SIAM Journal Research Articles

An inexact variable metric variant of incremental aggregated Forward-Backward method for the large-scale composite optimization problem

ABSTRACT This paper focuses on the large-scale composite optimization problem, which is the sum of a finite number of smooth (possibly nonconvex) functions and a convex (possibly nonsmooth) function. A common method to solve this problem is proximal incremental aggregated gradient (PIAG) method (N. D. Vanli, M. Gurbuzbalaban and A. Ozdaglar, SIAM Journal on Optimization, 2018, 28(2): 1282–1300), which exploits the gradient information from previous iterations to approximate the gradient at the current point. However, as many first-order algorithms, it may suffer from slow convergence. To address this issue, this paper proposes an inexact variable metric variant of incremental aggregated Forward-Backward (iVMFB-IAG) method. This methods incorporates a variable metric strategy to accelerate the PIAG method and introduces an inexact criterion to enhance its practicality. Under the Kurdyka-Łojasiewicz (KL) property and the uniformly bounded positive definiteness of the scaling matrix, the sequence generated by the iVMFB-IAG method converges globally to the critical point of the problem. Additionally, the convergence rate is established under the KL framework. Moreover, under certain restrictive conditions, the method exhibits local superlinear convergence. Through the numerical experiments on image reconstruction applications, we demonstrate the effectiveness of the iVMFB-IAG method by restoring two distinct images.

Optimal Incentives in a Limit Order Book: A SPDE Control Approach

With the fragmentation of electronic markets, exchanges are now competing in order to attract trading activity on their platform. Consequently, they developed several regulatory tools to control liquidity provision/consumption on their liquidity pool. In this paper, we study the problem of an exchange using incentives in order to increase market liquidity. We model the limit order book as the solution of a stochastic partial differential equation (SPDE) as in [R. Cont and M. S. Müller, 2021, A stochastic partial differential equation model for limit order book dynamics, SIAM Journal on Financial Mathematics 12(2), 744–787]. The incentives proposed to the market participants are functions of the time and the distance of their limit order to the mid-price. We formulate the control problem of the exchange who wishes to modify the shape of the order book by increasing the volume at specific limits. Due to the particular nature of the SPDE control problem, we are able to characterize the solution with a classic Feynman–Kac representation theorem. Moreover, when studying the asymptotic behavior of the solution, a specific penalty function enables the exchange to obtain closed-form incentives at each limit of the order book. We study numerically the form of the incentives and their impact on the shape of the order book, and analyze the sensitivity of the incentives to the market parameters.

Angular Momentum Conserving Limiters for the Euler Equations

Limiters that are used to stabilise the discontinuous Galerkin method fail to preserve its angular momentum conservation property. We propose a simple technique that restores the conservation of angular momentum while preserving the second-order accuracy of the scheme when paired with an explicit time integrator. Numerical experiments confirm the conservation property of the technique paired with two limiters [Giuliani and Krivodonova. 2018. “Analysis of Slope Limiters on Unstructured Triangular Meshes.” Journal of Computational Physics 374: 1–26; Giuliani and Krivodonova. 2019. “A Moment Limiter for the Discontinuous Galerkin Method on Unstructured Triangular Meshes.” SIAM Journal on Scientific Computing 41 (1): A508–A537]. We show that the technique eliminates numerical artifacts that arise from limiting solutions to the Euler equations.

On the Implied Volatility of Asian Options Under Stochastic Volatility Models

In this paper, we study the short-time behaviour of the at-the-money implied volatility for arithmetic Asian options with fixed strike price. The asset price is assumed to follow the Black–Scholes model with a general stochastic volatility process. Using techniques of the Malliavin calculus developed in Alòs, García-Lorite, and Muguruza [2022. On Smile Properties of Volatility Derivatives: Understanding the VIX Skew. SIAM Journal on Financial Mathematics. 13(1): 32–69. https://doi.org/10.1137/19M1269981], we give sufficient conditions on the stochastic volatility in order to compute the level of the implied volatility of the option when the maturity converges to zero. Then, we find a short maturity asymptotic formula for the skew slope of the implied volatility that depends on the correlation between prices and volatilities and the Hurst parameter of the volatility model. We apply our general results to the SABR and fractional Bergomi models, and provide numerical simulations that confirm the accurateness of the asymptotic formulas.

Short-Time Asymptotics for Non-Self-Similar Stochastic Volatility Models

We provide a short-time large deviation principle (LDP) for stochastic volatility models, where the volatility is expressed as a function of a Volterra process. This LDP does not require strict self-similarity assumptions on the Volterra process. For this reason, we are able to apply such an LDP to two notable examples of non-self-similar rough volatility models: models where the volatility is given as a function of a log-modulated fractional Brownian motion (Bayer, C., F. Harang, and P. Pigato. 2021. “Log-Modulated Rough Stochastic Volatility Models.” SIAM Journal on Financial Mathematics 12 (3): 1257–1284), and models where it is given as a function of a fractional Ornstein–Uhlenbeck (fOU) process (Gatheral, J., T. Jaisson, and M. Rosenbaum. 2018. “Volatility is Rough.” Quantitative Finance 18 (6): 933–949). In both cases, we derive consequences for short-maturity European option prices implied volatility surfaces and implied volatility skew. In the fOU case, we also discuss moderate deviations pricing and simulation results.

SIGEST

The SIGEST article in this issue is “Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors,” by Antoine Gautier, Francesco Tudisco, and Matthias Hein. Most computational and applied mathematicians will be aware of the results that Perron published in 1907 about the eigensystems of positive matrices, which were then extended by Frobenius in 1912 to the case of nonnegative matrices. This theory has impacted many areas of mathematics, including graph theory, Markov chains, and matrix computation, and it forms a fundamental component in the analysis of a range of models in areas such as demography, economics, wireless networking, and search engine optimization. Our SIGEST article, which first appeared in SIAM Journal on Matrix Analysis and Applications in 2019, extends Perron--Frobenius theory in two directions. First, the authors generalize from matrices to multidimensional arrays. This ties in with one of SIAM Review's most highly cited offerings: •Tensor decompositions and applications, T. G. Kolda and B. W. Bader, SIAM Review, 51 (3) (2009), pp. 455--500. It may also be viewed as extending the theory from graphs to hypergraphs---objects that are currently of much interest, as evidenced by several recent SIAM Review articles, including •Hypergraph cuts with general splitting functions, N. Veldt, A. R. Benson and J. Kleinberg, SIAM Review, 64 (3) (2022), pp. 650--685. By studying this higher-order setting, the authors open up new applications in network science, computer vision, and machine learning. The second major direction of the article is to develop and study nonlinear versions of the underlying spectral problems, and corresponding extensions of the traditional power method. This makes available new classes of iterations for which a comprehensive and satisfactory convergence theory is available. In preparing this SIGEST version, the authors have included new material. The introduction has been extended, and section 2 has been added to provide nontrivial examples of tensor eigenvalue problems in applications, including problems from computer vision and optimal transport. Moreover, subsection 4.1 includes a new nonlinear Perron--Frobenius theorem (Theorem 4.4) that builds on the previously known results in Theorems 4.2. and 4.3.

SIGEST

The SIGEST article in this issue, “Improbability of Collisions of Point-Vortices in Bounded Planar Domains,” by Martin Donati, is based on the 2022 SIAM Journal on Mathematics Analysis article “Two-Dimensional Point Vortex Dynamics in Bounded Domains: Global Existence for Almost Every Initial Data.” This work concerns point-vortex dynamics in a very general bounded domain in the plane. The main result is that the set of initial configurations which lead to finite-time collision, although nonempty, has Lebesgue measure zero. This is an elegant and highly nontrivial extension to general bounded domains with $C^{2, \alpha}$ boundary of a result previously known only for three domains: the full plane, a disk, and the complement of a disk. It is notable that the result also extends to point-vortex dynamics in multiply connected domains. In preparing this SIGEST version, the author has added a new introductory section that explains the context of the new results. Moreover, some of the technical details from the original article have been replaced by more accessible high-level explanations. Recent references on the topics of vortex collisions and desingularization problems are also included. This SIGEST article will be of particular interest to researchers in fluid mechanics, partial differential equations, and applied analysis.

Ranked Document Retrieval in External Memory

The ranked (or top- k ) document retrieval problem is defined as follows: preprocess a collection {T 1 ,T 2 ,… ,T d } of d strings (called documents) of total length n into a data structure, such that for any given query (P,k) , where P is a string (called pattern) of length p ≥ 1 and k ∈ [1,d] is an integer, the identifiers of those k documents that are most relevant to P can be reported, ideally in the sorted order of their relevance. The seminal work by Hon et al. [FOCS 2009 and Journal of the ACM 2014] presented an O(n) -space (in words) data structure with O(p+k log k) query time. The query time was later improved to O(p+k) [SODA 2012] and further to O(p/ log σn+k ) [SIAM Journal on Computing 2017] by Navarro and Nekrich, where σ is the alphabet size. We revisit this problem in the external memory model and present three data structures. The first one takes O(n) -space and answer queries in O(p/B + log B n + k/B+ log * (n/B) ) I/Os, where B is the block size. The second one takes O(n log * (n/B) ) space and answer queries in optimal O(p/B + log B n + k/B) I/Os. In both cases, the answers are reported in the unsorted order of relevance. To handle sorted top- k document retrieval, we present an O(n log (d/B)) space data structure with optimal query cost.

Multistability for a Reduced Nematic Liquid Crystal Model in the Exterior of 2D Polygons

We study nematic equilibria in an unbounded domain, with a two-dimensional regular polygonal hole with K edges, in a reduced Landau–de Gennes framework. This complements our previous work on the “interior problem” for nematic equilibria confined inside regular polygons (SIAM Journal on Applied Mathematics, 80(4):1678-1703, 2020). The two essential dimensionless model parameters are lambda -the ratio of the edge length of polygon hole to the nematic correlation length, and an additional degree of freedom, gamma ^*-the nematic director at infinity. In the lambda rightarrow 0 limit, the limiting profile has two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the limiting profile has either no interior defects or two line defects depending on gamma ^*, and for a triangular hole, there is a unique interior point defect outside the hole. In the lambda rightarrow infty limit, there are at least left( {begin{array}{c}K 2end{array}}right) stable states and the multistability is enhanced by gamma ^*, compared to the interior problem. Our work offers new insights into how to tune the existence, location, and dimensionality of defects.

Wasserstein model reduction approach for parametrized flow problems in porous media

The aim of this work is to build a reduced order model for parametrized porous media equations. The main challenge of this type of problems is that the Kolmogorov width of the solution manifold typically decays quite slowly and thus makes usual linear model order reduction methods inappropriate. In this work, we investigate an adaptation of the methodology proposed in [Ehrlacher et al., Nonlinear model reduction on metric spaces. Application to one-dimensional conservative PDEs in Wasserstein spaces, ESAIM: Mathematical Modelling and Numerical Analysis (2020)], based on the use of Wasserstein barycenters [Agueh & Carlier, Barycenters in the Wasserstein Space, SIAM Journal on Mathematical Analysis (2011)], to the case of non-conservative problems. Numerical examples in one-dimensional test cases illustrate the advantages and limitations of this approach and suggest further research directions that we intend to explore in the future.

Higher-order adaptive virtual element methods with contraction properties

<abstract><p>The realization of a standard Adaptive Finite Element Method (AFEM) preserves the mesh conformity by performing a completion step in the refinement loop: In addition to elements marked for refinement due to their contribution to the global error estimator, other elements are refined. In the new perspective opened by the introduction of Virtual Element Methods (VEM), elements with hanging nodes can be viewed as polygons with aligned edges, carrying virtual functions together with standard polynomial functions. The potential advantage is that all activated degrees of freedom are motivated by error reduction, not just by geometric reasons. This point of view is at the basis of the paper [L. Beirão da Veiga et al., "Adaptive VEM: stabilization-free a posteriori error analysis and contraction property", SIAM Journal on Numerical Analysis, vol. 61, 2023], devoted to the convergence analysis of an adaptive VEM generated by the successive newest-vertex bisections of triangular elements without applying completion, in the lowest-order case (polynomial degree $ k = 1 $). The purpose of this paper is to extend these results to the case of VEMs of order $ k\ge2 $ built on triangular meshes. The problem at hand is a variable-coefficient, second-order self-adjoint elliptic equation with Dirichlet boundary conditions; the data of the problem are assumed to be piecewise polynomials of degree $ k-1 $. By extending the concept of global index of a hanging node, under an admissibility assumption of the mesh, we derive a stabilization-free a posteriori error estimator. This is the sum of residual-type terms and certain virtual inconsistency terms (which vanish for $ k = 1 $). We define an adaptive VEM of order $ k $ based on this estimator, and we prove its convergence by establishing a contraction result for a linear combination of (squared) energy norm of the error, (squared) residual estimator, and (squared) virtual inconsistency estimator.</p></abstract>

Identifying codes in bipartite graphs of given maximum degree

An identifying code of a closed-twin-free graph G is a set S of vertices of G such that any two vertices in G have a distinct intersection between their closed neighborhoods and S. It was conjectured in [F. Foucaud, R. Klasing, A. Kosowski, A. Raspaud. On the size of identifying codes in triangle-free graphs. Discrete Applied Mathematics, 2012] that there exists an absolute constant c such that for every connected graph G of order n and maximum degree ∆, G admits an identifying code of size at most ∆-1/∆n+c. We provide significant support for this conjecture by proving it for the class of all bipartite graphs that do not contain any pairs of open-twins of degree at least 2. In particular, this class of bipartite graphs contains all trees and more generally, all bipartite graphs without 4-cycles. Moreover, our proof allows us to precisely determine the constant c for the considered class, and the list of graphs needing c ≥ 0. For ∆ = 2 (the graph is a path or a cycle), it is long known that c= 3/2 suffices. For connected graphs in the considered graph class, for each ∆ ≥ 3, we show that c= 1/∆ ≤ 1/3 suffices and that c is required to be positive only for a finite number of trees. In particular, for ∆ = 3, there are 12 trees with diameter at most 6 with a positive constant c and, for each ∆ ≥ 4, the only tree with positive constant c is the ∆-star. Our proof is based on induction and utilizes recent results from [F. Foucaud, T. Lehtilä. Revisiting and improving upper bounds for identifying codes. SIAM Journal on Discrete Mathematics, 2022].

Quantum approximate counting for Markov chains and collision counting

In this paper we show how to generalize the quantum approximate counting technique developed by Brassard, H{\o}yer and Tapp [ICALP 1998] to a more general setting: estimating the number of marked states of a Markov chain (a Markov chain can be seen as a random walk over a graph with weighted edges). This makes it possible to construct quantum approximate counting algorithms from quantum search algorithms based on the powerful ``quantum walk based search'' framework established by Magniez, Nayak, Roland and Santha [SIAM Journal on Computing 2011]. As an application, we apply this approach to the quantum element distinctness algorithm by Ambainis [SIAM Journal on Computing 2007]: for two injective functions over a set of $N$ elements, we obtain a quantum algorithm that estimates the number $m$ of collisions of the two functions within relative error $\epsilon$ by making $\tilde{O}\left(\frac{1}{\epsilon^{25/24}}\big(\frac{N}{\sqrt{m}}\big)^{2/3}\right)$ queries, which gives an improvement over the $\Theta\big(\frac{1}{\epsilon}\frac{N}{\sqrt{m}}\big)$-query classical algorithm based on random sampling when $m\ll N$.

SIGEST

The SIGEST article in this issue is “A Proximal Markov Chain Monte Carlo Method for Bayesian Inference in Imaging Inverse Problems: When Langevin Meets Moreau,” by Alain Durmus, Éric Moulines, and Marcelo Pereyra. The authors provide new algorithms to sample from high-dimensional log-concave probability measures, where they combine Moreau--Yosida envelopes with the Euler--Maruyama discretization of Langevin diffusions. This allows for an efficient Markov chain Monte Carlo methodology that is applicable to inverse problems arising in imaging sciences. Asymptotic and nonasymptotic convergence results are provided, along with extensive computational experiments on realistic imaging problems involving deconvolution and tomographic reconstruction. The original article, which appeared in the SIAM Journal on Imaging Sciences in 2018, has attracted substantial interest. In preparing this highlighted SIGEST version, the authors have expanded the introduction to make it accessible to a wide audience. The final section also discusses follow-up work arising from the original publication.

SIGEST

In this section we present “Minimizing the Repayment Cost of Federal Student Loans,” by Paolo Guasoni and Yu-Jui Huang. This is the highlighted SIGEST version of an article that first appeared in the SIAM Journal on Financial Mathematics (SIFIN) in 2021. The article studies optimal strategies for repayment of federal student loans in the USA. These loans, which can be used to cover tuition and living expenses, have specific features that make them tricky to evaluate. Sections 2 and 4, as in the original SIFIN article, take account of two key student loan features: capping repayment levels at a proportion of current income, and forgiving the debt after a sufficiently long period of good standing. Sections 3 and 5, which are new to the revised SIGEST version, give results that incorporate a third feature: the effect of simple interest. The modeling and analysis presented here shed light on an important, mathematically challenging financial product and give practical insights into a complicated and potentially daunting set of choices available to those who must negotiate their way through these contracts.

Existence of martingale solutions for stochastic flocking models with local alignment

We establish the existence of martingale solutions to a class of stochastic conservation equations. The underlying models correspond to random perturbations of kinetic models for collective motion such as the Cucker F, Smale S (IEEE Transactions on Automatic Control 52(5):852–862, 2007), Cucker F, Smale S (Japanese Journal of Mathematics 2(1):197–227, 2007) and Motsch S, Tadmor E (Journal of Statistical Physics 144(5):923–947, 2011) models. By regularizing the coefficients, we first construct approximate solutions obtained as the mean-field limit of the corresponding particle systems. We then establish the compactness in law of this family of solutions by relying on a stochastic averaging lemma. This extends the results obtained in Karper T, Mellet A, Trivisa K (Springer Proceedings in Mathematics & Statistics, 2012), Karper T, Mellet A, Trivisa K (SIAM Journal on Mathematical Analysis 45(1):215–243, 2013) in the deterministic case.

Improved bounds for the solutions of renewal equations

Sequences of non-decreasing (non-increasing) lower (upper) bounds for the renewal-type equation as well as for the renewal function which are improvements of the famous corresponding bounds of Marshal [(1973). Linear bounds on the renewal function. SIAM Journal on Applied Mathematics 24(2): 245–250] are given. Also, sequences such bounds converging to the ordinary renewal function are obtained for several reliability classes of the lifetime distributions of the inter-arrival times, which are refinements of all of the existing known corresponding bounds. For the first time, a lower bound for the renewal function with DMRL lifetimes is given. Finally, sequences of such improved bounds are given for the ordinary renewal density as well as for the right-tail of the distribution of the forward recurrence time.

Initial Data Identification for the One-Dimensional Burgers Equation

In this article, we study the problem of identification for the 1-D Burgers equation. This problem consists in identifying the set of initial data evolving to a given target at a final time. Due to the property of nonbackward uniqueness of the Burgers equation, there may exist multiple initial data leading to the same given target. In articles “Initial data identification in conservation laws and Hamilton-Jacobi equations” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">arXiv:1903.06448</i> , 2019) and “The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes” ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SIAM Journal on Mathematical Analysis</i> , vol. 52, the authors fully characterize the set of initial data leading to a given target using the classical Lax–Hopf formula. In this article, an alternative proof based only on generalized backward characteristics is given. This leads to the hope of investigating systems of conservation laws in 1-D, where the classical Lax–Hopf formula no more holds. Moreover, numerical illustrations are presented using as a target, a function optimized for minimum pressure rise in the context of sonic-boom minimization problems. All of initial data leading to this given target are constructed using a wavefront tracking algorithm.

SIGEST

The SIGEST article in this issue is Hamiltonicity of Cubic Planar Graphs with Bounded Face Sizes, by František Kardoš. The original version of this article appeared in the SIAM Journal on Discrete Mathematics in 2020. Barnette's Conjecture is a famous problem in graph theory which asserts that every cubic bipartite 3-connected planar graph has a Hamiltonian cycle. The conjecture stems from an attempt in 1880 by P. G. Tait to solve the Four Color Theorem by establishing that every cubic 3-connected planar graph has a Hamiltonian cycle. This is a stronger statement than the Four Color Theorem and it actually turns out to be false, as shown by W. T. Tutte in 1946. Barnette's Conjecture is equivalent to the statement that every plane cubic 3-connected graph with all faces of even size has a Hamiltonian cycle. Our SIGEST paper proves a closely related conjecture of Barnette and Goodey asserting that every plane cubic 3-connected graph with all faces of size at most six has a Hamiltonian cycle; this also solves an open problem from mathematical chemistry, which posited the existence of a Hamiltonian cycle under the stronger assumption that all faces have size five or six. The proof consists of several reduction steps and eventually boils the problem down to checking a finite number of configurations, which is handled with the assistance of a computer. The derivations are accompanied by high-quality graphical illustrations, and the author has provided computer code for the automated parts of the proof.

Finding Sparse Solutions for Packing and Covering Semidefinite Programs

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it may be desirable to obtain {\it sparse} dual solutions, i.e., those with support size (almost) independent of the number of primal constraints. In this paper, we give an algorithm that finds such solutions, which is an extension of a {\it logarithmic-potential} based algorithm of Grigoriadis, Khachiyan, Porkolab and Villavicencio (SIAM Journal of Optimization 41 (2001)) for packing/covering linear programs.