Comparison of an Improved Self-consistent Lower Bound Theory with Lehmann’s Method for Low-lying Eigenvalues

2021 ◽  
Author(s):  
Miklos Ronto ◽  
Eli Pollak ◽  
Rocco Martinazzo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Energy Levels ◽  
Upper Bounds ◽  
Hamiltonian Matrix ◽  
Lanczos Algorithm ◽  
Eigenvalue Estimates ◽  
Self Consistent ◽  
Lower Bound Theory ◽  
Bound Theory

Abstract Ritz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT. Using two lattice Hamiltonians, we compared the improved SCLBT with its previous implementation as well as with Lehmann’s lower bound theory. The novel SCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and SCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the SCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the SCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds.

scholarly journals Comparison of an improved self-consistent lower bound theory with Lehmann’s method for low-lying eigenvalues

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Miklos Ronto ◽  
Eli Pollak ◽  
Rocco Martinazzo
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Energy Levels ◽  
Upper Bounds ◽  
Hamiltonian Matrix ◽  
Lanczos Algorithm ◽  
Eigenvalue Estimates ◽  
Self Consistent ◽  
Lower Bound Theory ◽  
Bound Theory

AbstractRitz eigenvalues only provide upper bounds for the energy levels, while obtaining lower bounds requires at least the calculation of the variances associated with these eigenvalues. The well-known Weinstein and Temple lower bounds based on the eigenvalues and variances converge very slowly and their quality is considerably worse than that of the Ritz upper bounds. Lehmann presented a method that in principle optimizes Temple’s lower bounds with significantly improved results. We have recently formulated a Self-Consistent Lower Bound Theory (SCLBT), which improves upon Temple’s results. In this paper, we further improve the SCLBT and compare its quality with Lehmann’s theory. The Lánczos algorithm for constructing the Hamiltonian matrix simplifies Lehmann’s theory and is essential for the SCLBT method. Using two lattice Hamiltonians, we compared the improved SCLBT (iSCLBT) with its previous implementation as well as with Lehmann’s lower bound theory. The novel iSCLBT exhibits a significant improvement over the previous version. Both Lehmann’s theory and the SCLBT variants provide significantly better lower bounds than those obtained from Weinstein’s and Temple’s methods. Compared to each other, the Lehmann and iSCLBT theories exhibit similar performance in terms of the quality and convergence of the lower bounds. By increasing the number of states included in the calculations, the lower bounds are tighter and their quality becomes comparable with that of the Ritz upper bounds. Both methods are suitable for providing lower bounds for low-lying excited states as well. Compared to Lehmann’s theory, one of the advantages of the iSCLBT method is that it does not necessarily require the Weinstein lower bound for its initial input, but Ritz eigenvalue estimates can also be used. Especially owing to this property the iSCLBT method sometimes exhibits improved convergence compared to that of Lehmann’s lower bounds

A LOWER BOUND FOR ORDER-PRESERVING BROADCAST IN THE POSTAL MODEL

1993 ◽  
Vol 03 (04) ◽  
pp. 313-320 ◽  
Author(s):  
PHILIP D. MACKENZIE
Keyword(s):  
Communication Network ◽  
Lower Bound ◽  
Lower Bounds ◽  
Message Passing ◽  
Upper Bounds ◽  
Communication Latency ◽  
Multiple Messages ◽  
Postal Model ◽  
Parameter Ranges

In the postal model of message passing systems, the actual communication network between processors is abstracted by a single communication latency factor, which measures the inverse ratio of the time it takes for a processor to send a message and the time that passes until the recipient receives the message. In this paper we examine the problem of broadcasting multiple messages in an order-preserving fashion in the postal model. We prove lower bounds for all parameter ranges and show that these lower bounds are within a factor of seven of the best upper bounds. In some cases, our lower bounds show significant asymptotic improvements over the previous best lower bounds.

The Heine-Borel theorem in extended basic logic

1949 ◽  
Vol 14 (1) ◽  
pp. 9-15 ◽  
Author(s):  
Frederic B. Fitch
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bound ◽  
Fundamental Theorem ◽  
Upper Bounds ◽  
Basic Logic ◽  
Real Numbers ◽  
Consistent Theory

A demonstrably consistent theory of real numbers has been outlined by the writer in An extension of basic logic1 (hereafter referred to as EBL). This theory deals with non-negative real numbers, but it could be easily modified to deal with negative real numbers also. It was shown that the theory was adequate for proving a form of the fundamental theorem on least upper bounds and greatest lower bounds. More precisely, the following results were obtained in the terminology of EBL: If С is a class of U-reals and is completely represented in Κ′ and if some U-real is an upper bound of С, then there is a U-real which is a least upper bound of С. If D is a class of (U-reals and is completely represented in Κ′, then there is a U-real which is a greatest lower bound of D.

The Unrestricted Black-Box Complexity of Jump Functions

2016 ◽  
Vol 24 (4) ◽  
pp. 719-744 ◽  
Author(s):  
Maxim Buzdalov ◽  
Benjamin Doerr ◽  
Mikhail Kever
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Black Box ◽  
Problem Instance ◽  
Theory Approach ◽  
Function Classes ◽  
Fitness Value ◽  
Jump Function ◽  
Jump Sizes

We analyze the unrestricted black-box complexity of the Jump function classes for different jump sizes. For upper bounds, we present three algorithms for small, medium, and extreme jump sizes. We prove a matrix lower bound theorem which is capable of giving better lower bounds than the classic information theory approach. Using this theorem, we prove lower bounds that almost match the upper bounds. For the case of extreme jump functions, which apart from the optimum reveal only the middle fitness value(s), we use an additional lower bound argument to show that any black-box algorithm does not gain significant insight about the problem instance from the first [Formula: see text] fitness evaluations. This, together with our upper bound, shows that the black-box complexity of extreme jump functions is [Formula: see text].

scholarly journals Computability-theoretic complexity of effective Banach spaces

2022 ◽  
Author(s):  
◽  
Long Qian
Keyword(s):  
Banach Space ◽  
Lower Bound ◽  
Banach Spaces ◽  
Lower Bounds ◽  
Approximation Property ◽  
Upper Bounds ◽  
Space Theory ◽  
Approximation Properties ◽  
Open Problems

<p><b>We investigate the geometry of effective Banach spaces, namely a sequenceof approximation properties that lies in between a Banach space having a basis and the approximation property.</b></p> <p>We establish some upper bounds on suchproperties, as well as proving some arithmetical lower bounds. Unfortunately,the upper bounds obtained in some cases are far away from the lower bound.</p> <p>However, we will show that much tighter bounds will require genuinely newconstructions, and resolve long-standing open problems in Banach space theory.</p> <p>We also investigate the effectivisations of certain classical theorems in Banachspaces.</p>

scholarly journals A lower bound for general t-stack sortable permutations via pattern avoidance

2020 ◽  
Vol 22 ◽  
Author(s):  
Pranav Chinmay
Keyword(s):  
Lower Bound ◽  
Recurrence Relation ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Pattern Avoidance ◽  
Lower And Upper Bounds

There is no formula for general t-stack sortable permutations. Thus, we attempt to study them by establishing lower and upper bounds. Permutations that avoid certain pattern sets provide natural lower bounds. This paper presents a recurrence relation that counts the number of permutations that avoid the set (23451,24351,32451,34251,42351,43251). This establishes a lower bound on 3-stack sortable permutations. Additionally, the proof generalizes to provide lower bounds for all t-stack sortable permutations.

scholarly journals A new lower bound for the smallest singular value

Filomat ◽  
2019 ◽  
Vol 33 (9) ◽  
pp. 2711-2723
Author(s):  
Ksenija Doroslovacki ◽  
Ljiljana Cvetkovic ◽  
Ernest Sanca
Keyword(s):  
Lower Bound ◽  
Boundary Value Problems ◽  
Lower Bounds ◽  
Large Scale ◽  
Block Structure ◽  
Upper Bounds ◽  
Singular Value ◽  
Ecological Modelling ◽  
Infinity Norm ◽  
Smallest Singular Value

The aim of this paper is to obtain new lower bounds for the smallest singular value for some special subclasses of nonsingularH-matrices. This is done in two steps: first, unifying principle for deriving new upper bounds for the norm 1 of the inverse of an arbitrary nonsingular H-matrix is presented, and then, it is combined with some well-known upper bounds for the infinity norm of the inverse. The importance and efficiency of the results are illustrated by an example from ecological modelling, as well as on a type of large-scale matrices posessing a block structure, arising in boundary value problems.

Methods for calculating vibrational energy levels

2004 ◽  
Vol 82 (6) ◽  
pp. 900-914 ◽  
Author(s):  
Tucker Carrington
Keyword(s):  
Iterative Methods ◽  
Vibrational Energy ◽  
Energy Levels ◽  
Basis Functions ◽  
Basis Set ◽  
Hamiltonian Matrix ◽  
Lanczos Algorithm ◽  
New Ideas ◽  
Recent Method ◽  
Vibrational Energy Levels

This article reviews new methods for computing vibrational energy levels of small polyatomic molecules. The principal impediment to the calculation of energy levels is the size of the required basis set. If one uses a product basis the Hamiltonian matrix for a four-atom molecule is too large to store in core memory. We discuss iterative methods that enable one to use a product basis to compute energy levels (and spectra) without storing a Hamiltonian matrix. Despite the advantages of iterative methods it is not possible, using product basis functions, to calculate vibrational spectra of molecules with more than four atoms. A very recent method combining contracted basis functions and the Lanczos algorithm with which vibrational energy levels of methane have been computed is described. New ideas, based on exploiting preconditioning, for reducing the number of matrix-vector products required to converge energy levels of interest are also summarized.Key words: vibrational energy levels, kinetic energy operators, Lanczos algorithm, contracted basis functions, preconditioning.

scholarly journals Constraint and Satisfiability Reasoning for Graph Coloring

2020 ◽  
Vol 69 ◽  
pp. 33-65
Author(s):  
Emmanuel Hebrard ◽  
George Katsirelos
Keyword(s):  
Combinatorial Optimization ◽  
Lower Bound ◽  
Lower Bounds ◽  
Branch And Bound ◽  
Graph Coloring ◽  
Upper Bounds ◽  
Scheduling Problems ◽  
Marginal Costs ◽  
Free Tree ◽  
Pruning Techniques

Graph coloring is an important problem in combinatorial optimization and a major component of numerous allocation and scheduling problems. In this paper we introduce a hybrid CP/SAT approach to graph coloring based on the addition-contraction recurrence of Zykov. Decisions correspond to either adding an edge between two non-adjacent vertices or contracting these two vertices, hence enforcing inequality or equality, respectively. This scheme yields a symmetry-free tree and makes learnt clauses stronger by not committing to a particular color. We introduce a new lower bound for this problem based on Mycielskian graphs; a method to produce a clausal explanation of this bound for use in a CDCL algorithm; a branching heuristic emulating Br´elaz’ heuristic on the Zykov tree; and dedicated pruning techniques relying on marginal costs with respect to the bound and on reasoning about transitivity when unit propagating learnt clauses. The combination of these techniques in both a branch-and-bound and in a bottom-up search outperforms other SAT-based approaches and Dsatur on standard benchmarks both for finding upper bounds and for proving lower bounds.

Quantum collision-resistance of non-uniformly distributed functions: upper and lower bounds

2018 ◽  
Vol 18 (15&16) ◽  
pp. 1332-1349
Author(s):  
Ehsan Ebrahimi ◽  
Dominique Unruh
Keyword(s):  
Lower Bound ◽  
Uniform Distribution ◽  
Lower Bounds ◽  
Upper Bounds ◽  
Query Complexity ◽  
Upper And Lower Bounds ◽  
Collision Resistance ◽  
Quantum Query Complexity ◽  
Quantum Query

We study the quantum query complexity of finding a collision for a function f whose outputs are chosen according to a non-uniform distribution D. We derive some upper bounds and lower bounds depending on the min-entropy and the collision-entropy of D. In particular, we improve the previous lower bound by Ebrahimi Targhi et al. from \Omega(2^{k/9}) to \Omega(2^{k/5}) where k is the min-entropy of D.

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