Consider a process, stochastic or deterministic, obtained by using a numerical integration scheme, or from Monte-Carlo methods involving an approximation to an integral, or a Newton-Raphson iteration to approximate the root of an equation. We will assume that we can sample from the distribution of the process from time 0 to finite time n. We propose a scheme for unbiased estimation of the limiting value of the process, together with estimates of standard error and apply this to examples including numerical integrals, root-finding and option pricing in a Heston Stochastic Volatility model. This results in unbiased estimators in place of biased ones in many potential applications.
Numerical integration is a foundational technique in scientific computing and is at the core of many computer vision applications. Among these applications, neural volume rendering has recently been proposed as a new paradigm for view synthesis, achieving photorealistic image quality. However, a fundamental obstacle to making these methods practical is the extreme computational and memory requirements caused by the required volume integrations along the rendered rays during training and inference. Millions of rays, each requiring hundreds of forward passes through a neural network are needed to approximate those integrations with Monte Carlo sampling. Here, we propose automatic integration, a new framework for learning efficient, closed-form solutions to integrals using coordinate-based neural networks. For training, we instantiate the computational graph corresponding to the derivative of the network. The graph is fitted to the signal to integrate. After optimization, we reassemble the graph to obtain a network that represents the antiderivative. By the fundamental theorem of calculus, this enables the calculation of any definite integral in two evaluations of the network. Applying this approach to neural rendering, we improve a tradeoff between rendering speed and image quality: improving render times by greater than 10 times with a tradeoff of slightly reduced image quality.
We present a delay-compensating control method that transforms exponentially stabilizing controllers for an undelayed system into a sample-based predictive controller with numerical integration. Our method handles both first-order and transport delays in actuators and trades-off numerical accuracy with computation delay to guaranteed stability under hardware limitations. Through hybrid stability analysis and numerical simulation, we demonstrate the efficacy of our method from both theoretical and simulation perspectives.
Wideband direction of arrival (DOA) estimation techniques for sensors array have been studied extensively in the literature. Nevertheless, needing prior information on the number and directions of sources or demanding heavy computational load makes most of these techniques less useful in practice. In this paper, a low complexity subspace-based framework for DOA estimation of broadband signals, named as wideband modal orthogonality (WIMO), is proposed and accordingly two DOA estimators are developed. First, a closed-form approximation of spatial-temporal covariance matrix (STCM) in the uniform spectrum case is presented. The eigenvectors of STCM associated with non-zero eigenvalues are modal components of the wideband source in a given bandwidth and direction. WIMO idea is to extract these eigenvectors at desired DOAs from the approximated STCM and test their orthogonality to estimated noise subspace. In the non-uniform spectrum case, WIMO idea can be applied by approximating STCM through numerical integration. Fortunately, STCM approximation and modal extraction can be performed offline. WIMO provides DOA estimation without the conventional prerequisites, such as spectral decomposition, focusing procedure and, a priori information on the number of sources and their DOAs. Several numerical examples are conducted to compare the WIMO performance with the state-of-the-art methods. Simulations demonstrate that the two proposed DOA estimators achieve superior performance in terms of probability of resolution and estimation error along with orders of magnitude runtime speedup.
This paper investigates the chattering and deadlock behaviors of the proportional-integral (PI) controller with an anti-windup (AW) limiter recommended by the IEEE Standard 421.5-2016. Depending on the simulation method, the controller may enter a chattering or deadlock state in some combinations of parameters and inputs. Chattering and deadlock are analyzed in the context of three numerical integration approaches: explicit partitioned method (EPM), execution-list based method (ELM), and implicit trapezoidal method (ITM). This paper derives the chattering stop condition for EPM and ELP, and analyzes the impacts of step size and convergence tolerance for simultaneous method. The deduced chattering stop conditions and deadlock behavior is verified with numerical simulations.
We take up a recently proposed compartmental SEIQR model with delays, ignore loss of immunity in the context of a fast pandemic, extend the model to a network structured on infectivity, and consider the continuum limit of the same with a simple separable interaction model for the infectivities $\beta$. Numerical simulations show that the evolving dynamics of the network is effectively captured by a single scalar function of time, regardless of the distribution of $\beta$ in the population. The continuum limit of the network model allows a simple derivation of the simpler model, which is a single scalar delay differential equation (DDE), wherein the variation in $\beta$ appears through an integral closely related to the moment generating function of $u=\sqrt{\beta}$. If the first few moments of $u$ exist, the governing DDE can be expanded in a series that shows a direct correspondence with the original compartmental DDE with a single $\beta$. Even otherwise, the new scalar DDE can be solved using either numerical integration over $u$ at each time step, or with the analytical integral if available in some useful form. Our work provides a new academic example of complete dimensional collapse, ties up an underlying continuum model for a pandemic with a simpler-seeming compartmental model, and will hopefully lead to new analysis of continuum models for epidemics.
The multilevel Monte Carlo (MLMC) method is highly efficient for estimating expectations of a functional of a solution to a stochastic differential equation (SDE). However, MLMC estimators may be unstable and have a poor (noncanonical) complexity in the case of low regularity of the functional. To overcome this issue, we extend our previously introduced idea of numerical smoothing in (Quantitative Finance, 23(2), 209-227, 2023), in the context of deterministic quadrature methods to the MLMC setting. The numerical smoothing technique is based on root-finding methods combined with one-dimensional numerical integration with respect to a single well-chosen variable. This study is motivated by the computation of probabilities of events, pricing options with a discontinuous payoff, and density estimation problems for dynamics where the discretization of the underlying stochastic processes is necessary. The analysis and numerical experiments reveal that the numerical smoothing significantly improves the strong convergence, and consequently, the complexity and robustness (by making the kurtosis at deep levels bounded) of the MLMC method. In particular, we show that numerical smoothing enables recovering the MLMC complexities obtained for Lipschitz functionals due to the optimal variance decay rate when using the Euler--Maruyama scheme. For the Milstein scheme, numerical smoothing recovers the canonical MLMC complexity even for the nonsmooth integrand mentioned above. Finally, our approach efficiently estimates univariate and multivariate density functions.
We consider an investor, whose portfolio consists of a single risky asset and a risk free asset, who wants to maximize his expected utility of the portfolio subject to the Value at Risk assuming a heavy tail distribution of the stock prices return. We use Markov Decision Process and dynamic programming principle to get the optimal strategies and the value function which maximize the expected utility for parametric as well as non parametric distributions. Due to lack of explicit solution in the non parametric case, we use numerical integration for optimization
An accurate, closed-form expression evaluating the nonlinear interference (NLI) power in coherent optical transmission systems in the presence of inter-channel stimulated Raman scattering (ISRS) is derived. The analytical result enables a rapid estimate of the signal-to-noise ratio (SNR) and avoids the need for integral evaluations and split-step simulations. The formula also provides new insight into the underlying parameter dependence of ISRS on the NLI. The proposed result is applicable for dispersion unmanaged, ultra-wideband transmission systems that use optical bandwidths of up to 15 THz. The accuracy of the closed-form expression is compared to numerical integration of the ISRS Gaussian Noise model and split-step simulations in a point-to-point transmission, as well as in a mesh optical network scenario.
We propose a fast and accurate numerical method for pricing European swaptions in multi-factor Gaussian term structure models. Our method can be used to accelerate the calibration of such models to the volatility surface. The pricing of an interest rate option in such a model involves evaluating a multi-dimensional integral of the payoff of the claim on a domain where the payoff is positive. In our method, we approximate the exercise boundary of the state space by a hyperplane tangent to the maximum probability point on the boundary and simplify the multi-dimensional integration into an analytical form. The maximum probability point can be determined using the gradient descent method. We demonstrate that our method is superior to previous methods by comparing the results to the price obtained by numerical integration.
In this paper we derive a series expansion for the price of a continuously sampled arithmetic Asian option in the Black-Scholes setting. The expansion is based on polynomials that are orthogonal with respect to the log-normal distribution. All terms in the series are fully explicit and no numerical integration nor any special functions are involved. We provide sufficient conditions to guarantee convergence of the series. The moment indeterminacy of the log-normal distribution introduces an asymptotic bias in the series, however we show numerically that the bias can safely be ignored in practice.
The shape of allele-frequency clines maintained by migration-selection balance depends not only on the properties of migration and selection, but also on the dominance relations among alleles and on linkage to other loci under selection. We investigate a two-locus model in which two diallelic, recombining loci are subject to selection caused by an abrupt environmental change. The habitat is one-dimensional and unbounded, selection at each locus is modeled by step functions such that in one region one allele at each locus is advantageous and in the other deleterious. We admit an environmentally independent, intermediate degree of dominance at both loci, including complete dominance. First, we derive an explicit expression for the single-locus cline with dominance, thus generalizing classical results by Haldane (1948). We show that the slope of the cline in the center (at the step) or, equivalently, the width of the cline, is independent of the degree of dominance. Second, under the assumption of strong recombination relative to selection and migration, the first-order approximations of the allele-frequency clines at each of the loci and of the linkage disequilibrium are derived. This may be interpreted as the quasi-linkage-equilibrium approximation of the two-locus cline. Explicit asymptotic expressions for the clines are deduced as $x\to\pm\infty$. For equivalent loci, explicit expressions for the whole clines are derived. The influence of dominance and of linkage on the slope of the cline in the center and on a global measure of steepness are investigated. This global measure reflects the influence of dominance. Finally, the accuracy of the approximations and the dependence of the shape of the two-locus cline on the full range of recombination rates is explored by numerical integration of the underlying system of partial differential equations.
Over the last few decades, ecologists have come to appreciate that key ecological patterns, which describe ecological communities at relatively large spatial scales, are not only scale dependent, but also intimately intertwined. The relative abundance of species, which informs us about the commonness and rarity of species, changes its shape from small to large spatial scales. The average number of species as a function of area has a steep initial increase, followed by decreasing slopes at large scales. Finally, if we find a species in a given location, it is more likely we find an individual of the same species close-by, rather than farther apart. Such spatial turnover depends on the geographical distribution of species, which often are spatially aggregated. This reverberates on the abundances as well as the richness of species within a region, but so far it has been difficult to quantify such relationships. Within a neutral framework, which considers all individuals competitively equivalent, we introduce a spatial stochastic model, which phenomenologically accounts for birth, death, immigration and local dispersal of individuals. We calculate the pair correlation function, which encapsulates spatial turnover, and the conditional probability to find a species with a certain population within a given circular area. Also, we calculate the macro-ecological patterns, which we have referred to above, and compare the analytical formul{\ae} with the numerical integration of the model. Finally, we contrast the model predictions with the empirical data for two lowland tropical forest inventories, showing always a good agreement.
We consider the problem of pricing basket options in a multivariate Black Scholes or Variance Gamma model. From a numerical point of view, pricing such options corresponds to moderate and high dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster compared to Monte Carlo or Quasi Monte Carlo in dimensions up to 35.
Realised pay-offs for discretisation-invariant swaps are those which satisfy a restricted `aggregation property' of Neuberger [2012] for twice continuously differentiable deterministic functions of a multivariate martingale. They are initially characterised as solutions to a second-order system of PDEs, then those pay-offs based on martingale and log-martingale processes alone form a vector space. Hence there exist an infinite variety of other variance and higher-moment risk premia that are less prone to bias than standard variance swaps because their option replication portfolios have no discrete-monitoring or jump errors. Their fair values are also independent of the monitoring partition. A sub-class consists of pay-offs with fair values that are further free from numerical integration errors over option strikes. Here exact pricing and hedging is possible via dynamic trading strategies on a few vanilla puts and calls. An S&P 500 empirical study on higher-moment and other DI swaps concludes.
Boolean satisfiability is a propositional logic problem of interest in multiple fields, e.g., physics, mathematics, and computer science. Beyond a field of research, instances of the SAT problem, as it is known, require efficient solution methods in a variety of applications. It is the decision problem of determining whether a Boolean formula has a satisfying assignment, believed to require exponentially growing time for an algorithm to solve for the worst-case instances. Yet, the efficient solution of many classes of Boolean formulae eludes even the most successful algorithms, not only for the worst-case scenarios, but also for typical-case instances. Here, we introduce a memory-assisted physical system (a digital memcomputing machine) that, when its non-linear ordinary differential equations are integrated numerically, shows evidence for polynomially-bounded scalability while solving "hard" planted-solution instances of SAT, known to require exponential time to solve in the typical case for both complete and incomplete algorithms. Furthermore, we analytically demonstrate that the physical system can efficiently solve the SAT problem in continuous time, without the need to introduce chaos or an exponentially growing energy. The efficiency of the simulations is related to the collective dynamical properties of the original physical system that persist in the numerical integration to robustly guide the solution search even in the presence of numerical errors. We anticipate our results to broaden research directions in physics-inspired computing paradigms ranging from theory to application, from simulation to hardware implementation.
The randomized midpoint method, proposed by [SL19], has emerged as an optimal discretization procedure for simulating the continuous time Langevin diffusions. Focusing on the case of strong-convex and smooth potentials, in this paper, we analyze several probabilistic properties of the randomized midpoint discretization method for both overdamped and underdamped Langevin diffusions. We first characterize the stationary distribution of the discrete chain obtained with constant step-size discretization and show that it is biased away from the target distribution. Notably, the step-size needs to go to zero to obtain asymptotic unbiasedness. Next, we establish the asymptotic normality for numerical integration using the randomized midpoint method and highlight the relative advantages and disadvantages over other discretizations. Our results collectively provide several insights into the behavior of the randomized midpoint discretization method, including obtaining confidence intervals for numerical integrations.
In this paper, we develop a quadrature framework for large-scale kernel machines via a numerical integration representation. Considering that the integration domain and measure of typical kernels, e.g., Gaussian kernels, arc-cosine kernels, are fully symmetric, we leverage deterministic fully symmetric interpolatory rules to efficiently compute quadrature nodes and associated weights for kernel approximation. The developed interpolatory rules are able to reduce the number of needed nodes while retaining a high approximation accuracy. Further, we randomize the above deterministic rules by the classical Monte-Carlo sampling and control variates techniques with two merits: 1) The proposed stochastic rules make the dimension of the feature mapping flexibly varying, such that we can control the discrepancy between the original and approximate kernels by tuning the dimnension. 2) Our stochastic rules have nice statistical properties of unbiasedness and variance reduction with fast convergence rate. In addition, we elucidate the relationship between our deterministic/stochastic interpolatory rules and current quadrature rules for kernel approximation, including the sparse grids quadrature and stochastic spherical-radial rules, thereby unifying these methods under our framework. Experimental results on several benchmark datasets show that our methods compare favorably with other representative kernel approximation based methods.
Mesh-based simulations are central to modeling complex physical systems in many disciplines across science and engineering. Mesh representations support powerful numerical integration methods and their resolution can be adapted to strike favorable trade-offs between accuracy and efficiency. However, high-dimensional scientific simulations are very expensive to run, and solvers and parameters must often be tuned individually to each system studied. Here we introduce MeshGraphNets, a framework for learning mesh-based simulations using graph neural networks. Our model can be trained to pass messages on a mesh graph and to adapt the mesh discretization during forward simulation. Our results show it can accurately predict the dynamics of a wide range of physical systems, including aerodynamics, structural mechanics, and cloth. The model's adaptivity supports learning resolution-independent dynamics and can scale to more complex state spaces at test time. Our method is also highly efficient, running 1-2 orders of magnitude faster than the simulation on which it is trained. Our approach broadens the range of problems on which neural network simulators can operate and promises to improve the efficiency of complex, scientific modeling tasks.
This paper introduces methods and a novel toolbox that efficiently integrates any high-dimensional Neural Mass Models (NMMs) specified by two essential components. The first is the set of nonlinear Random Differential Equations of the dynamics of each neural mass. The second is the highly sparse three-dimensional Connectome Tensor (CT) that encodes the strength of the connections and the delays of information transfer along the axons of each connection. Semi-analytical integration of the RDE is done with the Local Linearization scheme for each neural mass model, which is the only scheme guaranteeing dynamical fidelity to the original continuous-time nonlinear dynamic. It also seamlessly allows modeling distributed delays CT with any level of complexity or realism, as shown by the Moore-Penrose diagram of the algorithm. This ease of implementation includes models with distributed-delay CTs. We achieve high computational efficiency by using a tensor representation of the model that leverages semi-analytic expressions to integrate the Random Differential Equations (RDEs) underlying the NMM. We discretized the state equation with Local Linearization via an algebraic formulation. This approach increases numerical integration speed and efficiency, a crucial aspect of large-scale NMM simulations. To illustrate the usefulness of the toolbox, we simulate both a single Zetterberg-Jansen-Rit (ZJR) cortical column and an interconnected population of such columns. These examples illustrate the consequence of modifying the CT in these models, especially by introducing distributed delays. We provide an open-source Matlab live script for the toolbox.