Optimal Approximation Research Articles

Optimal approximate minimization of one-letter weighted finite automata

Abstract In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory and bounds on the quality of the approximation in the spectral norm and $\ell ^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.

Algorithmic Persuasion with Evidence

In a game of persuasion with evidence, a sender has private information. By presenting evidence on the information, the sender wishes to persuade a receiver to take a single action (e.g., hire a job candidate, or convict a defendant). The sender’s utility depends solely on whether the receiver takes the action. The receiver’s utility depends on both the action and the sender’s private information. We study three natural variations. First, we consider the problem of computing an equilibrium of the game without commitment power. Second, we consider a persuasion variant, where the sender commits to a signaling scheme and the receiver, after seeing the evidence, takes the action or not. Third, we study a delegation variant, where the receiver first commits to taking the action if being presented certain evidence, and the sender presents evidence to maximize the probability the action is taken. We study these variants through the computational lens, and give hardness results, optimal approximation algorithms, and polynomial-time algorithms for special cases. Among our results is an approximation algorithm that rounds a semidefinite program that might be of independent interest, since, to the best of our knowledge, it is the first such approximation algorithm in algorithmic economics.

A global optimality result using coupled proximal contractions with stability analysis

In this work a multi-valued coupled proximal contraction mapping is used for determining the distance between two sets through determination of two pairs of points simultaneously. It is a global optimization problem by its very nature which is converted here into a problem of determining an optimal approximation of the solution of a multi-valued coupled fixed point problem where the actual solution does not exist in the most general cases. The result is well demonstrated with an example. In another two sections we derive a data dependence result and a weak stability result respectively for the problem considered here. For our purpose we define a notion of weak stability of the solution sets. The study is in the framework of metric spaces and broadly falls within the domain of set-valued optimization.

Average Cost Optimality of Partially Observed MDPs: Contraction of Nonlinear Filters and Existence of Optimal Solutions and Approximations

Average Cost Optimality of Partially Observed MDPs: Contraction of Nonlinear Filters and Existence of Optimal Solutions and Approximations

Optimal one-sided approximants of circular arc

The optimal one-sided parametric polynomial approximants of a circular arc are considered. More precisely, the approximant must be entirely in or out of the underlying circle of an arc. The natural restriction to an arc's approximants interpolating boundary points is assumed. However, the study of approximants, which additionally interpolate corresponding tangent directions and curvatures at the boundary of an arc, is also considered. Several low-degree polynomial approximants are studied in detail. When several solutions fulfilling the interpolation conditions exist, the optimal one is characterized, and a numerical algorithm for its construction is suggested. Theoretical results are demonstrated with several numerical examples and a comparison with general (i.e. non-one-sided) approximants are provided.

Low-rank quaternion tensor completion for color video inpainting via a novel factorization strategy

Recently, a quaternion tensor product named Qt-product was proposed, and then the singular value decomposition and the rank of a third-order quaternion tensor were given. From a more applicable perspective, we extend the Qt-product and propose a novel multiplication principle for third-order quaternion tensor named gQt-product. With the gQt-product, we introduce a brand-new singular value decomposition for third-order quaternion tensors named gQt-SVD and then define gQt-rank and multi-gQt-rank. We prove that the optimal low-rank approximation of a third-order quaternion tensor exists and some numerical experiments demonstrate the low-rankness of color videos. So, we apply the low-rank quaternion tensor completion to color video inpainting problems and present alternating least-square algorithms to solve the proposed low gQt-rank and multi-gQt-rank quaternion tensor completion models. The convergence analyses of the proposed algorithms are established and some numerical experiments on various color video datasets show the high recovery accuracy and computational efficiency of our methods.

Temporal second-order two-grid finite element method for semilinear time-fractional Rayleigh-Stokes equations

In this paper, we have developed a temporal second-order two-grid FEM to solve the semilinear time-fractional Rayleigh-Stokes equations. The proposed two-grid FEM uses the L2-1σ scheme and second order scheme to approximate the Caputo fractional derivative and the time first-order derivative in temporal direction and the standard FEM in spatial direction. The L2-norm and H1-norm stability and error estimates for the standard finite element solution and the two-grid solution are derived. The results shown that as long as the mesh sizes satisfy H=h12 and H=hr2r+2 respectively, the two-grid algorithm can achieve asymptotically optimal approximation. Furthermore, the non-uniform L2-1σ scheme was applied for temporal discretization to handle the weak singularity of the solution. Finally, the theoretical findings were confirmed by numerical results, and the effectiveness of the two-grid algorithm was demonstrated.

Optimal approximations for the free boundary problems of the space-time fractional Black-Scholes equations using a combined physics-informed neural network

The combined physics-informed neural network is employed to deal with the free boundary problems of fractional Black-Scholes equations. The solution assumption and the loss function are determined, the transfer learning is borrowed, the combined neural network with data enhancement layer is designed, then the classical Black-Scholes model is numerically solved and the comparative analysis of numerical results under different neural networks is made. For further insight into the long-term memory of fluctuation, the free boundary problems of the space-time Black-Scholes equations under Caputo, Caputo-Fabrizio and Atangana-Baleanu-Caputo fractional derivatives are studied. The corresponding empirical analyses are presented and the optimal exercise boundaries of American put option are simulated. The market analysis shows that introducing fractional calculus tools and neural network algorithms into American put option pricing can yield more realistic prediction results. The work provides a viable method for subsequent researchers to study American option pricing using fractional calculus and neural networks combined with true market data and to deal with the free boundary problems in other research fields.

Requirements for Standard and I/O Cell Library Domestic Characterization System Development

The article is devoted to define requirements and discover specific featuresfor domestic standard cell and I/O libraries characterization EDA tool. A general structure of the library characterization system is presented. The basic requirements indicted for the system are considered and analyzed in the article. A comparative analysis of correspondence to these requirements for domestic and foreign characterization tools is done. The specific features the domestic library characterization system should satisfy are formulated. Among the main features the following are outlined in the article: compatibility and support of domestic OS, support of well-known foreign and domestic Spice-simulators, Russian language support for documentation, scalability, ensuring confidence of the output data. A brief review of typical tasks, intrinsic for characterization tool development and corresponding algorithms, required for its resolving is given. The following issues along with appropriate proposals for its solution are mentioned: automated definition of optimal input waveforms for sequential cells initialization, searching for maximum capacitance load for the cell, setup and hold parameters calculation for sequential cells, method of optimal output current PWL approximation for CCS parameters calculation. As a conclusion of the investigation the current status of domestic characterization systems development is analyzed. In the outcome a problem of domestic library characterization tool that aligned with all the requirements and support the specific features defined above has been raised.

Green Rate‐Splitting Multiple Access for Multicell and Rate Fairness Maximization

ABSTRACTThis paper investigates the green design of rate‐splitting multiple access (RSMA) for multicell, multiantenna downlink communication systems to maximize rate fairness under quality of service (QoS) constraints. An optimization framework is developed to minimize the weighted sum power consumption by jointly optimizing transmit beamformers, power allocation, and common/private rate split while ensuring per‐user minimum rate requirements. The algorithm design explicitly considers the rate fairness among users via a newly proposed green fairness index. The nonconvex transceiver design problems are transformed into tractable forms and efficiently solved by applying tools from alternating optimization and successive convex approximation. Extensive simulations demonstrate that the proposed green RSMA optimization framework can effectively balance the throughput among cell‐edge and cell‐center users, hence combating the near‐far effect. Simulation results show that at 20‐dB SNR, green RSMA achieves over 20% and 50% sum rate improvement compared with NOMA and OMA, respectively. Moreover, green RSMA maintains high fairness among users, even when the channel conditions are disparate. When the weakest user is located 120 m from the base station, green RSMA exhibits a 21.2% higher fairness index than NOMA, showcasing its ability to ensure equitable service quality across users. The framework also achieves substantial energy efficiency gains, consuming less power while providing higher data rates than baseline schemes. These results demonstrate the effectiveness of green RSMA in enhancing spectral efficiency, user fairness, and energy efficiency in multicell, multiantenna communication systems, making it a promising solution for sustainable and high‐performance wireless networks.

Comparison of Adams-Bashforth-Moulton and Dormand-Prince Methods in Lengyel-Epstein Reaction Model Forming Zinc Oxide Nanostructures

This study adopts a numerical approach in the Lengyel-Epstein reaction model for forming zinc oxide (ZnO) nanostructures. It aims to determine the optimal approximation technique to analyze the growth of ion concentrations in ZnO nanostructures. For this purpose, ordinary differential equations are developed using the Dormand-Prince method in the Lengyel-Epstein reaction model. The results obtained from this technique are compared with the results obtained from the Adams-Bashforth-Moulton (ABM) method. After a comparative analysis of both methods, the results showed that the ABM method performs better than the Dormand-Prince method. The accuracy and stability of the ABM method are higher than those of the Dormand-Prince method. Furthermore, error analysis for both methods confirms that the results obtained from the former are more optimized. Moreover, this method also validates the results obtained from the experimental procedure by using the aqueous chemical growth (ACG) method to form the nanostructures of ZnO.

Correlation mining of multimodal features based on higher-order partial least squares for emotion recognition in conversations

In fields requiring an understanding of emotions, such as digital human interaction and public opinion analysis, achieving a dependable and interpretable model for mining correlations among multimodal features remains a primary objective. However, current deep learning methods often lack transparency and suffer from low interpretability. To address these challenges, we propose a novel Correlation Mining method based on Higher-Order Partial Least Squares (HOPLS) for multimodal Emotion Recognition in conversations (CMHER) in this paper. CMHER innovatively combines HOPLS with transformers and Gated Recurrent Units (GRUs) to compute correlation matrices within unimodal data streams and between cross-modal sources. HOPLS projects source data into a latent space to predict target data via correlation matrix computations, eliminating the need for Graphical Processing Unit (GPU) acceleration and making it suitable for experimental and edge systems. The integration of HOPLS with deep neural networks involves preprocessing multimodal features into suitable dimensions and latent representations, followed by HOPLS computing correlation matrices for cross-modal latent vectors and final labels through optimal joint subspace approximation, which aims at the improvements of both interpretability and reliability. Additionally, a generalization error fitting module further refines the predicted correlation matrices to improve predictive capability and overall model performance. Experiments on two public datasets validate the superiority of our proposed method.

Decomposition and graphical correspondence analysis of checkerboard copulas

Abstract We analyze optimal low-rank approximations and correspondence analysis of the dependence structure given by arbitrary bivariate checkerboard copulas. Methodologically, we make use of the truncation of singular value decompositions of doubly stochastic matrices representing the copulas. The resulting (truncated) representations of the dependence structures are sparse, in particular, compared to the number of squares on the checkerboard. The additive structure of the decomposition carries through to statistical functionals of the copula, such as Kendall’s τ \tau or Spearman’s ρ \rho , and also motivates similarity measures for checkerboard copulas. We link our analysis to continuous decompositions of copula densities and copula-generating algorithms and discuss further general properties of the decomposition and its truncation. For example, truncated series might lack nonnegativity, and approximation errors increase for monotonicity-like copulas. We provide algorithms and extensions that account for and counteract these properties. The low-rank representation is illustrated for various copula examples, and some analytical results are derived. The resulting correspondence analysis profile plots are analyzed, providing graphical insights into the dependence structure implied by the copula. An illustration is provided with an empirical data set on fuel injector spray characteristics in jet engines.

On Newton–Cotes Formula-Type Inequalities for Multiplicative Generalized Convex Functions via Riemann–Liouville Fractional Integrals with Applications to Quadrature Formulas and Computational Analysis

In this article, we develop multiplicative fractional versions of Simpson’s and Newton’s formula-type inequalities for differentiable generalized convex functions with the help of established identities. The main motivation for using generalized convex functions lies in their ability to extend results beyond traditional convex functions, encompassing a broader class of functions, and providing optimal approximations for both lower and upper bounds. These inequalities are very useful in finding the error bounds for the numerical integration formulas in multiplicative calculus. Applying these results to the Quadrature formulas demonstrates their practical utility in numerical integration. Furthermore, numerical analysis provides empirical evidence of the effectiveness of the derived findings. It is also demonstrated that the newly proven inequalities extend certain existing results in the literature.

Optimal approximation of a large matrix by a sum of projected linear mappings on prescribed subspaces

We propose and justify a matrix reduction method for calculating the optimal approximation of an observed matrix $A \in \mathbb{C}^{m \times n}$ by a sum $\sum_{i=1}^p \sum_{j=1}^q B_iX_{ij}C_j$ of matrix products where each $B_i \in \mathbb{C}^{m \times g_i}$ and $C_j \in \mathbb{C}^{h_j \times n}$ is known and where the unknown matrix kernels $X_{ij}$ are determined by minimizing the Frobenius norm of the error. The sum can be represented as a bounded linear mapping $BXC$ with unknown kernel $X$ from a prescribed subspace ${\mathcal T} \subseteq \mathbb{C}^n$ onto a prescribed subspace ${\mathcal S} \subseteq \mathbb{C}^m$ defined, respectively, by the collective domains and ranges of the given matrices $C_1,\ldots,C_q$ and $B_1,\ldots,B_p$. We show that the optimal kernel is $X = B^{\dagger}AC^{\dagger}$ and that the optimal approximation $BB^{\dagger}AC^{\dagger}C$ is the projection of the observed mapping $A$ onto a mapping from ${\mathcal T}$ to ${\mathcal S}$. If $A$ is large, $B$ and $C$ may also be large and direct calculation of $B^{\dagger}$ and $C^{\dagger}$ becomes unwieldy and inefficient. The proposed method avoids this difficulty by reducing the solution process to finding the pseudo-inverses of a collection of much smaller matrices. This significantly reduces the computational burden.

High-Resolution and Robust One-Bit Direct-of-Arrival Estimation via Reweighted Atomic Norm Estimation.

In recent years, one-bit quantization has attracted widespread attention in the field of direction-of-arrival (DOA) estimation as a low-cost and low-power solution. Many researchers have proposed various estimation algorithms for one-bit DOA estimation, among which atomic norm minimization algorithms exhibit particularly attractive performance as gridless estimation algorithms. However, current one-bit DOA algorithms with atomic norm minimization typically rely on approximating the trace function, which is not the optimal approximation and introduces errors, along with resolution limitations. To date, there have been few studies on how to enhance resolution under the framework of one-bit DOA estimation. This paper aims to improve the resolution constraints of one-bit DOA estimation. The log-det heuristic is applied to approximate and solve the atomic norm minimization problem. In particular, a reweighted binary atomic norm minimization with noise assumption constraints is proposed to achieve high-resolution and robust one-bit DOA estimation. Finally, the alternating direction method of multipliers algorithm is employed to solve the established optimization problem. Simulations are conducted to demonstrate that the proposed algorithm can effectively enhance the resolution.

A mini immersed finite element method for two-phase Stokes problems on Cartesian meshes

Abstract This paper presents a mini immersed finite element (IFE) method for solving two- and three-dimensional two-phase Stokes problems on Cartesian meshes. The IFE space is constructed from the conventional mini element, with shape functions modified on interface elements according to interface jump conditions while keeping the degrees of freedom unchanged. Both discontinuous viscosity coefficients and surface forces are taken into account in the construction. The interface is approximated using discrete level set functions, and explicit formulas for IFE basis functions and correction functions are derived, facilitating ease of implementation.The inf-sup stability and the optimal a priori error estimate of the IFE method, along with the optimal approximation capabilities of the IFE space, are derived rigorously, with constants that are independent of the mesh size and the manner in which the interface intersects the mesh, but may depend on the discontinuous viscosity coefficients. Additionally, it is proved that the condition number has the usual bound independent of the interface. Numerical experiments are provided to confirm the theoretical results.

Radial Basis Function Neural Network and Feedforward Active Disturbance Rejection Control of Permanent Magnet Synchronous Motor

A composite control strategy is proposed to improve the position-tracking performance and anti-interference capabilities of permanent magnet synchronous motors (PMSMs). This strategy integrates an active disturbance rejection controller (ADRC) and a radial basis function neural network (RBFNN) with feedforward control. Initially, the flexibility and robustness of the ADRC are utilized in the position loop control. Subsequently, the parameters of the extended state observer (ESO) within the ADRC are optimized, benefiting from the fast convergence speed and optimal approximation provided by the RBFNN. To further enhance the dynamic tracking performance, a differential feedforward link is introduced between the desired speed and the output signal. The simulation and experimental results demonstrate that when the expected electrical angle inputs are sinusoidal and pulse signals, the incorporation of the feedforward link and the adjustment of parameters in the ADRC lead to improved position-tracking capabilities and greater adaptability to load disturbances.

Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods

Abstract In this article, we present a scheme for solving two-dimensional hyperbolic equation using an expanded mixed finite element method. To solve the resulting nonlinear expanded mixed finite element system more efficiently, we propose a two-step two-grid algorithm. Numerical stability and error estimate are proved on both the coarse grid and fine grid. It is shown that the two-grid method can achieve asymptotically optimal approximation as long as the coarse grid size H H and the fine grid size h h satisfy h = O ( H ( 2 k + 1 ) ⁄ ( k + 1 ) ) h={\mathcal{O}}\left({H}^{\left(2k+1)/\left(k+1)}) ( k ≥ 1 k\ge 1 ), where k k is the degree of the approximating space for the primary variable. Numerical experiment is presented to demonstrate the accuracy and the efficiency of the proposed method.

An improved accelerated 3-term conjugate gradient algorithm with second-order Hessian approximation for nonlinear least-squares optimization

An improved accelerated 3-term conjugate gradient algorithm with second-order Hessian approximation for nonlinear least-squares optimization