Uniform Interpolation Research Articles

Numerical simulations of three-dimensional airfoil icing are computationally intensive, with icing complexities on swept wings surpassing those on straight wings. To enable rapid and accurate ice formation predictions on swept wings, this study proposes a prediction methodology integrating proper orthogonal decomposition (POD) and Kriging surrogate modelling. This approach incorporates key physical parameters influencing ice formation, including flight altitude, flight speed, ambient temperature, liquid water content, and median volume diameter. First, an optimized Latin hypercube sampling method (OLHS) was employed to generate 120 icing conditions under both continuous and intermittent maximum icing scenarios. Numerical simulations were then conducted to establish an icing dataset, which was subsequently transformed into one-dimensional ice height data for various two-dimensional airfoil sections. Next, surrogate models for two-dimensional airfoils were developed using POD and Kriging interpolation to establish relationships between meteorological and flight conditions and the corresponding icing shapes. Finally, three-dimensional ice geometries were reconstructed through uniform interpolation of multiple two-dimensional icing profiles. Validation results demonstrated a strong agreement between surrogate model predictions and numerical simulations, enabling rapid and accurate real-time ice shape estimations across various conditions. The predicted ice shape similarity exceeded 94% for rime ice and 89% for glaze ice. This methodology provides valuable insights for aircraft anti-icing and de-icing design while also contributing to the development of optimized ice-tolerant aerodynamic strategies.

In this study, a Gaussian process model is utilized to study the Fredholm integral equations of the first kind (FIEFKs). Based on the H–Hk formulation, two cases of FIEFKs are under consideration with respect to the right-hand term: discrete data and analytical expressions. In the former case, explicit approximate solutions with minimum norm are obtained via a Gaussian process model. In the latter case, the exact solutions with minimum norm in operator forms are given, which can also be numerically solved via Gaussian process interpolation. The interpolation points are selected sequentially by minimizing the posterior variance of the right-hand term, i.e., minimizing the maximum uncertainty. Compared with uniform interpolation points, the approximate solutions converge faster at sequential points. In particular, for solvable degenerate kernel equations, the exact solutions with minimum norm can be easily obtained using our proposed sequential method. Finally, the efficacy and feasibility of the proposed method are demonstrated through illustrative examples provided in this paper.

Abstract We study interpolation properties for Shavrukov’s bimodal logic $$\textbf{GR}$$ GR of usual and Rosser provability predicates. For this purpose, we introduce a new sublogic $$\textbf{GR}^\circ $$ GR ∘ of $$\textbf{GR}$$ GR and its relational semantics. Based on our new semantics, we prove that $$\textbf{GR}^\circ $$ GR ∘ and $$\textbf{GR}$$ GR enjoy Lyndon interpolation property and uniform interpolation property. As a consequence of our proofs, we obtain the completeness and the finite frame property of $$\textbf{GR}^\circ $$ GR ∘ and $$\textbf{GR}$$ GR with respect to our new semantics.

Reduced-order modeling stands as a pivotal method in curbing the computational expenses linked with expansive fluid dynamics quandaries by employing proxy numerical simulations. Within this realm, downscaling and reconstruction methods serve as fundamental constituents of reduced-order modeling. The traditional intrinsic orthogonal decomposition relies on linear mapping, often relinquishing a substantial amount of nonlinear flow information within the flow field. Meanwhile, autoencoders equipped with fully-connected structures, maybe encounter a parameter explosion when handling larger-scale flow field meshes, impeding effective training. Convolutional autoencoders necessitate uniform interpolation across the flow field to attain a uniform flow field snapshot, yet this process frequently introduces interpolation errors and unwarranted temporal overheads. This paper introduces an innovative solution: a non-interpolated convolutional autoencoder, designed to extract nonlinear features from the flow field while curbing parameter count, evading interpolation errors, and mitigating additional computational burdens. Illustratively, in a two-dimensional cylindrical winding flow scenario, both the reduced dimensional reconstruction display root mean square errors of approximately 1×10-3. Notably, the velocity cloud and absolute error cloud vividly exhibit the non-interpolated convolutional autoencoder's remarkable prowess in reconstruction.

Abstract A modular proof-theoretic framework was recently developed to prove Craig interpolation for normal modal logics based on generalizations of sequent calculi (e.g. nested sequents, hypersequents and labelled sequents). In this paper, we turn to uniform interpolation, which is stronger than Craig interpolation. We develop a constructive method for proving uniform interpolation via nested sequents and apply it to reprove the uniform interpolation property for normal modal logics $\textsf{K}$, $\textsf{D}$ and $\textsf{T}$. We then use the know-how developed for nested sequents to apply the same method to hypersequents and obtain the first direct proof of uniform interpolation for $\textsf{S5}$ via a cut-free sequent-like calculus. While our method is proof-theoretic, the definition of uniform interpolation for nested sequents and hypersequents also uses semantic notions, including bisimulation modulo an atomic proposition.

Abstract In this paper, a proof-theoretic method to prove uniform Lyndon interpolation (ULIP) for non-normal modal and conditional logics is introduced and applied to show that the logics, $\textsf{E}$, $\textsf{M}$, $\textsf{EN}$, $\textsf{MN}$, $\textsf{MC}$, $\textsf{K}$, and their conditional versions, $\textsf{CE}$, $\textsf{CM}$, $\textsf{CEN}$, $\textsf{CMN}$, $\textsf{CMC}$, $\textsf{CK}$, in addition to $\textsf{CKID}$ have that property. In particular, it implies that these logics have uniform interpolation (UIP). Although for some of them the latter is known, the fact that they have uniform LIP is new. Also, the proof-theoretic proofs of these facts are new, as well as the constructive way to explicitly compute the interpolants that they provide. On the negative side, it is shown that the logics $\textsf{CKCEM}$ and $\textsf{CKCEMID}$ enjoy UIP but not uniform LIP. Moreover, it is proved that the non-normal modal logics, $\textsf{EC}$ and $\textsf{ECN}$, and their conditional versions, $\textsf{CEC}$ and $\textsf{CECN}$, do not have Craig interpolation, and whence no uniform (Lyndon) interpolation.

Machine learning predictive model for dynamic response of rising bubbles impacting on a horizontal wall

Abstract A logic $\mathcal{L}$ is said to satisfy the descending chain condition, DCC, if any descending chain of formulas in $\mathcal{L}$ with ordering induced by $\vdash _{\mathcal{L}};$ eventually stops. In this short note, we first establish a general theorem, which states that if a propositional logic $\mathcal{L}$ satisfies both DCC and has the Craig Interpolation Property, CIP, then it satisfies the Uniform Interpolation Property, UIP, as well. As a result, by using the Nishimura lattice, we give a new simply proof of uniform interpolation for $\textbf{IPL}_2$, the two-variable fragment of Intuitionistic Propositional Logic; and one-variable uniform interpolation for $\textbf{IPL}$. Also, we will see that the modal logics $\textbf{S}_4$ and $\textbf{K}_4$ do not satisfy atomic DCC.

A type of manipulator configuration with four-fingers is put forward, for grasping fragile hollow workpieces, with adjustable finger length and finger pad shape. Within a certain range, the manipulator can be used to grasp the internal cross section of fragile objects with different size of circular and oval shape. The design idea and structure of the manipulator are introduced, while the join positions between the finger pad and the finger body, as determined by uniform interpolation or Chebyshev interpolation, are comparatively analyzed and researched regarding forming force contact with workpiece. During the process of grasping, the internal forces and deformations between finger pad and workpiece are analyzed, based on a constructed finite element analysis model. The calculation example shows that, under the same grasping parameters, the maximum impact force on the workpiece is reduced by 63%, when the curvature adjustment points for the finger pad are distributed according to the Chebyshev interpolation, compared to their equal spacing distribution. Research has provided a theoretical basis for the design optimization of the finger pad structure and the connection point positions. For using manipulator to grasp objects with different size of circular and oval shape, the working space of the proposed manipulator is studied. The experiments show that, the manipulator structure, as presented in this article, can meet the requirements of relevant tasks.

Barycentric Lagrange Interpolation Methods for Evaluating Singular Integrals

Many results from cyclic triaxial experiments indicate that porous media, such as clays, exhibit various long-term behaviours under different cyclic stress ratios (CSRs). These can be classified into three main categories, namely, cyclic shakedown, cyclic stable and cyclic failure. Modelling these soil deformation responses, along with pore pressure and other fundamental cyclic aspects, such as closed hysteresis cycles and degradation, is still an open challenge, and research to date is limited. In order to properly describe and capture these characteristics, an enhanced plasticity model, based on the bounding surface and stress distance concepts, is developed here. In detail, a new uniform interpolation function of the plastic modulus, suitable for all loading stages, is proposed, and a new damage factor associated with the plastic shear strain and the deformation type parameter, is also incorporated into the plastic modulus. Accordingly, cyclic shakedown and cyclic failure can be distinguished, and degradation is achieved. Closed hysteresis loops, typical of clays, are obtained through a radial mapping rule along with a moving projection centre, located by the stress reversal points. Comparisons between the obtained numerical results and the experimental ones from literature confirm the suitability of the constitutive approach, which is capable of correctly capturing and reproducing the key aspects of clays’ cyclic behaviour.

In this study, an immersed boundary method developed for compressible viscous flows (Ramakrishnan, R., Girdhar, A., & Ghosh, S. (2016). Immersed Boundary Methods for Compressible Laminar Flows) is modified to improve their stability and robustness. The embedded object is represented as a set of line segments in two dimensions with their outward unit normal vectors specified. A forcing method that leverages the finite volume approach is used, wherein the solution at the cell interfaces that lie near the boundaries of the embedded solid is reconstructed to implicitly satisfy boundary conditions at the immersed surface. The proposed immersed boundary method is validated for transonic inviscid flow past a bump in a channel, supersonic flow past a circular cylinder, transonic viscous flow past a NACA0012 airfoil, and supersonic viscous flow past a circular cylinder. The results are compared with simulations from the literature using contours of flow properties, surface pressure, or Mach number plots and show good agreement.

In this paper by using a model-theoretic approach, we prove Craig interpolation property for Formal Propositional Logic, FPL, Basic propositional logic, BPL and the uniform left-interpolation property for FPL. We also show that there are countably infinite extensions of FPL with the uniform interpolation property.

We present the concept of a disjunctive basis as a generic framework for normal forms in modal logic based on coalgebra. Disjunctive bases were defined in previous work on completeness for modal fixpoint logics, where they played a central role in the proof of a generic completeness theorem for coalgebraic mu-calculi. Believing the concept has a much wider significance, here we investigate it more thoroughly in its own right. We show that the presence of a disjunctive basis at the "one-step" level entails a number of good properties for a coalgebraic mu-calculus, in particular, a simulation theorem showing that every alternating automaton can be transformed into an equivalent nondeterministic one. Based on this, we prove a Lyndon theorem for the full fixpoint logic, its fixpoint-free fragment and its one-step fragment, and a Uniform Interpolation result, for both the full mu-calculus and its fixpoint-free fragment. We also raise the questions, when a disjunctive basis exists, and how disjunctive bases are related to Moss' coalgebraic "nabla" modalities. Nabla formulas provide disjunctive bases for many coalgebraic modal logics, but there are cases where disjunctive bases give useful normal forms even when nabla formulas fail to do so, our prime example being graded modal logic. We also show that disjunctive bases are preserved by forming sums, products and compositions of coalgebraic modal logics, providing tools for modular construction of modal logics admitting disjunctive bases. Finally, we consider the problem of giving a category-theoretic formulation of disjunctive bases, and provide a partial solution.Comment: This is a corrected version of the paper arXiv:1710.10706 published originally on 26/3, 2019

Admissibility of Π2-Inference Rules: interpolation, model completion, and contact algebras

Uniform interpolants were largely studied in non-classical propositional logics since the nineties, and their connection to model completeness was pointed out in the literature. A successive parallel research line inside the automated reasoning community investigated uniform quantifier-free interpolants (sometimes referred to as “covers”) in first-order theories. In this paper, we investigate cover transfer to theory combinations in the disjoint signatures case. We prove that, for convex theories, cover algorithms can be transferred to theory combinations under the same hypothesis needed to transfer quantifier-free interpolation (i.e., the equality interpolating property, aka strong amalgamation property). The key feature of our algorithm relies on the extensive usage of the Beth definability property for primitive fragments to convert implicitly defined variables into their explicitly defining terms. In the non-convex case, we show by a counterexample that covers may not exist in the combined theories, even in case combined quantifier-free interpolants do exist. However, we exhibit a cover transfer algorithm operating also in the non-convex case for special kinds of theory combinations; these combinations (called ‘tame combinations’) concern multi-sorted theories arising in many model-checking applications (in particular, the ones oriented to verification of data-aware processes).

The concept of uniform interpolant for a quantifier-free formula from a given formula with a list of symbols, while well-known in the logic literature, has been unknown to the formal methods and automated reasoning community for a long time. This concept is precisely defined. Two algorithms for computing quantifier-free uniform interpolants in the theory of equality over uninterpreted symbols (EUF) endowed with a list of symbols to be eliminated are proposed. The first algorithm is non-deterministic and generates a uniform interpolant expressed as a disjunction of conjunctions of literals, whereas the second algorithm gives a compact representation of a uniform interpolant as a conjunction of Horn clauses. Both algorithms exploit efficient dedicated DAG representations of terms. Correctness and completeness proofs are supplied, using arguments combining rewrite techniques with model theory.

In this survey, we report our recent work concerning combination results for interpolation and uniform interpolation in the context of quantifier-free fragments of first-order theories. We stress model-theoretic and algebraic aspects connecting this topic with amalgamation, strong amalgamation, and model-completeness. We give sufficient (and, in relevant situations, also necessary) conditions for the transfer of the quantifier-free interpolation property to combined first-order theories; we also investigate the non-disjoint signature case under the assumption that the shared theory is universal Horn. For convex, strong-amalgamating, stably infinite theories over disjoint signatures, we also provide a modular transfer result for the existence of uniform interpolants. Model completions play a key role in the whole paper: They enter into transfer results in the non-disjoint signature case and also represent a semantic counterpart of uniform interpolants.

Recently with the development of deep learning on data representation and generation, how to sampling on a data manifold becomes a crucial problem for research. In this paper, we propose a method to learn a minimizing geodesic within a data manifold. Along the learned geodesic, our method is able to generate high-quality uniform interpolations with the shortest path between two given data samples. Specifically, we use an autoencoder network to map data samples into the latent space and perform interpolation in the latent space via an interpolation network. We add prior geometric information to regularize our autoencoder for a flat latent embedding. The Riemannian metric on the data manifold is induced by the canonical metric in the Euclidean space in which the data manifold is isometrically immersed. Based on this defined Riemannian metric, we introduce a constant-speed loss and a minimizing geodesic loss to regularize the interpolation network to generate uniform interpolations along the learned geodesic on the manifold. We provide a theoretical analysis of our model and use image interpolation as an example to demonstrate the effectiveness of our method.

The solution of initial, boundary, and mixed value problems through the integral equation method yields certain boundary singular integral equations. In many scientific and industrial applications in artificial intelligence, biological systems, scattering, radiation, and image processing, there is a need to evaluate such types of singular integrals, especially the weakly singular kernels. This study presents a new numerical method for evaluating weakly singular kernels based on some advanced matrix-vector barycentric Lagrange interpolation formulas. We developed these formulas to be applied to numerically evaluating singular integrals. At the same time, we created three computational rules to determine the optimal locations for the distribution of the interpolation nodes to be within the integration domain and never be outside for any value of the interpolant degree. These rules are devised so that the equidistant nodes depend on the step-sizes, which are defined as functions of the interpolant degree by some small real number greater than or equal to zero. Thus, we overcame the singularity of the kernels on the whole integration domain and obtained uniform interpolation. Moreover, the presented method gives the kernel's values and the kernel's integral values at the singular points, whereas the numerical or exact values do not exist. The solutions to the illustrated four examples are shown in the given tables and figures. The interpolant solutions which we obtain by low-degree interpolants are faster to converge to the numerical or exact ones (if they exist). This confirms the originality of the presented method and its effectiveness in obtaining high-precision results.