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A squared smoothing Newton method for semidefinite programming
Get PDFThis paper proposes a squared smoothing Newton method via the Huber smoothing function for solving semidefinite programming problems (SDPs). We first study the fundamental properties of the matrix-valued mapping defined upon the Huber function. Using these results and existing ones in the literature, we then conduct rigorous convergence analysis and establish convergence properties for the proposed algorithm. In particular, we show that the proposed method is well-defined and admits global convergence. Moreover, under suitable regularity conditions, i.e., the primal and dual constraint nondegenerate conditions, the proposed method is shown to have a superlinear convergence rate. To evaluate the practical performance of the algorithm, we conduct extensive numerical experiments for solving various classes of SDPs. Comparison with the state-of-the-art SDP solver {\tt {\tt SDPNAL+}} demonstrates that our method is also efficient for computing accurate solutions of SDPs
Optimal temperature distribution for a nonisothermal Cahn--Hilliard system with source term
Get PDFIn this note, we study the optimal control of a nonisothermal phase field system of Cahn--Hilliard type that constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. The system couples a Cahn--Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter, typical of the classic Cahn--Hilliard equation, is no longer satisfied. In this paper, we analyze the case that the double-well potential driving the evolution of the phase transition is differentiable, either (in the regular case) on the whole set of reals or (in the singular logarithmic case) on a finite open interval; nondifferentiable cases like the double obstacle potential are excluded from the analysis. We prove the Fréchet differentiability of the control-to-state operator between suitable Banach spaces for both the regular and the logarithmic cases and establish the solvability of the corresponding adjoint systems in order to derive the associated first-order necessary optimality conditions for the optimal control problem. Crucial for the whole analysis to work is the so-called ``strict separation property'', which states that the order parameter attains its values in a compact subset of the interior of the effective domain of the nonlinearity. While this separation property turns out to be generally valid for regular potentials in three dimensions of space, it can be shown for the logarithmic case only in two dimensions
Sharp-interface limits of Cahn--Hilliard models and mechanics with moving contact lines
Get PDFWe construct gradient structures for free boundary problems with moving capillary interfaces with nonlinear (hyper)elasticity and study the impact of moving contact lines. In this context, we numerically analyze how phase-field models converge to certain sharp-interface models when the interface thickness tends to zero. In particular, we study the scaling of the Cahn--Hilliard mobility with certain powers of the interfacial thickness. In the presence of interfaces, it is known that the intended sharp-interface limit holds only for a particular range of powers However, in the presence of moving contact lines we show that some scalings that are valid for interfaces produce significant errors and the effective range of valid powers of the interfacial thickness in the mobility reduces
Foundational competencies and responsibilities of a research software engineer
Get PDFThe term Research Software Engineer, or RSE, emerged a little over 10 years ago as a way to represent individuals working in the research community but focusing on software development. The term has been widely adopted and there are a number of high-level definitions of what an RSE is. However, the roles of RSEs vary depending on the institutional context they work in. At one end of the spectrum, RSE roles may look similar to a traditional research role. At the other extreme, they resemble that of a software engineer in industry. Most RSE roles inhabit the space between these two extremes. Therefore, providing a straightforward, comprehensive definition of what an RSE does and what experience, skills and competencies are required to become one is challenging. In this community paper we define the broad notion of what an RSE is, explore the different types of work they undertake, and define a list of foundational competencies as well as values that outline the general profile of an RSE. These foundational skills are encountered to a large extent within the skill sets of current RSEs in Germany and beyond, and we propose them as a starting point for aspiring RSEs to shape their technical profile. Further research and training can build upon this foundation of skills and focus on various aspects in greater detail. We expect that graduates and practitioners will have a larger and more diverse set of skills than outlined here. On this basis, we elaborate on the progression of these skills along different dimensions. We look at specific types of RSE roles, propose recommendations for organisations, and give examples of future specialisations. An appendix details how existing curricula fit into this framework
Weighted sparsity and sparse tensor networks for least squares approximation
Get PDFThe approximation of high-dimensional functions is a ubiquitous problem in many scientific fields that is only feasible practically if advantageous structural properties can be exploited. One prominent structure is sparsity relatively to some basis. For the analysis of these best n-term approximations a relevant tool is the Stechkin's lemma. In its standard form, however, this lemma does not allow to explain convergence rates for a wide range of relevant function classes. This work presents a new weighted version of Stechkin's lemma that improves the best n-term rates for weighted ℓp-spaces and associated function classes such as Sobolev or Besov spaces. For the class of holomorphic functions, which for example occur as solutions of common high-dimensional parameter dependent PDEs, we recover exponential rates that are not directly obtainable with Stechkin's lemma. This sparsity can be used to devise weighted sparse least squares approximation algorithms as known from compressed sensing. However, in high-dimensional settings, classical algorithms for sparse approximation suffer the curse of dimensionality. We demonstrate that sparse approximations can be encoded efficiently using tensor networks with sparse component tensors. This representation gives rise to a new alternating algorithm for best n-term approximation with a complexity scaling polynomially in n and the dimension. We also demonstrate that weighted ℓpsummability not only induces sparsity of the tensor but also low ranks. This is not exploited by the previous format. We thus propose a new low-rank tensor train format with a single weighted sparse core tensor and an ad-hoc algorithm for approximation in this format. To analyse the sample complexity for this new model class we derive a novel result of independent interest that allows to transfer the restricted isometry property from one set to another sufficiently close set. We then prove that the new model class is close enough to the set of weighted sparse vectors such that the restricted isometry property transfers. Numerical examples illustrate the theoretical results for a benchmark problem from uncertainty quantification. Although they lead up to the analysis of our final model class, our contributions on weighted Stechkin and the restricted isometry property are of independent interest and can be read independently
The power of first-order smooth optimization for black-box non-smooth problems
Get PDFGradient-free/zeroth-order methods for black-box convex optimization have been extensively studied in the last decade with the main focus on oracle calls complexity. In this paper, besides the oracle complexity, we focus also on iteration complexity, and propose a generic approach that, based on optimal first-order methods, allows to obtain in a black-box fashion new zeroth-order algorithms for non-smooth convex optimization problems. Our approach not only leads to optimal oracle complexity, but also allows to obtain iteration complexity similar to first-order methods, which, in turn, allows to exploit parallel computations to accelerate the convergence of our algorithms. We also elaborate on extensions for stochastic optimization problems, saddle-point problems, and distributed optimization
A Large-Deviations Principle for all the Components in a Sparse Inhomogeneous Random Graph
No full textWe study an inhomogeneous sparse random graph, GN , on [N]={1,..., N} as introduced in a seminal paper by Bollobás et al. (Random Struct Algorithms 31(1):3–122, 2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞, we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of GN . In particular, we recover the criterion for the existence of the phase transition given in Bollobás et al. (2007
Efficient option pricing in the rough Heston model using weak simulation schemes
Get PDFWe provide an efficient and accurate simulation scheme for the rough Heston model in the standard () as well as the hyper-rough regime (). The scheme is based on low-dimensional Markovian approximations of the rough Heston process derived in [Bayer and Breneis, arXiv:2309.07023], and provides weak approximation to the rough Heston process. Numerical experiments show that the new scheme exhibits second order weak convergence, while the computational cost increases linear with respect to the number of time steps. In comparison, existing schemes based on discretization of the underlying stochastic Volterra integrals such as Gatheral's HQE scheme show a quadratic dependence of the computational cost. Extensive numerical tests for standard and path-dependent European options and Bermudan options show the method's accuracy and efficiency
Connection Intervals in Multi-Scale Infrastructure-Augmented Dynamic Networks
No full textWe consider a hybrid spatial communication system in which mobile nodes can connect to static sinks in a bounded number of intermediate relaying hops. We describe the distribution of the connection intervals of a typical mobile node, i.e., the intervals of uninterrupted connection to the family of sinks. This is achieved in the limit of many hops, sparse sinks and growing time horizons. We identify three regimes reflecting various degrees of sink densities. Namely, (1) a regime of dense sinks, in which the limit is deterministic and given as an expectation with respect to percolation clusters, (2) a regime of sparse sinks, in which the limit depends on a random number of reachable sinks, and (3) an intermediate critical regime