Arbitrary Constraints Research Articles

Numerical Generalized Randomized HMC processes for restricted domains

We propose a generic approach for numerically efficient simulation from analytically intractable distributions with constrained support. Our approach relies upon Generalized Randomized Hamiltonian Monte Carlo (GRHMC) processes and combines these with a randomized transition kernel that appropriately adjusts the Hamiltonian flow at the boundary of the constrained domain, ensuring that it remains within the domain. The numerical implementation of this constrained GRHMC process exploits the sparsity of the randomized transition kernel and the specific structure of the constraints so that the proposed approach is numerically accurate, computationally fast and operational even in high-dimensional applications. We illustrate this approach with posterior distributions of several Bayesian models with challenging parameter domain constraints in applications to real-word data sets. Building on the capability of GRHMC processes to efficiently explore otherwise challenging and high-dimensional posteriors, the proposed method expands the set of Bayesian models that can be analyzed by using the standard Markov-Chain Monte-Carlo (MCMC) methodology, As such, it can advance the development and use of Bayesian models with useful constrained priors, which are difficult to handle with existing methods. The article is accompanied by an R-package (https://github.com/torekleppe/pdmphmc), which allows for automatically implementing GRHMC processes for arbitrary target distributions and domain constraints.

Statistical disaggregation—A Monte Carlo approach for imputation under constraints

AbstractEquality‐constrained models naturally arise in problems in which the measurements are taken at different levels of resolution. The challenge in this setting is that the models usually induce a joint distribution which is intractable. Resorting to instead sampling from the joint distribution by means of a Monte Carlo approach is also challenging. For example, a naive rejection sampler does not work when the probability mass of the constraint is zero. A typical example of such constrained problems is to learn energy consumption for a higher resolution level based on data at a lower resolution, for example, to decompose a daily reading into readings at a finer level. We introduce a novel Monte Carlo sampling algorithm based on Langevin diffusions and rejection sampling to solve the problem of sampling from equality‐constrained models. Our method has the advantage of being exact for linear constraints and naturally deals with multimodal distributions on arbitrary constraints. We test our method on statistical disaggregation problems for electricity consumption datasets, and our approach provides better uncertainty estimation and accuracy in data imputation compared with other naive/unconstrained methods.

Wave interaction with multiple floating elastic plates with arbitrary constraints near a sloping beach

The problem of wave interaction with multiple elastic plates floating near a sloping beach is considered, particularly resembling the case of a floating solar farm near a coast. The linearized shallow water theory is adopted to describe the motion of fluid. The Kirchhoff–Love plate theory is used to model the elastic plates. A highly efficient domain decomposition approach is applied to derive the solution. Particularly, the eigenfunction expansions are employed to establish the velocity potential in free surface fluid domains, while the Green function method is used to construct the velocity potential of the fluid domain covered by floating elastic plates. This approach can significantly reduce the number of unknowns in the velocity potential, especially when a large number of plates are involved. Extensive results and discussions are provided for the wave run-up on the beach, maximum deflection, and principal strain on the elastic plates. In particular, based on a wide space approximated solution, the oscillatory behaviour of the wave run-up vs the incident wavenumber is analysed, along with the corresponding physical mechanisms. Furthermore, apart from the frequency-domain results, time-domain analyses are also conducted based on a Fourier transform approach. Two different types of incident impulses are considered to interact with floating elastic plates near a beach, namely a Gaussian wave packet and a storm-type incident wave.

Construction of Uniform Designs over a Domain with Linear Constraints

Uniform design is a powerful and robust experimental methodology that is particularly advantageous for multidimensional numerical integration and high-level experiments. As its applications expand across diverse disciplines, the theoretical foundation of uniform design continues to evolve. In real-world scenarios, experimental factors are often subject to one or more linear constraints, which pose challenges in constructing efficient designs within constrained high-dimensional experimental spaces. These challenges typically require sophisticated algorithms, which may compromise uniformity and robustness. Addressing these constraints is critical for reducing costs, improving model accuracy, and identifying global optima in optimization problems. However, existing research primarily focuses on unconstrained or minimally constrained hypercubes, leaving a gap in constructing designs tailored to arbitrary linear constraints. This study bridges this gap by extending the inverse Rosenblatt transformation framework to develop innovative methods for constructing uniform designs over arbitrary hyperplanes and hyperspheres within unit hypercubes. Explicit construction formulas for these constrained domains are derived, offering simplified calculations for practitioners and providing a practical solution applicable to a wide range of experimental scenarios. Numerical simulations demonstrate the feasibility and effectiveness of these methods, setting a new benchmark for uniform design in constrained experimental regions.

Composable Constraint Models for Permutation Enumeration

Constraint programming (CP) is a powerful tool for modeling mathematical concepts and objects and finding both solutions or counter examples. One of the major strengths of CP is that problems can easily be combined or expanded. In this paper, we illustrate that this versatility makes CP an ideal tool for exploring problems in permutation patterns. We declaratively define permutation properties, permutation pattern avoidance and containment constraints using CP and show how this allows us to solve a wide range of problems. We show how this approach enables the arbitrary composition of these conditions, and also allows the easy addition of extra conditions. We demonstrate the effectiveness of our techniques by modelling the containment and avoidance of six permutation patterns, eight permutation properties and measuring five statistics on the resulting permutations. In addition to calculating properties and statistics for the generated permutations, we show that arbitrary additional constraints can also be easily and efficiently added. This approach enables mathematicians to investigate permutation pattern problems in a quick and efficient manner. We demonstrate the utility of constraint programming for permutation patterns by showing how we can easily and efficiently extend the known permutation counts for a conjecture involving the class of $1324$ avoiding permutations. For this problem, we expand the enumeration of $1324$-avoiding permutations with a fixed number of inversions to permutations of length 16 and show for the first time that in the enumeration there is a pattern occurring which follows a unique sequence on the Online Encyclopedia of Integer Sequences.

Scarcity Poetics: Christian Bök’s Eunoia and the Economics of Literary Value

Christian Bök’s Canadian bestseller Eunoia (2001) is an ideal study in how the romantic notion of literary value actually abides economic theories. Eunoia is a collection of five prose-poems each written using only one vowel grapheme (A, E, I, O, or U). These arbitrary material production constraints work just like economic sanctions, artificially inflating the scarcity—and hence value—of the text, both commercially and literarily. Discourse analysis of surrounding debates reveals that detractions and praises alike abide the same basic supply-and-demand logic: e.g., economic theories of (relative) marginal utility, as applied by Lee Erickson in The Economy of Literary Forms (1996). Building on Erickson’s thesis of how publication media costs shaped literary form and content, on Mary Poovey’s history of literary value’s origin in economic value, and on other efforts to combine literary study with economics, this essay applies economic theory to a rare opportunity to generalize the relation of literary material, form, and content. Conclusively, scarcity poetics/tactics are not just unique to Eunoia, but fundamental to all literary form. Eunoia’s extreme manipulation of linguistic materials isolates the general mechanism by which literary value is produced.

The Research on the Differences Between the Markowitz Model and the Index Model under Different Constraints

Investment portfolio optimization has been a commonly discussed topic. This paper compared the differences between two established models regarding such topic, the Markowitz Model and the Index Model, by comparing the optimized portfolio outcomes formed by ten stocks, one equity index (the S&P 500 Index), and a proxy for the riskless asset (1-month Fed Funds Rate) under five different scenarios that each simulates a particular real-life circumstance, and concluded that while the Markowitz Model produced more optimistic results under situations with no constraints, with an arbitrary box constraint, and with a constraint of excluding the broad index, the Index Model outperformed the Markowitz Model under the constraints that simulate Regulation T and the U.S. mutual funds industry. This paper provided an analysis of the strengths and weaknesses of each model under different schemes by comparing the total returns, risks, and reward-to-volatility rate of each portfolio. It may also guide the investors who aim to optimize their portfolios to some extent.

Analytical solutions for large-deflection bending of variable-thickness inhomogeneous functional graded composite circular plates with parameterized boundaries under hygro-thermo-mechanical loads

Analytical solutions for large-deflection bending of variable-thickness inhomogeneous functional graded composite circular plates with parameterized boundaries under hygro-thermo-mechanical loads

Nonlinear thermal flutter of variable angle tow curved panels with elastically restrained edges in supersonic airflow by generalized Galerkin method

Variable Angle Tows (VAT) composite materials can effectively transfer major stresses from the interior region to the edges through in-plane load redistribution, thereby significantly enhancing their structural performance. Therefore, correct handling of boundary conditions is crucial in analyzing VAT structures. Considering that most practical engineering cases involve non-ideal or non-classical boundary support, this paper proposes an efficient method to address the nonlinear thermal flutter problem of elastic supported VAT curved panels in supersonic airflow. Assuming the fiber orientation angle varies in four different ways along the x-direction, the basic equations are established based on the von Kármán large deformation plate theory and the first-order piston theory. The generalized Galerkin method, capable of handling arbitrary elastic boundary constraints, is used to solve the problem in the space domain, while the Runge-Kutta method is applied in the time domain. Numerical examples illustrate how elastically restrained conditions, fiber orientation angles, height-rise ratio, and thermal loads, along with aerodynamic pressure, influence flutter responses of VAT panels. The presented method can be used for the aeroelastic stability design of VAT curved panels under arbitrary boundary conditions.

Mathematical foundation of the master‐slave elimination for arbitrary nonlinear multi‐point constraints

AbstractNonlinear multi‐point constraints are essential in modeling various engineering problems, for example, joints undergoing large rotations or coupling of different element types in finite element analysis. They can be handled using Lagrange multipliers, the penalty method, or master‐slave elimination. The latter satisfies the constraints exactly and reduces the dimension of the resulting system of equations, which is particularly advantageous when a large number of constraints have to be considered. However, the existing schemes in literature are limited to linear constraints. Therefore, the authors introduced an extension of the method to arbitrary nonlinear constraints. A mathematically rigorous derivation of this new method is presented. The starting point is the optimization problem with constraints. It is transformed into a modified optimization problem without constraints using the implicit function theorem. For this, an appropriate selection of slave degrees of freedom (dofs) is crucial, ensuring that the Jacobian constraints meets specific conditions. This implies the consideration of three aspects: The handling of redundant constraints, the automatic selection of slave dofs, and the change of slave dofs in the context of large deformations. Additionally, it allows for the combination of the new method with existing constraint methods.

Exact solutions for anisotropic beams with arbitrary distributed loads

Exact solutions for anisotropic beams with arbitrary distributed loads

Elasticity solutions for functionally graded beams with arbitrary distributed loads

Elasticity solutions for functionally graded beams with arbitrary distributed loads

Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas

Leveraging physics-informed neural computing for transport simulations of nuclear fusion plasmas

Adjusting for covariates representing potential confounders, mediators, or competing predictors in the presence of measurement error: Dispelling a potential misapprehension and insights for optimal study design with nutritional epidemiology examples

Background Variables such as dietary intake are measured with error yet frequently used in observational epidemiology. Although this limitation is sometimes noted, these variables are still often modeled as covariates without formal correction or sincere dialogue about measurement unreliability potentially weakening the validity of statistical conclusions. Further, larger sample sizes increase power (bias) to detect spurious correlations. Counterintuitively, recent work suggested a non-monotonic relationship between confounder unreliability and how much controlling for the confounder reduces (or induces) bias when testing for an exposure-outcome association. If true, such non-monotonicity would be especially concerning for applications such as nutrition, where measurement reliability varies substantially, and large sample sizes are common. Methods We offer a detailed derivations of the square partial correlation between the outcome and exposure, controlling for the confounder. In our derivation, the measurement reliabilities of exposures and confounders are not arbitrarily constrained to be equal. Further, our theoretical results are investigated using simulations. Results Reassuringly, these derivations and simulations show that the counterintuitive non-monotonicity relationship between confounder unreliability and how much controlling for the confounder reduces (or induces) bias when testing for an exposure-outcome association is an artifact of the arbitrary constraint which forces the measurement reliabilities of exposures and confounders to be equal, which that does not always hold. Conclusions The profound and manifold effects of measurement error on estimation and statistical conclusion validity in realistic scenarios indicate that merely mentioning measurement error as a limitation and then dispensing with it is not an adequate response. We also explore questions for optimal study design subject to resource constraints when considering reliability of exposures, covariates, and outcomes.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

We establish an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case.

Unconstraining Multi-Robot Manipulation: Enabling Arbitrary Constraints in ECBS with Bounded Sub-Optimality

Multi-Robot-Arm Motion Planning (M-RAMP) is a challenging problem featuring complex single-agent planning and multi-agent coordination. Recent advancements in extending the popular Conflict-Based Search (CBS) algorithm have made large strides in solving Multi-Agent Path Finding (MAPF) problems. However, fundamental challenges remain in applying CBS to M-RAMP. A core challenge is the existing reliance of the CBS framework on conservative "complete" constraints. These constraints ensure solution guarantees but often result in slow pruning of the search space -causing repeated expensive single-agent planning calls. Therefore, even though it is possible to leverage domain knowledge and design incomplete M-RAMP-specific CBS constraints to more efficiently prune the search, using these constraints would render the algorithm itself incomplete. This forces practitioners to choose between efficiency and completeness. In light of these challenges, we propose a novel algorithm, Generalized ECBS, aimed at removing the burden of choice between completeness and efficiency in MAPF algorithms. Our approach enables the use of arbitrary constraints in conflict-based algorithms while preserving completeness and bounding sub-optimality. This enables practitioners to capitalize on the benefits of arbitrary constraints and opens a new space for constraint design in MAPF that has not been explored. We provide a theoretical analysis of our algorithms, propose new "incomplete" constraints, and demonstrate their effectiveness through experiments in M-RAMP.

Master–slave elimination scheme for arbitrary smooth nonlinear multi-point constraints

Nonlinear multi-point constraints are essential in modeling various engineering problems, for example in the context of (a) linking individual degrees of freedom of multiple nodes to model nonlinear joints, (b) coupling different element types in finite element analysis, (c) enforcing various types of rigidity in parts of the mesh and (d) considering deformation-dependent Dirichlet boundary conditions. One method for addressing constraints is the master–slave elimination, which offers the benefit of reducing the problem dimension as opposed to Lagrange multipliers and the penalty method. However, the existing master–slave elimination method is limited to linear constraints. In this paper, we introduce a new master–slave elimination method for handling arbitrary smooth nonlinear multi-point constraints in the system of equations of the discretized system. We present a rigorous mathematical derivation of the method. Within this method, new constraints can be easily considered as an item of a “constraint library”; i.e. no case-by-case-programming is required. In addition to the theoretical aspects, we also provide helpful remarks on the efficient implementation. Among others, we show that the new method results in a reduced computational complexity compared to the existing methods. The study also places emphasis on comparing the new approach with existing methods via numerical examples. We have developed innovative benchmarks which encompass all relevant computational properties, and provide analytical and reference solutions. Our findings demonstrate that our new method is as accurate, robust and flexible as the Lagrange multipliers, and more efficient due to the reduction of the total number of degrees of freedom, which is particularly advantageous when a large number of constraints have to be considered.

Multi-dimensional State Space Collapse in Non-complete Resource Pooling Scenarios

The present paper establishes an explicit multi-dimensional state space collapse (SSC) for parallel-processing systems with arbitrary compatibility constraints between servers and job types. This breaks major new ground beyond the SSC results and queue length asymptotics in the literature which are largely restricted to complete resource pooling (CRP) scenarios where the steady-state queue length vector concentrates around a line in heavy traffic. The multi-dimensional SSC that we establish reveals heavy-traffic behavior which is also far more tractable than the pre-limit queue length distribution, yet exhibits a fundamentally more intricate structure than in the one-dimensional case, providing useful insight into the system dynamics. In particular, we prove that the limiting queue length vector lives in a K-dimensional cone of which the set of spanning vectors is random in general, capturing the delicate interplay between the various job types and servers. For a broad class of systems we provide a further simplification which shows that the collection of random cones constitutes a fixed K-dimensional cone, resulting in a K-dimensional SSC. The dimension~K represents the number of critically loaded subsystems, or equivalently, capacity bottlenecks in heavy-traffic, with K=1 corresponding to conventional CRP scenarios. Our approach leverages probability generating function (PGF) expressions for Markovian systems operating under redundancy policies.

Wave interaction with multiple adjacent floating solar panels with arbitrary constraints

The problem of wave interaction with multiple adjacent floating solar panels with arbitrary types and numbers of constraints is considered. All the solar panels are assumed to be homogeneous, with the same physical properties, as well as modeled by using the Kirchhoff-Love plate theory. The motion of the fluid is described by the linear velocity potential theory. The domain decomposition method is employed to obtain the solutions. In particular, the entire fluid domain is divided into two types, the one below the free surface, and the other below elastic plates. The velocity potential in the free surface domain is expressed into a series of eigenfunctions. By contrast, the boundary integral equation and the Green function are employed to construct the velocity potential of fluid beneath the entire elastic cover, with unknowns distributed along two interfaces and jumps of physical parameters of the plates. All these unknowns are solved from the system of linear equations, which is established from the matching conditions of velocity potentials and edge conditions. This approach is confirmed with much higher computational efficiency compared with the one only involving eigenfunction expansion for the fluid beneath each plate. Extensive results and discussions are provided for the reflection and transmission coefficients of water waves, maximum deflection, and principal strain of the elastic plates; especially, the influence of different types and numbers of edge constraints are investigated in detail.

Resource engines

In this paper we aim to push the analogy between thermodynamics and quantum resource theories one step further. Previous inspirations were based predominantly on thermodynamic considerations concerning scenarios with a single heat bath, neglecting an important part of thermodynamics that studies heat engines operating between two baths at different temperatures. Here, we investigate the performance of resource engines, which replace the access to two heat baths at different temperatures with two arbitrary constraints on state transformations. The idea is to imitate the action of a two–stroke heat engine, where the system is sent to two agents (Alice and Bob) in turns, and they can transform it using their constrained sets of free operations. We raise and address several questions, including whether or not a resource engine can generate a full set of quantum operations or all possible state transformations, and how many strokes are needed for that. We also explain how the resource engine picture provides a natural way to fuse two or more resource theories, and we discuss in detail the fusion of two resource theories of thermodynamics with two different temperatures, and two resource theories of coherence with respect to two different bases.