Gradient Of Cost Function Research Articles

2D Magnetotelluric Inversion Using Linear Finite Element Methods and a Discretize‐Last Strategy With First and Second–Order Anisotropic Regularization

AbstractWe present a new inversion scheme for 2D magnetotelluric data. In contrast to established approaches, it is based on a mesh‐free formulation of the Quasi‐Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) iteration which uses the cost function gradient to implicitly construct approximations of the Hessian inverse to update the unknown conductivity. We introduce conventional first–order regularization as well as second–order regularization where inversions based on the latter are more appropriate for sparse data and can be read as maximum likelihood estimation of the unknown conductivity. We apply first–order finite element method (FEM) discretizations of the inversion scheme, forward and adjoint problems, where the latter is required for the construction of the cost function gradients. We allow for unstructured first–order triangular meshes supporting an enhanced ground level resolution including topographical features and coarsening at the far field leading to significant reduction in computational costs from using structured mesh. Formulating the inversion iteration in continuous form prior to discretization eliminates bias due to local refinements in the mesh and gives way for computationally efficient sparse matrix techniques in the implementation. A keystone in the new scheme is the multi‐grid approximation of the Hessian of the regularizations to construct efficient preconditioning for the inversion iteration. The method is applied to the Commeni4 benchmark and two field data sets. Tests show that for both first and second–order regularization an anisotropic approach is important to address the vast differences in horizontal and vertical spatial scale which in conventional approaches is implicitly introduced through the elongated shape of grid cells.

A hybrid quantum-classical classification model based on branching multi-scale entanglement renormalization ansatz

Tensor networks are emerging architectures for implementing quantum classification models. The branching multi-scale entanglement renormalization ansatz (BMERA) is a tensor network known for its enhanced entanglement properties. This paper introduces a hybrid quantum-classical classification model based on BMERA and explores the correlation between circuit layout, expressiveness, and classification accuracy. Additionally, we present an autodifferentiation method for computing the cost function gradient, which serves as a viable option for other hybrid quantum-classical models. Through numerical experiments, we demonstrate the accuracy and robustness of our classification model in tasks such as image recognition and cluster excitation discrimination, offering a novel approach for designing quantum classification models.

Provably Trainable Rotationally Equivariant Quantum Machine Learning

Exploiting the power of quantum computation to realize superior machine learning algorithms has been a major research focus of recent years, but the prospects of quantum machine learning (QML) remain dampened by considerable technical challenges. A particularly significant issue is that generic QML models suffer from so-called barren plateaus in their training landscapes—large regions where cost function gradients vanish exponentially in the number of qubits employed, rendering large models effectively untrainable. A leading strategy for combating this effect is to build problem-specific models that take into account the symmetries of their data in order to focus on a smaller, relevant subset of Hilbert space. In this work, we introduce a family of rotationally equivariant QML models built upon the quantum Fourier transform, and leverage recent insights from the Lie-algebraic study of QML models to prove that (a subset of) our models do not exhibit barren plateaus. In addition to our analytical results we numerically test our rotationally equivariant models on a dataset of simulated scanning tunneling microscope images of phosphorus impurities in silicon, where rotational symmetry naturally arises, and find that they dramatically outperform their generic counterparts in practice. Published by the American Physical Society 2024

Reliability-based design optimization for a vertical-type breakwater with multiple limit-state equations under Korean marine environments varying from sea to sea

In this study, as part of basic research aimed at enhancing the accuracy of load and resistance coefficients to ensure their suitability for practical design and promoting the application of underutilized reliability-based design in Korea, the author conducts optimal design based on the reliability analysis of a vertical-type breakwater in the seas off of Haeundae, Yeosu, Mokpo, Gunsan, and Incheon—representative ports in Korea. In doing so, the author utilized the double-loop approach, which simultaneously addresses a reliability problem nested within an optimization process, employing the Polak–He optimization algorithm. To mitigate the substantial numerical effort required by the double-loop approach based on the Polak–He optimization algorithm, which necessitates the gradients of both cost and constraint functions, the subset simulation method was employed. In this process, the author deliberately refrained from using design waves of a specific return period and linear probabilistic models such as the Gaussian distribution, especially concerning wave and lift forces, often viewed as barriers to the widespread application of reliability-based design in Korea. Instead, the author focused on characterizing the uncertainties associated with the wave force, lift force, and overturning moment—variables that significantly impact the integrity of vertical-type breakwaters—by developing probabilistic models for these random variables directly from long-term in situ wave data. These models capture the varied characteristics of the Korean marine environment from sea to sea. In this way, the need for additional assumptions concerning the interrelationship between significant wave and maximum wave heights, along with the wave period, can be eliminated. Following Occam's razor principle, which suggests that explanations constructed with the smallest possible set of assumptions are superior, the reliability-based design optimization of a vertical-type breakwater presented in this study demonstrates promise in terms of simplicity and practicality. The limit state of the vertical-type breakwater was defined to encompass sliding, overturning, and collapse failures, and the strong interrelations between the wave force, lift force, and overturning moment were described using the Nataf joint distribution. As anticipated, simulation results show that solely considering sliding failure, as in the current reliability-based design platform in Korea, leads to an underestimated failure probability. Furthermore, ensuring a consistent failure probability for vertical-type breakwaters using design waves with a specific return period, as in past studies, is not feasible. In contrast, this study demonstrated that breakwaters optimally designed to meet the reliability index requirement of β = 3.5–4 consistently maintain a target failure probability in all sea areas.

Reconstruction of Cosmological Initial Density Field with Observations from the Epoch of Reionization

The initial density distribution provides a basis for understanding the complete evolution of cosmological density fluctuations. While reconstruction in our local Universe exploits the observations of galaxy surveys with large volumes, observations of high-redshift galaxies are performed with a small field of view and therefore can hardly be used for reconstruction. Here, we propose reconstructing the initial density field using the H i 21 cm and CO line intensity maps from the epoch of reionization. Observations of these two intensity maps provide complementary information on the density field—the H i 21 cm field is a proxy of matter distributions in the neutral regions, while the CO line intensity maps are sensitive to the high-density, star-forming regions that host the sources for reionization. Technically, we employ the conjugate gradient method and develop the machinery for minimizing the cost function for the intensity mapping observations. Analytical expressions for the gradient of cost function are derived explicitly. We show that the resimulated intensity maps match the input maps of mock observations using semi-numerical simulations of reionization with an rms error ≲7% at all stages of reionization. This reconstruction is also robust with an rms error of ∼10% when an optimistic level of shot noise is applied to the CO map or white noise at the level of ≲10% of the standard deviation is applied to each map. Our proof-of-concept work demonstrates the robustness of the reconstruction method, thereby providing an effective technique for reconstructing the cosmological initial density distribution from high-redshift observations.

Non-ergodic linear convergence property of the delayed gradient descent under the strongly convexity and the Polyak–Łojasiewicz condition

In this work, we establish the linear convergence estimate for the gradient descent involving the delay [Formula: see text] when the cost function is [Formula: see text]-strongly convex and [Formula: see text]-smooth. This result improves upon the well-known estimates in [Y. Arjevani, O. Shamir and N. Srebro, A tight convergence analysis for stochastic gradient descent with delayed updates, Proc. Mach. Learn. Res. 117 (2020) 111–132; S. U. Stich and S. P. Karimireddy, The error-feedback framework: Better rates for SGD with delayed gradients and compressed updates, J. Mach. Learn. Res. 21(1) (2020) 9613–9648] in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate [Formula: see text] can be extended from [Formula: see text] to [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak–Łojasiewicz[Formula: see text](PL) condition, for which the available choice of learning rate is further improved as [Formula: see text] for the large delay [Formula: see text]. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results.

MiPhDUO: microwave imaging via physics-informed deep unrolled optimization

Microwave imaging (MWI) is a non-invasive technique that can identify unknown scatterer objects’ features while offering advantages such as low cost and portable devices with respect to other imaging methods. However, MWI faces challenges in solving the underlying inverse scattering problem, which involves recovering target properties from its scattered fields. Existing methods include linearized and non-linear optimization approaches, but they have limitations respectively in terms of range of validity and computational complexity (in view of the possible occurrence of ‘false solutions’). In recent years, learning-based approaches have emerged as they can allow real-time imaging but usually lack generalizability and a direct connection to the underlying physics. This paper proposes a physics-informed approach that combines convolutional neural networks with physics-based calculations. It is based on a few cascaded operations, making use of the gradient of the relevant cost function, and successively improving the estimation of the unknown target. The proposed approach is assessed using simulated as well as experimental Fresnel data. The results show that the integration of physics with deep learning can contribute to improve reconstruction accuracy, generalizability, and computational efficiency in MWI.

Optimal control of three-dimensional unsteady partial differential equations with convection term in continuous casting

In this paper, we investigate an optimal control problem (OCP) for three-dimensional (3D) unsteady partial differential equations with convection term (UPDECT), extensively applied to the solidification process of billets. Firstly, we introduce the mathematical model of OCP and adopt a life cycle model method (LCMM) to numerically solve the 3D UPDECT. Secondly, the Lipschitz continuity of the cost function gradient is proved using the adjoint approach as a numerical method for designing optimal control systems. The cost function gradient of OCP for 3D UPDECT is analyzed based on a Hamiltonian costate system. Thirdly, we propose a modified trinomial conjugate gradient method (MTCGM) to solve the OCP for 3D UPDECT and prove the global convergence of MTCGM. Finally, the MTCGM is applied to the experimental simulation of continuous casting. According to experiment results, the optimizer based on MTCGM can significantly accelerate the convergence speed and reduce the iterative numbers, and the MTCGM provides a more stable temperature control effect.

Efficient management of HVAC systems through coordinated operation of parallel chiller units: An economic predictive control approach

In developed countries, air conditioning systems have become major contributors to energy consumption in buildings. Cooling installations made up of independent chiller units connected in parallel pose a challenge in finding the most energy-efficient management strategy. This article proposes a novel economic model predictive control strategy to optimize the operation of multiple-chiller HVAC systems according to an economic cost comprising energy consumption as well as a thermal comfort index. Provided that the gradient of the economic cost function can be calculated or estimated, the proposed controller only entails the solution of a single quadratic programming (QP) problem at each sampling period, reducing computational requirements and thus facilitating the deployment on commercial embedded HVAC management platforms. The proposed controller improves economic performance. As our numerical analysis shows, optimal sequencing is achieved for the case of dual-chiller plants. Moreover, the controller adapts to possible variations in the economic criteria, enabling the system to respond to changes in electricity price and user preferences. A realistic case study, using a high fidelity model of a building simulated in TRNSYS, demonstrates the effectiveness of the proposed methodology. In particular, the proposed controller demonstrates to be more efficient than state-of-the-art QP-based economic predictive controllers in achieving a reduction of the energy consumption up to 5.19%, which is in line with the targeted reduction of 7.9% by 2030 in the EU Green Deal's ‘Fit for 55’ package.

A distributed finite‐time optimization algorithm for directed networks of continuous‐time agents

AbstractThis article proposes a new distributed finite‐time optimization algorithm for agents under directed graphs. By employing the nonsmooth technique and graph theory, a distributed discontinuous algorithm for continuous‐time agents subject to strongly convex local cost functions is first designed with a finite‐time distributed estimator, where the gradients of the local cost functions are estimated in finite time. It is shown that for a strongly connected graph and arbitrary initial conditions, the proposed algorithms can achieve consensus, and the systems can converge to the optimal point in finite time. Then, a two‐step approach is proposed to achieve finite‐time optimization of high‐order agents with disturbances under directed graphs. Finally, the validity of the proposed finite‐time optimization algorithm is verified by two numerical examples.

Assimilation of GOES-16 ABI All-Sky Radiance Observations in RRFS Using EnVar: Methodology, System Development, and Impacts for a Severe Convective Event

Abstract The Advanced Baseline Imager (ABI) aboard the GOES-16 and GOES-17 satellites provides high-resolution observations of cloud structures that could be highly beneficial for convective-scale DA. However, only clear-air radiance observations are typically assimilated at operational centers due to a variety of problems associated with cloudy radiance data. As such, many questions remain about how to best assimilate all-sky radiance data, especially when using hybrid DA systems such as EnVar wherein a nonlinear observation operator can lead to cost function gradient imbalance and slow minimization. Here, we develop new methods for assimilating all-sky radiance observations in EnVar using the novel Rapid Refresh Forecasting System (RRFS) that utilizes the Finite-Volume Cubed-Sphere (FV3) model. We first modify the EnVar solver by directly including brightness temperature (Tb) as a state variable. This modification improves the balance of the cost function gradient and speeds up minimization. Including Tb as a state variable also improves the model fit to observations and increases forecast skill compared to utilizing a standard state vector configuration. We also evaluate the impact of assimilating ABI all-sky radiances in RRFS for a severe convective event in the central Great Plains. Assimilating the radiance observations results in better spinup of a tornadic supercell. These data also aid in suppressing spurious convection by reducing the snow hydrometeor content near the tropopause and weakening spurious anvil clouds. The all-sky radiance observations pair well with reflectivity observations that remove primarily liquid hydrometeors (i.e., rain) closer to the surface. Additionally, the benefits of assimilating the ABI observations continue into the forecast period, especially for localized convective events.

Output feedback distributed optimization algorithms of second‐order Lipschitz nonlinear multi‐agent systems

AbstractThis paper introduces output feedback distributed optimization algorithms designed specifically for second‐order nonlinear multi‐agent systems. The agents are allowed to have heterogeneous dynamics, characterized by distinct nonlinearities, as long as they satisfy the Lipschitz continuity condition. For the case with unknown states, nonlinear state observers are designed first for each agent to reconstruct agents' unknown states. It is proven that the agents' unknown states are estimated accurately by the developed state observers. Then, based on the agents' state estimates and the gradient of each agent local cost function, a kind of output feedback distributed optimization algorithms are proposed for the considered multi‐agent systems. Under the proposed distributed optimization algorithms, all the agents' outputs asymptotically approach the minimizer of the global cost function which is the sum of all the local cost functions. By using Lyapunov stability theory, convex analysis, and input‐to‐state stability theory, the asymptotical convergence of the output feedback distributed optimization closed‐loop system is proven. Simulations are conducted to validate the efficacy of the proposed algorithms.

Fully Distributed Algorithm for Resource Allocation Over Unbalanced Directed Networks Without Global Lipschitz Condition

This note investigates the distributed optimal resource allocation problem of multi-agent systems over unbalanced directed networks under the relaxed condition that the gradients of local cost functions are locally Lipschitz. The objective is to cooperatively drive the decision variables of the agents to the optimal solution, which minimizes the sum of the local cost functions, while ensuring that the network resource constraints and local feasibility constraints are satisfied. A novel distributed algorithm is developed over unbalanced directed network topologies based on the topology balancing technique and adaptive control approach. The developed algorithm is fully distributed in the sense that it depends on neither the global Lipschitz continuity of the gradients nor prior global information about the network connectivity. By regarding the proposed algorithm as a perturbed system, its input-to-state stability with a vanishing perturbation is first established, and asymptotic convergence of the decision variables toward the optimal solution is then proved. The effectiveness of the proposed fully distributed algorithm is illustrated with two examples.

A distributed proximal gradient method with time-varying delays for solving additive convex optimizations

We consider the problem of minimizing a finite sum of differentiable and nondifferentiable convex functions in the setting of finite-dimensional Euclidean space. We propose and analyze a distributed proximal gradient method with computational delays. The occurrence of local delays when computing local gradient of each differentiable cost function allows the use of out-of-date iterates when generating the next estimates, which benefits a situation where the cost of gradient computation is expensive so that it cannot be done within a limited time constraints. We provide a condition on control parameter to guarantee that the sequences generated by the proposed method converge to the unique solution. We finally illustrate the presented theoretical results by performing some numerical experiments on binary image classification.

Barren plateaus in quantum tensor network optimization

We analyze the barren plateau phenomenon in the variational optimization of quantum circuits inspired by matrix product states (qMPS), tree tensor networks (qTTN), and the multiscale entanglement renormalization ansatz (qMERA). We consider as the cost function the expectation value of a Hamiltonian that is a sum of local terms. For randomly chosen variational parameters we show that the variance of the cost function gradient decreases exponentially with the distance of a Hamiltonian term from the canonical centre in the quantum tensor network. Therefore, as a function of qubit count, for qMPS most gradient variances decrease exponentially and for qTTN as well as qMERA they decrease polynomially. We also show that the calculation of these gradients is exponentially more efficient on a classical computer than on a quantum computer.

A Gradient‐Cost Multiobjective Alternate Framework for Variational Quantum Eigensolver with Variable Ansatz

AbstractVariational quantum eigensolver (VQE), aiming at determining the ground state energy of a quantum system described by a Hamiltonian on noisy intermediate scale quantum (NISQ) devices. However, the accuracy and trainability of the current VQE algorithm are significantly influenced due to the barren plateau (BP), the non‐negligible gate error and limited coherence time in NISQ devices. To tackle these issues, a gradient‐cost multiobjective alternate framework with variable ansatz is proposed. A theoretical framework is first proposed for VQE with variable ansatz (VA‐VQE) via alternately solving a multiobjective optimization problem defined by cost function values and gradient magnitudes and the original VQE problem. Then, a novel implementation method based on the double ε‐greedy strategy with the candidate tree and modified multiobjective genetic algorithm is proposed. As a result, the local optima are avoided both in ansatz and parameter perspectives, the BP phenomenon is alleviated, and the stability of output ansatz is enhanced. The experimental results indicate that this framework shows considerably average improvement of the error and stability by 59.6% and 78.8% compared with the state‐of‐the‐art cost‐criterion‐based VA‐VQE implementation, and 54.1% and 73.5% compared with the gradient‐criterion‐based counterpart, respectively, with lower quantum costs.

Fast model-based scenario optimization in NSTX-U enabled by analytic gradient computation

Model-based optimization offers a systematic approach to advanced scenario planning. In this case, the feedforward-control inputs (actuator trajectories) that are needed to attain and sustain a desired scenario are obtained by solving a nonlinear constrained optimization problem. This class of problems generally minimize a cost function that measures the difference between desired and actual plasma states. Several numerical optimization algorithms, such as sequential quadratic programming, require repeated calculation of the cost function gradients with respect to the input trajectories. Calculating these gradients numerically can be computationally intensive, increasing the time needed to solve the feedforward-control optimization problem. This work introduces a method to analytically calculate these cost function gradients from the current profile evolution model. This can significantly reduce the computational time and allow for fast feedforward-control optimization, which would eventually enable optimal scenario planning between discharges. The performance of the feedforward optimizer with analytical gradients is compared to a traditional optimization algorithm based on numerical gradients for different NSTX-U scenarios. The plasma dynamics in the optimization algorithm are simulated using the Control Oriented Transport SIMulator (COTSIM). Results of the work show that analytical gradients consistently reduce the computation time while achieving trajectories that are comparable to those obtained by traditional optimization algorithms based on numerical gradients.

New DY-HS hybrid conjugate gradient algorithm for solving optimization problem of unsteady partial differential equations with convection term

This paper studies an optimization problem for the unsteady partial differential equations (PDEs) with convection term, widely used in continuous casting process. Considering the change of casting speed, a dynamic optimization method based on new DY-HS hybrid conjugate gradient algorithm (DY-HSHCGA) is proposed. In the DY-HSHCGA, the Dai–Yuan and the Hestenes–Stiefel conjugate gradient algorithms are convex combined, and a new conjugate parameter θk is obtained through the condition of quasi-Newton direction. Moreover, Lipschitz continuity of the gradient of cost function, as an important conditions for convergence, is analyzed in this paper. On the basis on this condition, the global convergence of DY-HSHCGA is proved. Finally, the effectiveness of DY-HSHCGA is verified by some instances from the steel plant. Comparing with other algorithms DY-HSHCGA obviously accelerates the convergence rate and reduces the number of iteration. The optimizer based on the DY-HSHCGA shows a more stable results.

A distributed extremum seeking based resource allocation algorithm over switching networks

AbstractIn this article, a distributed resource allocation problem for second‐order multi‐agent systems over undirected switching topologies is investigated. For the distributed resource allocation problem, most of the existing algorithms depend on the gradient of cost function with specific expression, which may be unknown to agents. To deal with unknown cost functions, the extremum seeking control is introduced in the designed distributed resource allocation algorithm to estimate the gradient of the cost function only by the value of the cost. Besides, an estimator is proposed to estimate the optimal Lagrange multiplier associated with the equality constraint. The communication graph is supposed to be switched among a set of undirected subgraphs, which may be connected or unconnected. For undirected and connected switching graphs, the average dwell‐time constraint is used to describe the switching frequency. For the undirected graphs, containing connected and unconnected ones, both the average dwell‐time and the time‐ratio constraints are used to model the switching signal. For the two cases, the semi‐globally practically asymptotical stability of the designed algorithm is established. The obtained results are verified by an example.

Quasi-Newton extremum seeking control based on incremental recursive identification

This paper proposes a novel quasi-Newton extremum seeking control method based on incremental recursive identification for a real-time optimization problem. The optimization of static map and dynamical system are both considered where the dynamics of the nonlinear system and the cost function to be minimized are unknown. Firstly, the proposed method only employs the measured function values to effectively estimate the time-varying gradient of the unknown cost function by using incremental recursive least square identification. Then, an extremum seeking control (ESC) approach based on the quasi-Newton direction is developed to improve convergence speed, which only requires the estimated gradient of the objective function without Hessian information. Furthermore, the convergence properties of the proposed ESC method are established for static map and dynamical system. Finally, we demonstrate the superiority and effectiveness of the proposed method using three examples.