Saddle Point Structure Research Articles

A conforming mixed finite element method for a coupled Navier–Stokes/transport system modeling reverse osmosis processes

We consider the coupled Navier–Stokes/transport equations with nonlinear transmission conditions, which constitute one of the most common models utilized to simulate a reverse osmosis effect in water desalination processes when considering feed and permeate channels coupled through a semi-permeate membrane. The variational formulation consists of a set of equations where the velocities, the concentrations, along with tensors and vector fields introduced as auxiliary unknowns and two Lagrange multipliers are the main unknowns of the system. The latter are introduced to deal with the trace of functions that do not have enough regularity to be restricted to the boundary. In addition, the pressures can be recovered afterwards by a postprocessing formula. As a consequence, we obtain a nonlinear Banach spaces-based mixed formulation, which has a perturbed saddle point structure. We analyze the continuous and discrete solvability of this problem by linearizing the perturbation and applying the classical Banach fixed point theorem along with the Banach–Nečas–Babuška result. Regarding the discrete scheme, feasible choices of finite element subspaces that can be used include Raviart–Thomas spaces for the auxiliary tensor and vector unknowns, piecewise polynomials for the velocities and the concentrations, and continuous polynomial space of lowest order for the traces, yielding stable discrete schemes. An optimal a priori error estimate is derived, and numerical results illustrating both, the performance of the scheme confirming the theoretical rates of convergence, and its applicability, are reported.

Finite Element Formulations for Maxwell’s Eigenvalue Problem Using Continuous Lagrangian Interpolations

Abstract We consider nodal-based Lagrangian interpolations for the finite element approximation of the Maxwell eigenvalue problem. The first approach introduced is a standard Galerkin method on Powell–Sabin meshes, which has recently been shown to yield convergent approximations in two dimensions, whereas the other two are stabilized formulations that can be motivated by a variational multiscale approach. For the latter, a mixed formulation equivalent to the original problem is used, in which the operator has a saddle point structure. The Lagrange multiplier introduced to enforce the divergence constraint vanishes in an appropriate functional setting. The first stabilized method consists of an augmented formulation including a mesh dependent term that can be regarded as the Laplacian of the divergence constraint multiplier. The second formulation is based on orthogonal projections, which can be recast as a residual based stabilization technique. We rely on the classical spectral theory to analyze the approximating methods for the eigenproblem. The stability and convergence aspects are inherited from the associated source problems together with an assumption which is discussed numerically. We investigate the performance of the proposed formulations and provide some convergence results validating the theoretical ones for several benchmark tests, including ones with smooth and singular solutions.

On the invertibility of matrices with a double saddle-point structure

We establish necessary and sufficient conditions for invertibility of symmetric three-by-three block matrices having a double saddle-point structure that guarantee the unique solvability of double saddle-point systems. We consider various scenarios, including the case where all diagonal blocks are allowed to be rank deficient. Under certain conditions related to the nullity of the blocks and intersections of their kernels, an explicit formula for the inverse is derived.

Charge-breaking domain walls separating neutral vacua in multi-Higgs models

The scalar potential of a multi-Higgs model can possess a rich structure of minima and saddle points, which evolves in an intricate way as the parameters change. In the hot early Universe, it could trigger multistep phase transitions, with exotic intermediate phases and peculiar domain wall configurations. Here, we provide a glimpse into this richness with the example of the three-Higgs-doublet model with the symmetry group Σ(36), either exact or softly broken. We present its phase diagram tracking not only the global minimum but all of its extrema. In particular, we reveal parameter space regions in which the deepest saddle point is charge breaking. This naturally leads to phase transitions between neutral vacua which involve expanding and colliding charge-breaking bubble walls. We also comment on opportunities of multistep phase transitions, on charge-breaking intermediate phases, and on phase transitions between different charge-breaking vacua. We illustrate the general discussion with two benchmark models, one of which possesses competing saddle points and allows for emergence of bubbles of the same true vacuum but with bubble walls of different nature. The intriguing cosmological implications of all these possibilities deserve dedicated study. Published by the American Physical Society 2024

Impact of the T296S mutation in P450 GcoA for aryl-O-demethylation: a QM/MM study.

Lignin, a complex plant cell wall component, holds promise as a renewable aromatic carbon feedstock. p-Vanillin is a key product of lignin depolymerization and a precursor of protocatechuic acid (PCA) that has tremendous potential for biofuel production. While the GcoAB enzyme, native to Amycolatopsis sp., naturally catalyzes aryl-O-demethylation toward guaiacol, recent research introduced a single mutation, T296S, into the GcoAP450 enzyme, enabling it to catalyze aryl-O-demethylation of p-vanillin. This structural modification increases the efficiency of GcoAP450 for the natural substrate while being active for p-vanillin. This study reveals the increased flexibility of p-vanillin and its ability to adapt a favorable conformation by aligning the methoxy group in close proximity to Fe(IV) = O of Cpd I in the active site of the T296S variant. The QM/MM calculations in accordance with the experimental data validated that the rate-limiting step for the oxidation of p-vanillin is hydrogen atom abstraction and provided a detailed geometric structure of stationary and saddle points for the oxidation of p-vanillin.

Effect of Pb 6s2 lone pair on the potential flattening of fluoride-ion conduction in perovskite-type fluoride

Cubic CsPb0.9K0.1F2.9 exhibits an exceptionally low activation energy of 7.9 kJ mol−1 for fluoride-ion conductivity. DFT analysis has revealed that the localised lone pair in the PbF5 polyhedron stabilises the saddle-point structure.

Symbol based convergence analysis in block multigrid methods with applications for Stokes problems

The main focus of this paper is the study of efficient multigrid methods for large linear systems with a particular saddle-point structure. Indeed, when the system matrix is symmetric, but indefinite, the variational convergence theory that is usually used to prove multigrid convergence cannot be directly applied.However, different algebraic approaches analyze properly preconditioned saddle-point problems, proving convergence of the Two-Grid method. In particular, this is efficient when the blocks of the coefficient matrix possess a Toeplitz or circulant structure. Indeed, it is possible to derive sufficient conditions for convergence and provide optimal parameters for the preconditioning of the saddle-point problem in terms of the associated symbols. In this paper, we propose a symbol based convergence analysis for problems that have a hidden block Toeplitz structure. Then, they can be investigated focusing on the properties of the associated generating function f, which consequently is a matrix-valued function with dimension depending on the block size of the problem. As numerical tests we focus on the matrix sequence stemming from the finite element approximation of the Stokes problem. We show the efficiency of the methods studying the hidden 9-by-9 block multilevel structure of the obtained matrix sequence. Moreover we propose an efficient algebraic multigrid method with convergence rate independent of the matrix size. Finally, we present several numerical tests comparing the results with state-of-the-art strategies.

Instanton gas approach to the Hubbard model

In this paper, we consider a path integral formulation of the Hubbard model based on a Hubbard-Stratonovich transformation that couples the auxiliary field to the local electronic density. This decoupling is known to have a saddle-point structure that shows a remarkable regularity: The field configuration at each saddle point can be understood in terms of a set of elementary field configurations localized in space and imaginary time which we coin instantons. The interaction between instantons is short ranged. Here, we formulate a classical partition function for the instanton gas that has predictive power. For a given set of physical parameters, we can predict the distribution of instantons and show that the instanton number is sharply defined in the thermodynamic limit, thereby defining a unique dominant saddle point. Decoupling in the charge channel conserves SU(2) spin symmetry for each field configurations. Hence, the instanton approach provides an SU(2) spin-symmetric approximation to the Hubbard model. It fails, however, to capture the magnetic transition inherent to the Hubbard model on the honeycomb lattice despite being able to describe local moment formation. In fact, the instanton itself corresponds to local moment formation and concomitant short-ranged antiferromagnetic correlations. This aspect is also seen in the single particle spectral function that shows clear signs of the upper and lower Hubbard bands. Our instanton approach bears remarkable similarities to local dynamical approaches, such as dynamical mean-field theory, in the sense that it has the unique property of allowing for local moment formation without breaking the SU(2) spin symmetry. In contrast to local approaches, it captures short-ranged magnetic fluctuations. Furthermore, it also offers possibilities for systematic improvements by taking into account fluctuations around the dominant saddle point. Finally, we show that the saddle point structure depends upon the choice of lattice geometry. For the square lattice at half filling, the saddle-point structure reflects the itinerant to localized nature of the magnetism as a function of the coupling strength. The implications of our results for Lefschetz thimble approaches to alleviate the sign problem are also discussed.

First-principles study of impurity diffusion coefficients in Niobium

Nb based alloys have potential applications in aerospace due to their good high temperature strength and low temperature plasticity. It is very important to study the atomic diffusion properties for understanding the dynamic process of Nb-based alloys. However, the diffusion data of some elements in Nb are still unknown. In the present work, the impurity diffusion coefficients of ten different elements (Si, Ti, V, Ta, Ru, Cr, Te, Tc, Ir, and Y) in BCC Nb was studied by first-principles calculation based on density functional theory and nine frequency model. The climbing image nudged elastic band (CI-NEB) method was used to find the minimum energy path and the related saddle point structure during diffusion. The atomic jump frequency was calculated through the Vineyard’ s transition state theory. The results are consistent well with the exiting experimental and previous calculation results.

Sella, an Open-Source Automation-Friendly Molecular Saddle Point Optimizer.

We present a new algorithm for the optimization of molecular structures to saddle points on the potential energy surface using a redundant internal coordinate system. This algorithm automates the procedure of defining the internal coordinate system, including the handling of linear bending angles, for example, through the addition of dummy atoms. Additionally, the algorithm supports constrained optimization using the null-space sequential quadratic programming formalism. Our algorithm determines the direction of the reaction coordinate through iterative diagonalization of the Hessian matrix and does not require evaluation of the full Hessian matrix. Geometry optimization steps are chosen using the restricted step partitioned rational function optimization method, and displacements are realized using a high-performance geodesic stepping algorithm. This results in a robust and efficient optimization algorithm suitable for use in automated frameworks. We have implemented our algorithm in Sella, an open-source software package designed to optimize atomic systems to saddle point structures. We also introduce a new benchmark test comprising 500 molecular structures that approximate saddle point geometries and show that our saddle point optimization algorithm outperforms the algorithms implemented in several leading electronic structure theory packages.

Microwave and Computational Study of Pivalic Sulfuric Anhydride and the Pivalic Acid Monomer: Mechanistic Insights into the RCOOH + SO3 Reaction.

The microwave spectrum of pivalic sulfuric anhydride, (CH3)3CCOOSO2OH (PivSA), has been observed by rotational spectroscopy. The compound was formed by the reaction of SO3 with (CH3)3CCOOH (pivalic acid) in a supersonic jet in a manner analogous to that previously observed with other carboxylic acids. Computational analysis indicates that the reaction is best described as a pericyclic process coupled with a 60° rotation of the t-butyl group. Product formation can occur through either a sequential (two-step) or a concerted (one-step) pathway. The former involves an internal rotation of the t-butyl group through a 0.11 kcal/mol barrier followed by the pericyclic reaction that joins the moieties. The latter passes through a second-order saddle point in which the internal rotation and pericyclic reaction occur simultaneously. This path is the most energetically favorable, as the zero-point corrected energy at the saddle point structure is 0.16 kcal/mol below that of a putative (CH3)3CCOOH-SO3 precursor complex. Additional computational work involving a series of carboxylic acids is reported, which explores the effects of gas-phase acidity and basicity of the RCOOH reactant on reaction energetics. These calculations, together with prior experimental and theoretical studies of the acetic and trifluoroacetic derivatives, demonstrate that the basicity of the carbonyl oxygen, not the acidity of the COOH proton, is the important driving factor for the reaction. As a precursor to the experimental work on the title molecule, microwave spectra of the parent and OD forms of the pivalic acid monomer were recorded and are reported here as well. A convenient synthesis of SO3 is also described.

The performance of exchange-correlation functionals in describing electron density parameters of saddle point structures along chemical reactions.

This work is focused on evaluating the performance of exchange-correlation functionals from density functional theory in providing descriptor values derived from the electron density of saddle point structures (transition states) in chemical reactions. The properties investigated were obtained from the quantum theory of atoms in molecules, including atomic charges and electron density topological data at the bond critical points. In addition, parameters from the Interacting quantum atom energy partition were used as well in this comparative study. The reference values are attained in coupled cluster calculations with iterative single and double excitations (CCSD). Six elementary reactions are considered here: CO + H2 ↔H2 CO, CO + H2 O↔HCOOH, HCN↔HNC, H + F2 ↔HF + F, H + N2 ↔HN2 , and H + CO↔HCO. In general, the BB1K functional (hybrid-meta-generalized gradient approximation) provides the best description of these properties. Our study indicates that an intermediate percentage of nonlocal exact exchange, around 40%-55% (perhaps even larger), is probably required for attaining more accurate values with actual functionals, although this condition is not able of explaining all the trends observed.

A novel multigrid method for elliptic distributed control problems

Large linear systems of saddle-point type have arisen in a wide variety of applications throughout computational science and engineering. The discretizations of distributed control problems have a saddle-point structure. The numerical solution of saddle-point problems has attracted considerable interest in recent years. In this work, we propose a novel Braess–Sarazin multigrid relaxation scheme for finite element discretizations of the distributed control problems, where we use the stiffness matrix obtained from the five-point finite difference method for the Laplacian to approximate the inverse of the mass matrix arising in the saddle-point system. We apply local Fourier analysis to examine the smoothing properties of the Braess–Sarazin multigrid relaxation. From our analysis, the optimal smoothing factor for Braess–Sarazin relaxation is derived. Numerical experiments validate our theoretical results. The relaxation scheme considered here shows its high efficiency and robustness with respect to the regularization parameter and grid size.

Landscape Analysis for Shallow Neural Networks: Complete Classification of Critical Points for Affine Target Functions

In this paper, we analyze the landscape of the true loss of neural networks with one hidden layer and ReLU, leaky ReLU, or quadratic activation. In all three cases, we provide a complete classification of the critical points in the case where the target function is affine and one-dimensional. In particular, we show that there exist no local maxima and clarify the structure of saddle points. Moreover, we prove that non-global local minima can only be caused by ‘dead’ ReLU neurons. In particular, they do not appear in the case of leaky ReLU or quadratic activation. Our approach is of a combinatorial nature and builds on a careful analysis of the different types of hidden neurons that can occur.

Accelerated gradient sliding for structured convex optimization

Our main goal in this paper is to show that one can skip gradient computations for gradient descent type methods applied to certain structured convex programming (CP) problems. To this end, we first present an accelerated gradient sliding (AGS) method for minimizing the summation of two smooth convex functions with different Lipschitz constants. We show that the AGS method can skip the gradient computation for one of these smooth components without slowing down the overall optimal rate of convergence. This result is much sharper than the classic black-box CP complexity results especially when the difference between the two Lipschitz constants associated with these components is large. We then consider an important class of bilinear saddle point problem whose objective function is given by the summation of a smooth component and a nonsmooth one with a bilinear saddle point structure. Using the aforementioned AGS method for smooth composite optimization and Nesterov’s smoothing technique, we show that one only needs \({{\mathcal{O}}}(1/\sqrt{\varepsilon })\) gradient computations for the smooth component while still preserving the optimal \({{\mathcal{O}}}(1/\varepsilon )\) overall iteration complexity for solving these saddle point problems. We demonstrate that even more significant savings on gradient computations can be obtained for strongly convex smooth and bilinear saddle point problems.

Continuous Lambertian shape from shading: A primal-dual algorithm

The continuous Lambertian shape from shading is studied using a PDE approach in terms of Hamilton–Jacobi equations. The latter will then be characterized by a maximization problem. In this paper we show the convergence of discretization and propose to use the well-known Chambolle–Pock primal-dual algorithm to solve numerically the shape from shading problem. The saddle-point structure of the problem makes the Chambolle–Pock algorithm suitable to approximate solutions of the discretized problems.

Application of Optimal Control to Inversion of Self-Potential Data: Theory and Synthetic Examples

Last decades, there has been an increased interest in the use of the self-potential (SP) method in hydrogeophysics. In response to this strong interest, we develop a novel approach to the inversion of SP data. Mathematically, the SP inverse problem is the source identification problem for the Poisson equation. Our approach substantially differs from the standard regularization approach, which explicitly includes the forward-problem operator into the cost functional. We formulated the inverse problem is as an optimal control problem and then translate it into a variational system. The system is approximated in suitable finite-element spaces giving rise to an algebraic problem with the saddle-point structure. In contrast to the standard approach, which leads to a dense linear system, our method results in a system with a sparse block matrix. It can be efficiently solved by either direct sparse solvers or preconditioned iterative solvers. In this paper, we present the formulation of the problem and its finite-element approximation. We discuss the iterative solution and preconditioning strategies. Our software implementation is based on an industrial finite-element package. We also present a numerical experiment with node-based linear finite elements on tetrahedral grids. Our results suggest that the proposed approach may serve as a rapid and reliable tool for large-scale SP inverse problems. Moreover, the same technique can easily be extended to a wide range of geophysical linear inverse problems, such as inversions of magnetic and gravity data.

Fast image inpainting strategy based on the space-fractional modified Cahn-Hilliard equations

The solution strategy of the space-fractional modified Cahn-Hilliard equation as a tool for the gray value image inpainting model is studied. The existing strategies solve the convexity splitting scheme of the vector-valued Cahn-Hilliard model by Fourier spectral method. In this paper, we constructed a fast solver for the discretized linear systems possessing the saddle-point structure within block-Toeplitz-Toeplitz-block (BTTB) structure arising from the 2D space-fractional modified Cahn-Hilliard equation. The new solver enjoys computational advantage since circulant approximation and fast Fourier transforms (FFTs) can be used for solving the involved linear subsystems. Theoretical analysis shows the spectrum of the preconditioned matrix clusters around 1, which implies the fast convergence rate of the proposed preconditioner. Numerical examples are given to confirm the effectiveness of our method.

Critical point theory for sparse recovery

We study the problem of sparse recovery in the context of compressed sensing. This is to minimize the sensing error of linear measurements by sparse vectors with at most s non-zero entries. We develop the so-called critical point theory for sparse recovery. This is done by introducing nondegenerate M-stationary points which adequately describe the global structure of this non-convex optimization problem. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic sparse recovery problem. Additionally, the equivalence of strong stability and nondegeneracy for M-stationary points is shown. We claim that the appearance of saddle points – these are M-stationary points with exactly s−1 non-zero entries – cannot be neglected. For this purpose, we derive a so-called Morse relation, which gives a lower bound on the number of saddle points in terms of the number of local minimizers. The relatively involved structure of saddle points can be seen as a source of well-known difficulty by solving the problem of sparse recovery to global optimality.

On nondegenerate M-stationary points for sparsity constrained nonlinear optimization

We study sparsity constrained nonlinear optimization (SCNO) from a topological point of view. Special focus will be on M-stationary points from Burdakov et al. (SIAM J Optim 26:397–425, 2016), also introduced as N^C-stationary points in Pan et al. (J Oper Res Soc China 3:421–439, 2015). We introduce nondegenerate M-stationary points and define their M-index. We show that all M-stationary points are generically nondegenerate. In particular, the sparsity constraint is active at all local minimizers of a generic SCNO. Some relations to other stationarity concepts, such as S-stationarity, basic feasibility, and CW-minimality, are discussed in detail. By doing so, the issues of instability and degeneracy of points due to different stationarity concepts are highlighted. The concept of M-stationarity allows to adequately describe the global structure of SCNO along the lines of Morse theory. For that, we study topological changes of lower level sets while passing an M-stationary point. As novelty for SCNO, multiple cells of dimension equal to the M-index are needed to be attached. This intriguing fact is in strong contrast with other optimization problems considered before, where just one cell suffices. As a consequence, we derive a Morse relation for SCNO, which relates the numbers of local minimizers and M-stationary points of M-index equal to one. The appearance of such saddle points cannot be thus neglected from the perspective of global optimization. Due to the multiplicity phenomenon in cell-attachment, a saddle point may lead to more than two different local minimizers. We conclude that the relatively involved structure of saddle points is the source of well-known difficulty if solving SCNO to global optimality.