orbitals (PAOs). The evaluation and transformation of the two-electron integrals use the same parallel local density fitting techniques as recently described for linear-scaling PNO-LMP2 (local second-order Moller-Plesset perturbation theory). The implementation of the amplitude equations, which are solved iteratively, employs the local integrated tensor framework. The efficiency and accuracy of the method are tested for excitation energies and correlation energies. It is demonstrated that the errors introduced by the local approximations are very small. They can be well controlled by few parameters for the distant pair approximation, initial PAO domains, and the PNO domains.

The Mutual Information in Random Linear Estimation

Abstract: We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections, a problem relevant in compressed sensing, sparse superposition codes or code division multiple access just to cite few. There has been a number of works considering the mutual information for this problem using the heuristic replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical cases, we present a converse lower bound via a method that uses spatial coupling, state evolution analysis and the I-MMSE theorem. This yields, in particular, a single letter formula for the mutual information and the minimal-mean-square error for random Gaussian linear estimation of all discrete bounded signals.

A Mixed Prinmal-Dual Bases Algorithm for Regression under Inequality Constraints. Application to Concave regression

Abstract: In some statistical non-parametric models the mean of the random variable y has to satisfy specific constraints. We consider the case where the set defined by the constraints is a closed polyhedral cone K in Rk. For example, when the mean is required to be concave in x, the set of acceptable means is a closed convex cone defined by k-2 linear inequalities in Rk. The least squares estimate of the mean is then the projection of the data point y on the cone K. In this paper, we present an algorithm to find the least square estimate of the mean in a finite number of steps. Other algorithms to solve this problem have been given before. The successive approximations in such algorithms are usually points on the faces of K. The solution here is reached by following a fixed line joining an abitrary but suitably chosen initial point y0 to the data point y. The 1- dimensional subspace spanned by the generators of the cone K is divided into 2' regions which can be described as the set of points with non-negative coordinates in mixed primal-dual bases relative to the cone K. Any point y belongs to one and only one of these regions SJ with corresponding basis 3j. The projection of y on K is immediately obtained from the expression of y in Oj by dropping the dual component of y.

The mutual information in random linear estimation

Abstract: We consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections, a problem relevant in compressed sensing, sparse superposition codes or code division multiple access just to cite few. There has been a number of works considering the mutual information for this problem using the heuristic replica method from statistical physics. Here we put these considerations on a firm rigorous basis. First, we show, using a Guerra-type interpolation, that the replica formula yields an upper bound to the exact mutual information. Secondly, for many relevant practical cases, we present a converse lower bound via a method that uses spatial coupling, state evolution analysis and the I-MMSE theorem. This yields, in particular, a single letter formula for the mutual information and the minimal-mean-square error for random Gaussian linear estimation of all discrete bounded signals.

Robust Density-Based Clustering To Identify Metastable Conformational States of Proteins

Abstract: A density-based clustering method is proposed that is deterministic, computationally efficient, and self-consistent in its parameter choice. By calculating a geometric coordinate space density for every point of a given data set, a local free energy is defined. On the basis of these free energy estimates, the frames are lumped into local free energy minima, ultimately forming microstates separated by local free energy barriers. The algorithm is embedded into a complete workflow to robustly generate Markov state models from molecular dynamics trajectories. It consists of (i) preprocessing of the data via principal component analysis in order to reduce the dimensionality of the problem, (ii) proposed density-based clustering to generate microstates, and (iii) dynamical clustering via the most probable path algorithm to construct metastable states. To characterize the resulting state-resolved conformational distribution, dihedral angle content color plots are introduced which identify structural differences ...

Large-Scale Cubic-Scaling Random Phase Approximation Correlation Energy Calculations Using a Gaussian Basis.

Abstract: We present an algorithm for computing the correlation energy in the random phase approximation (RPA) in a Gaussian basis requiring O(N3) operations and O(N2) memory. The method is based on the resolution of the identity (RI) with the overlap metric, a reformulation of RI-RPA in the Gaussian basis, imaginary time, and imaginary frequency integration techniques, and the use of sparse linear algebra. Additional memory reduction without extra computations can be achieved by an iterative scheme that overcomes the memory bottleneck of canonical RPA implementations. We report a massively parallel implementation that is the key for the application to large systems. Finally, cubic-scaling RPA is applied to a thousand water molecules using a correlation-consistent triple-ζ quality basis.

A guaranteed transient performance-based adaptive neural control scheme with low-complexity computation for flexible air-breathing hypersonic vehicles

Abstract: A robust adaptive neural control scheme is addressed for a generic flexible air-breathing hypersonic vehicle, capable of guaranteeing velocity and altitude tracking errors with desired transient performance. Different from the back-stepping design, a novel neural approximation controller is explored for the altitude subsystem based on a quite simple normal output-feedback formulation rather than a strict-feedback one, while there is no need of the complex recursive design procedure of virtual control laws. Furthermore, on the basis of the minimal learning parameter technique, the updating parameters are reduced greatly. Thus, the exploited strategy exhibits good low-complexity computation. In particular, a new finite-time-convergent differentiator is devised to estimate the newly generated states and it is also employed to provide the necessary high-order time derivatives of reference commands, based on which the proposed control methodology becomes achievable. Finally, the effectiveness of the design is confirmed by simulation results.

A Multiscale Restriction-Smoothed Basis Method for Compressible Black-Oil Models

Abstract: Simulation problems encountered in reservoir management are often computationally expensive because of the complex fluid physics for multiphase flow and the large number of grid cells required to honor geological heterogeneity. Multiscale methods have been proposed as a computationally inexpensive alternative to traditional fine-scale solvers for computing conservative approximations of the pressure and velocity fields on high-resolution geo-cellular models. Although a wide variety of such multiscale methods have been discussed in the literature, these methods have not yet seen widespread use in industry. One reason may be that no method has been presented so far that handles the combination of realistic flow physics and industrystandard grid formats in their full complexity. Herein, we present a multiscale method that handles both the most wide-spread type of flow physics (black-oil type models) and standard grid formats like corner-point, stair-stepped, PEBI, as well as general unstructured, polyhedral grids. Our approach is based on a finite-volume formulation in which the basis functions are constructed using restricted smoothing to effectively capture the local features of the permeability. The method can also easily be formulated for other types of flow models, provided one has a reliable (iterative) solution strategy that computes flow and transport in separate steps. The proposed method is implemented as open-source software and validated on a number of two and three-phase test cases with significant compressibility and gas dissolution. The test cases include both synthetic models and models of real fields with complex wells, faults, and inactive and degenerate cells. Through a prescribed tolerance, the solver can be set to either converge to a sequential or the fully implicit solution, in both cases with a significant speedup compared to a fine-scale multigrid solver. Altogether, this ensures that one can easily and systematically trade accuracy for efficiency, or vice versa.

Multilevel and weighted reduced basis method for stochastic optimal control problems constrained by Stokes equations

Abstract: In this paper we develop and analyze a multilevel weighted reduced basis method for solving stochastic optimal control problems constrained by Stokes equations. We prove the analytic regularity of the optimal solution in the probability space under certain assumptions on the random input data. The finite element method and the stochastic collocation method are employed for the numerical approximation of the problem in the deterministic space and the probability space, respectively, resulting in many large-scale optimality systems to solve. In order to reduce the unaffordable computational effort, we propose a reduced basis method using a multilevel greedy algorithm in combination with isotropic and anisotropic sparse-grid techniques. A weighted a posteriori error bound highlights the contribution stemming from each method. Numerical tests on stochastic dimensions ranging from 10 to 100 demonstrate that our method is very efficient, especially for solving high-dimensional and large-scale optimization problems.

Reduced basis method and domain decomposition for elliptic problems in networks and complex parametrized geometries

Abstract: The aim of this work is to solve parametrized partial differential equations in computational domains represented by networks of repetitive geometries by combining reduced basis and domain decomposition techniques. The main idea behind this approach is to compute once, locally and for few reference shapes, some representative finite element solutions for different values of the parameters and with a set of different suitable boundary conditions on the boundaries: these "functions will represent the basis of a reduced space where the global solution is sought for. The continuity of the latter is assured by a classical domain decomposition approach. Test results on Poisson problem show the flexibility of the proposed method in which accuracy and computational time may be tuned by varying the number of reduced basis functions employed, or the set of boundary conditions used for defining locally the basis functions. The proposed approach simplifies the pre-computation of the reduced basis space by splitting the global problem into smaller local subproblems. Thanks to this feature, it allows dealing with arbitrarily complex network and features more flexibility than a classical global reduced basis approximation where the topology of the geometry is fixed. (C) 2015 Elsevier Ltd. All rights reserved.

Accelerating molecular property calculations with nonorthonormal Krylov space methods

Abstract: We formulate Krylov space methods for large eigenvalue problems and linear equation systems that take advantage of decreasing residual norms to reduce the cost of matrix-vector multiplication. The residuals are used as subspace basis without prior orthonormalization, which leads to generalized eigenvalue problems or linear equation systems on the Krylov space. These nonorthonormal Krylov space (nKs) algorithms are favorable for large matrices with irregular sparsity patterns whose elements are computed on the fly, because fewer operations are necessary as the residual norm decreases as compared to the conventional method, while errors in the desired eigenpairs and solution vectors remain small. We consider real symmetric and symplectic eigenvalue problems as well as linear equation systems and Sylvester equations as they appear in configuration interaction and response theory. The nKs method can be implemented in existing electronic structure codes with minor modifications and yields speed-ups of 1.2-1.8 in typical time-dependent Hartree-Fock and density functional applications without accuracy loss. The algorithm can compute entire linear subspaces simultaneously which benefits electronic spectra and force constant calculations requiring many eigenpairs or solution vectors. The nKs approach is related to difference density methods in electronic ground state calculations and particularly efficient for integral direct computations of exchange-type contractions. By combination with resolution-of-the-identity methods for Coulomb contractions, three- to fivefold speed-ups of hybrid time-dependent density functional excited state and response calculations are achieved.

Two-loop QED corrections to the Altarelli-Parisi splitting functions

Abstract: We compute the two-loop QED corrections to the Altarelli-Parisi (AP) splitting functions by using a deconstructive algorithmic Abelianization of the well-known NLO QCD corrections. We present explicit results for the full set of splitting kernels in a basis that includes the leptonic distribution functions that, starting from this order in the QED coupling, couple to the partonic densities. Finally, we perform a phenomenological analysis of the impact of these corrections in the splitting functions.

Efficient heat-bath sampling in Fock space

Abstract: We introduce an algorithm for sampling many-body quantum states in Fock space The algorithm efficiently samples states with probability approximately proportional to an arbitrary function of the second-quantized Hamiltonian matrix element connecting the sampled state to the current state We apply the new sampling algorithm to the recently developed semistochastic full configuration interaction quantum Monte Carlo (S-FCIQMC) method, a semistochastic implementation of the power method for projecting out the ground state energy in a basis of Slater determinants Our new sampling method requires modest additional computational time and memory compared to uniform sampling but results in newly spawned weights that are approximately of the same magnitude, thereby greatly improving the efficiency of projection A comparison in efficiency between our sampling algorithm and uniform sampling is performed on the all-electron nitrogen dimer at equilibrium in Dunning's cc-pVXZ basis sets with X ∈ {D, T, Q, 5}, demonstrating a large gain in efficiency that increases with basis set size In addition, a comparison in efficiency is performed on three all-electron first-row dimers, B2, N2, and F2, in a cc-pVQZ basis, demonstrating that the gain in efficiency compared to uniform sampling also increases dramatically with the number of electrons

On the properties of positive spanning sets and positive bases

Abstract: The concepts of positive span and positive basis are important in derivative-free optimization. In fact, a well-known result is that if the gradient of a continuously differentiable objective function on \(\mathbb{R}^n\) is nonzero at a point, then one of the vectors in any positive basis (or any positive spanning set) of \(\mathbb{R}^n\) is a descent direction for the objective function from that point. This article summarizes the basic results and explores additional properties of positive spanning sets, positively independent sets and positive bases that are potentially useful in the design of derivative-free optimization algorithms. In particular, it provides construction procedures for these special sets of vectors that were not previously mentioned in the literature. It also proves that invertible linear transformations preserve positive independence and the positive spanning property. Moreover, this article introduces the notion of linear equivalence between positive spanning sets and between positively independent sets to simplify the analysis of their structures. Linear equivalence turns out to be a generalization of the concept of structural equivalence between positive bases that was introduced by Coope and Price (SIAM J Optim 11:859–869, 2001). Furthermore, this article clarifies which properties of linearly independent sets, spanning sets and ordinary bases carry over to positively independent sets, positive spanning sets, and positive bases. For example, a linearly independent set can always be extended to a basis of a linear space but a positively independent set cannot always be extended to a positive basis. Also, the maximum size of a linearly independent set in \(R^n\) is n but there is no limit to the size of a positively independent set in \(\mathbb{R}^n\) when \(n \ge 3\). Whenever possible, the results are proved for the more general case of frames of convex cones instead of focusing only on positive bases of linear spaces. In addition, this article discusses some algorithms for determining whether a given set of vectors is positively independent or whether it positively spans a linear subspace of \(\mathbb{R}^n\). Finally, it provides an algorithm for extending any finite set of vectors to a positive spanning set of \(\mathbb{R}^n\) using only a relatively small number of additional vectors.

Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system.

Abstract: As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.

A Nonorthogonal State-Interaction Approach for Matrix Product State Wave Functions

Abstract: We present a state-interaction approach for matrix product state (MPS) wave functions in a nonorthogonal molecular orbital basis. Our approach allows us to calculate, for example, transition and spin–orbit coupling matrix elements between arbitrary electronic states, provided that they share the same one-electron basis functions and size of the active orbital space, respectively. The key element is the transformation of the MPS wave functions of different states from a nonorthogonal to a biorthonormal molecular orbital basis representation, by exploiting a sequence of nonunitary transformations, following a proposal by Malmqvist [Int. J. Quantum Chem. 1986, 30, 479]. This is well-known for traditional wave function parametrizations but has not yet been exploited for MPS wave functions.

Linear Solution Scheme for Microstructure Design with Process Constraints

Abstract: This paper addresses a two-step linear solution scheme to find an optimum metallic microstructure satisfying performance needs and manufacturability constraints. The microstructure is quantified using the orientation distribution function, which determines the volume densities of crystals that make up the polycrystal microstructure. The orientation distribution function of polycrystalline alloys is represented in a discrete form using finite elements, and the volume-averaged properties are computed. The first step of the solution approach identifies the orientation distribution functions that lead to the set of optimal engineering properties using linear programming. This step leads to multiple solutions, of which only a few can be manufactured using traditional processing routes such as rolling and forging. In the second step, textures from a given process are represented in a space of reduced basis coefficients called the process plane. This step involves generation of orthogonal basis functions for rep...

Using an expanding nondirect product harmonic basis with an iterative eigensolver to compute vibrational energy levels with as many as seven atoms.

Abstract: We demonstrate that it is possible to use a variational method to compute 50 vibrational levels of ethylene oxide (a seven-atom molecule) with convergence errors less than 0.01 cm−1. This is done by beginning with a small basis and expanding it to include product basis functions that are deemed to be important. For ethylene oxide a basis with fewer than 3 × 106 functions is large enough. Because the resulting basis has no exploitable structure we use a mapping to evaluate the matrix-vector products required to use an iterative eigensolver. The expanded basis is compared to bases obtained from pre-determined pruning condition. Similar calculations are presented for molecules with 3, 4, 5, and 6 atoms. For the 6-atom molecule, CH3CH, the required expanded basis has about 106 000 functions and is about an order of magnitude smaller than bases made with a pre-determined pruning condition.

Efficient molecular quantum dynamics in coordinate and phase space using pruned bases

Abstract: We present an efficient implementation of dynamically pruned quantum dynamics, both in coordinate space and in phase space. We combine the ideas behind the biorthogonal von Neumann basis (PvB) with the orthogonalized momentum-symmetrized Gaussians (Weylets) to create a new basis, projected Weylets, that takes the best from both methods. We benchmark pruned time-dependent dynamics using phase-space-localized PvB, projected Weylets, and coordinate-space-localized DVR bases, with real-world examples in up to six dimensions. For the examples studied, coordinate-space localization is the most important factor for efficient pruning and the pruned dynamics is much faster than the unpruned, exact dynamics. Phase-space localization is useful for more demanding dynamics where many basis functions are required. There, projected Weylets offer a more compact representation than pruned DVR bases.

Push it to the limit: Characterizing the convergence of common sequences of basis sets for intermolecular interactions as described by density functional theory

Abstract: With the aim of systematically characterizing the convergence of common families of basis sets such that general recommendations for basis sets can be made, we have tested a wide variety of basis sets against complete-basis binding energies across the S22 set of intermolecular interactions—noncovalent interactions of small and medium-sized molecules consisting of first- and second-row atoms—with three distinct density functional approximations: SPW92, a form of local-density approximation; B3LYP, a global hybrid generalized gradient approximation; and B97M-V, a meta-generalized gradient approximation with nonlocal correlation. We have found that it is remarkably difficult to reach the basis set limit; for the methods and systems examined, the most complete basis is Jensen’s pc-4. The Dunning correlation-consistent sequence of basis sets converges slowly relative to the Jensen sequence. The Karlsruhe basis sets are quite cost effective, particularly when a correction for basis set superposition error is applied: counterpoise-corrected def2-SVPD binding energies are better than corresponding energies computed in comparably sized Dunning and Jensen bases, and on par with uncorrected results in basis sets 3-4 times larger. These trends are exhibited regardless of the level of density functional approximation employed. A sense of the magnitude of the intrinsic incompleteness error of each basis set not only provides a foundation for guiding basis set choice in future studies but also facilitates quantitative comparison of existing studies on similar types of systems.