Publications
100 results found
Crisan D, Holm DD, Luesink E, et al., 2023, Theoretical and computational analysis of the thermal quasi-geostrophic model, Journal of Nonlinear Science, Vol: 33, ISSN: 0938-8974
This work involves theoretical and numerical analysis of the thermal quasi-geostrophic(TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, theTQG model involves thermal geostrophic balance, in which the Rossby number, theFroude number and the stratification parameter are all of the same asymptotic order.The main analytical contribution of this paper is to construct local-in-time uniquestrong solutions for the TQG model. For this, we show that solutions of its regularisedversion α-TQG converge to solutions of TQG as its smoothing parameter α → 0and we obtain blow-up criteria for the α-TQG model. The main contribution of thecomputational analysis is to verify the rate of convergence of α-TQG solutions to TQGsolutions as α → 0, for example, simulations in appropriate GFD regimes.
Crisan D, Holm DD, Korn P, 2023, An implementation of Hasselmann's paradigm for stochastic climate modelling based on stochastic Lie transport *, Nonlinearity, Vol: 36, Pages: 4862-4903, ISSN: 0951-7715
A generic approach to stochastic climate modelling is developed for the example of an idealised Atmosphere-Ocean model that rests upon Hasselmann's paradigm for stochastic climate models. Namely, stochasticity is incorporated into the fast moving atmospheric component of an idealised coupled model by means of stochastic Lie transport, while the slow moving ocean model remains deterministic. More specifically the stochastic model stochastic advection by Lie transport (SALT) is constructed by introducing stochastic transport into the material loop in Kelvin's circulation theorem. The resulting stochastic model preserves circulation, as does the underlying deterministic climate model. A variant of SALT called Lagrangian-averaged (LA)-SALT is introduced in this paper. In LA-SALT, we replace the drift velocity of the stochastic vector field by its expected value. The remarkable property of LA-SALT is that the evolution of its higher moments are governed by deterministic equations. Our modelling approach is substantiated by establishing local existence results, first, for the deterministic climate model that couples compressible atmospheric equations to incompressible ocean equation, and second, for the two stochastic SALT and LA-SALT models.
Crisan D, Holm DD, Lang O, et al., 2023, Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model, STOCHASTICS AND DYNAMICS, ISSN: 0219-4937
Lang O, Crisan D, 2023, Well-posedness for a stochastic 2D Euler equation with transport noise, Stochastics and Partial Differential Equations: Analysis and Computations, Vol: 11, Pages: 433-480, ISSN: 2194-0401
We prove the existence of a unique global strong solution for a stochastic two-dimensional Euler vorticity equation for incompressible flows with noise of transport type. In particular, we show that the initial smoothness of the solution is preserved. The arguments are based on approximating the solution of the Euler equation with a family of viscous solutions which is proved to be relatively compact using a tightness criterion by Kurtz.
Lang O, Crisan D, Memin E, 2023, Analytical Properties for a Stochastic Rotating Shallow Water Model Under Location Uncertainty, JOURNAL OF MATHEMATICAL FLUID MECHANICS, Vol: 25, ISSN: 1422-6928
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- Citations: 1
Chassagneux JF, Chotai H, Crisan D, 2023, Modelling Multiperiod Carbon Markets Using Singular Forward-Backward SDEs, Mathematics of Operations Research, Vol: 48, Pages: 463-497, ISSN: 0364-765X
We introduce a model for the evolution of emissions and the price of emissions allowances in a carbon market, such as the European Union Emissions Trading System (EU ETS). The model accounts for multiple trading periods, or phases, with multiple times at which compliance can occur. At the end of each trading period, the participating firms must surrender allowances for their emissions made during that period, and additional allowances can be used for compliance in the following periods. We show that the multiperiod allowance pricing problem is well-posed for various mechanisms (such as banking, borrowing, and withdrawal of allowances) linking the trading periods. The results are based on the analysis of a forward-backward stochastic differential equation with coupled forward and backward components, a discontinuous terminal condition, and a forward component that is degenerate. We also introduce an infinite-period model for a carbon market with a sequence of compliance times and with no end date. We show that, under appropriate conditions, the value function for the multiperiod pricing problem converges, as the number of periods increases, to a value function for this infinite-period model and that such functions are unique. Finally, we present a numerical example that demonstrates empirically the convergence of the multiperiod pricing problem.
Crisan D, Ghil M, 2023, Asymptotic behavior of the forecast-assimilation process with unstable dynamics, CHAOS, Vol: 33, ISSN: 1054-1500
Crisan D, Ghil M, 2023, Asymptotic behavior of the forecast-assimilation process with unstable dynamics., Chaos, Vol: 33
Extensive numerical evidence shows that the assimilation of observations has a stabilizing effect on unstable dynamics, in numerical weather prediction, and elsewhere. In this paper, we apply mathematically rigorous methods to show why this is so. Our stabilization results do not assume a full set of observations and we provide examples where it suffices to observe the model's unstable degrees of freedom.
Crisan D, Lang O, 2023, Well-Posedness Properties for a Stochastic Rotating Shallow Water Model, Journal of Dynamics and Differential Equations, ISSN: 1040-7294
In this paper, we study the well-posedness properties of a stochastic rotating shallow water system. An inviscid version of this model has first been derived in Holm (Proc R Soc A 471:20140963, 2015) and the noise is chosen according to the Stochastic Advection by Lie Transport theory presented in Holm (Proc R Soc A 471:20140963, 2015). The system is perturbed by noise modulated by a function that is not Lipschitz in the norm where the well-posedness is sought. We show that the system admits a unique maximal solution which depends continuously on the initial condition. We also show that the interval of existence is strictly positive and the solution is global with positive probability.
Goodair D, Crisan D, Lang O, 2023, Existence and uniqueness of maximal solutions to SPDEs with applications to viscous fluid equations, Stochastics and Partial Differential Equations: Analysis and Computations, ISSN: 2194-0401
We present two criteria for the existence and uniqueness of a maximal strong solution for a general class of stochastic partial differential equations. Each criterion has its corresponding set of assumptions and can be applied to viscous fluid equations with additive, multiplicative or a general transport type noise. In particular, we apply these criteria to demonstrate well-posedness results for the 3D SALT [Stochastic Advection by Lie Transport, (Holm in Proc R Soc A Math Phys Eng Sci 471:20140963, 2015)] Navier–Stokes Equation in velocity and vorticity form, on the torus and the bounded domain respectively.
Ruzayqat H, Beskos A, Crisan D, et al., 2023, Unbiased estimation using a class of diffusion processes, Journal of Computational Physics, Vol: 472, ISSN: 0021-9991
We study the problem of unbiased estimation of expectations with respect to(w.r.t.) $\pi$ a given, general probability measure on$(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ that is absolutely continuous withrespect to a standard Gaussian measure. We focus on simulation associated to aparticular class of diffusion processes, sometimes termed theSchr\"odinger-F\"ollmer Sampler, which is a simulation technique thatapproximates the law of a particular diffusion bridge process $\{X_t\}_{t\in[0,1]}$ on $\mathbb{R}^d$, $d\in \mathbb{N}_0$. This latter process isconstructed such that, starting at $X_0=0$, one has $X_1\sim \pi$. Typically,the drift of the diffusion is intractable and, even if it were not, exactsampling of the associated diffusion is not possible. As a result,\cite{sf_orig,jiao} consider a stochastic Euler-Maruyama scheme that allows thedevelopment of biased estimators for expectations w.r.t.~$\pi$. We show thatfor this methodology to achieve a mean square error of$\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$, the associated cost is$\mathcal{O}(\epsilon^{-5})$. We then introduce an alternative approach thatprovides unbiased estimates of expectations w.r.t.~$\pi$, that is, it does notsuffer from the time discretization bias or the bias related with theapproximation of the drift function. We prove that to achieve a mean squareerror of $\mathcal{O}(\epsilon^2)$, the associated cost is, with highprobability, $\mathcal{O}(\epsilon^{-2}|\log(\epsilon)|^{2+\delta})$, for any$\delta>0$. We implement our method on several examples including Bayesianinverse problems.
Crisan D, Del Moral P, Jasra A, et al., 2022, LOG-NORMALIZATION CONSTANT ESTIMATION USING THE ENSEMBLE KALMAN-BUCY FILTER WITH APPLICATION TO HIGH-DIMENSIONAL MODELS, ADVANCES IN APPLIED PROBABILITY, Vol: 54, Pages: 1139-1163, ISSN: 0001-8678
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- Citations: 2
Crisan D, Street OD, 2022, On the analytical aspects of inertial particle motion, Journal of Mathematical Analysis and Applications, Vol: 516, Pages: 1-30, ISSN: 0022-247X
In their seminal 1983 paper, M. Maxey and J. Riley introduced an equation for the motion of a sphere through a fluid. Since this equation features the Basset history integral, the popularity of this equation has broadened the use of a certain form of fractional differential equation to study inertial particle motion. In this paper, we give a comprehensive theoretical analysis of the Maxey-Riley equation. In particular, we build on previous local in time existence and uniqueness results to prove that solutions of the Maxey-Riley equation are global in time. In doing so, we also prove that the notion of a maximal solution extends to this equation. We furthermore prove conditions under which solutions are differentiable at the initial time. By considering the derivative of the solution with respect to the initial conditions, we perform a sensitivity analysis and demonstrate that two inertial trajectories can not meet, as well as provide a control on the growth of the distance between a pair of inertial particles. The properties we prove here for the Maxey-Riley equations are also possessed, mutatis mutandis, by a broader class of fractional differential equations of a similar form.
Lang O, van Leeuwen PJ, Crisan D, et al., 2022, Bayesian inference for fluid dynamics: A case study for the stochastic rotating shallow water model, Frontiers in Applied Mathematics and Statistics, Vol: 8
In this work, we use a tempering-based adaptive particle filter to infer from a partially observed stochastic rotating shallow water (SRSW) model which has been derived using the Stochastic Advection by Lie Transport (SALT) approach. The methodology we present here validates the applicability of tempering and sample regeneration using a Metropolis-Hastings procedure to high-dimensional models appearing in geophysical fluid dynamics problems. The methodology is tested on the Lorenz 63 model with both full and partial observations. We then study the efficiency of the particle filter for the SRSW model in a configuration simulating the atmospheric Jetstream.
Crisan D, Lobbe A, Ortiz-Latorre S, 2022, An application of the splitting-up method for the computation of a neural network representation for the solution for the filtering equations, STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS, Vol: 10, Pages: 1050-1081, ISSN: 2194-0401
Paulin D, Jasra A, Beskos A, et al., 2022, A 4D-Var method with flow-dependent background covariances for the shallow-water equations, STATISTICS AND COMPUTING, Vol: 32, ISSN: 0960-3174
Dufee B, Memin E, Crisan D, 2022, Stochastic parametrization: An alternative to inflation in ensemble Kalman filters, QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Vol: 148, Pages: 1075-1091, ISSN: 0035-9009
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- Citations: 4
Crisan D, Holm DD, Leahy J-M, et al., 2022, Solution properties of the incompressible Euler system with rough path advection, arXiv
We consider the Euler equations for the incompressible flow of an ideal fluidwith an additional rough-in-time, divergence-free, Lie-advecting vector field.In recent work, we have demonstrated that this system arises from Clebsch andHamilton-Pontryagin variational principles with a perturbative geometric roughpath Lie-advection constraint. In this paper, we prove local well-posedness ofthe system in $L^2$-Sobolev spaces $H^m$ with integer regularity $m\ge \lfloord/2\rfloor+2$ and establish a Beale-Kato-Majda (BKM) blow-up criterion in termsof the $L^1_tL^\infty_x$-norm of the vorticity. In dimension two, we show thatthe $L^p$-norms of the vorticity are conserved, which yields globalwell-posedness and a Wong-Zakai approximation theorem for the stochasticversion of the equation.
Crisan D, Holm DD, Leahy J-M, et al., 2022, Variational principles for fluid dynamics on rough paths, arXiv
In this paper, we introduce a new framework for parametrization schemes (PS)in GFD. Using the theory of controlled rough paths, we derive a class of roughgeophysical fluid dynamics (RGFD) models as critical points of rough actionfunctionals. These RGFD models characterize Lagrangian trajectories in fluiddynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps.Three constrained variational approaches are formulated for the derivation ofthese models. The first is the Clebsch formulation, in which the constraintsare imposed as rough advection laws. The second is the Hamilton-Pontryaginformulation, in which the constraints are imposed as right-invariant roughvector fields. The third is the Euler--Poincar\'e formulation in which thevariations are constrained. These variational principles lead directly to theLie--Poisson Hamiltonian formulation of fluid dynamics on geometric roughpaths. The GRP framework preserves the geometric structure of fluid dynamicsobtained by using Lie group reduction to pass from Lagrangian to Eulerianvariational principles, thereby yielding a rough formulation of the Kelvincirculation theorem. The rough-path variational approach includes non-Markovianperturbations of the Lagrangian fluid trajectories. In particular, memoryeffects can be introduced through this formulation through a judicious choiceof the rough path (e.g. a realization of a fractional Brownian motion). In thespecial case when the rough path is a realization of a semimartingale, werecover the SGFD models in Holm (2015). However, by eliminating the need forstochastic variational tools, we retain a pathwise interpretation of theLagrangian trajectories. In contrast, the Lagrangian trajectories in thestochastic framework are described by stochastic integrals which do not have apathwise interpretation. Thus, the rough path formulation restores thisproperty.
Chassagneux JF, Crisan D, Delarue F, 2022, A Probabilistic Approach to Classical Solutions of the Master Equation for Large Population Equilibria, Memoirs of the American Mathematical Society, Vol: 280, ISSN: 0065-9266
We analyze a class of nonlinear partial differential equations (PDEs) defined on Rd × P2pRdq, where P2pRdq is the Wasserstein space of probability measures on Rd with a finite second-order moment. We show that such equations admit a classical solutions for sufficiently small time intervals. Under additional constraints, we prove that their solution can be extended to arbitrary large intervals. These nonlinear PDEs arise in the recent developments in the theory of large population stochastic control. More precisely they are the so-called master equations corresponding to asymptotic equilibria for a large population of controlled players with mean-field interaction and subject to minimization constraints. The results in the paper are deduced by exploiting this connection. In particular, we study the differentiability with respect to the initial condition of the flow generated by a forward-backward stochastic system of McKean-Vlasov type. As a byproduct, we prove that the decoupling field generated by the forward-backward system is a classical solution of the corresponding master equation. Finally, we give several applications to mean-field games and to the control of McKean-Vlasov diffusion processes.
Crisan D, Lobbe A, Ortiz-Latorre S, 2022, Pathwise Approximations for the Solution of the Non-Linear Filtering Problem, Stochastic Analysis, Filtering, and Stochastic Optimization: A Commemorative Volume to Honor Mark H. A. Davis's Contributions, Pages: 79-100, ISBN: 9783030985189
We consider high order approximations of the solution of the stochastic filtering problem, derive their pathwise representation in the spirit of the earlier work of Clark [2] and Davis [10, 11] and prove their robustness property. In particular, we show that the high order discretised filtering functionals can be represented by Lipschitz continuous functions defined on the observation path space. This property is important from the practical point of view as it is in fact the pathwise version of the filtering functional that is sought in numerical applications. Moreover, the pathwise viewpointwill be a stepping stone into the rigorous development ofmachine learning methods for the filtering problem. This work is a cotinuation of [5] where a discretisation of the solution of the filtering problem of arbitrary order has been established. We expand the work in [5] by showing that robust approximations can be derived from the discretisations therein.
Ruzayqat H, Er-Rai A, Beskos A, et al., 2022, A Lagged Particle Filter for Stable Filtering of Certain High-Dimensional State-Space Models, SIAM-ASA Journal on Uncertainty Quantification, Vol: 10, Pages: 1130-1161
We consider the problem of high-dimensional filtering of state-space models (SSMs) at discrete times. This problem is particularly challenging as analytical solutions are typically not available and many numerical approximation methods can have a cost that scales exponentially with the dimension of the hidden state. Inspired by lag-approximation methods for the smoothing problem [G. Kitagawa and S. Sato, Monte Carlo smoothing and self-organising state-space model, in Sequential Monte Carlo Methods in Practice, Springer, New York, 2001, pp. 178{195; J. Olsson et al., Bernoulli, 14 (2008), pp. 155{179], we introduce a lagged approximation of the smoothing distribution that is necessarily biased. For certain classes of SSMs, particularly those that forget the initial condition exponentially fast in time, the bias of our approximation is shown to be uniformly controlled in the dimension and exponentially small in time. We develop a sequential Monte Carlo (SMC) method to recursively estimate expectations with respect to our biased filtering distributions. Moreover, we prove for a class of SSMs that can contain dependencies amongst coordinates that as the dimension d ! 1 the cost to achieve a stable mean square error in estimation, for classes of expectations, is of O(Nd2) per unit time, where N is the number of simulated samples in the SMC algorithm. Our methodology is implemented on several challenging high-dimensional examples including the conservative shallow-water model.
Crisan D, Holm DD, Street OD, 2021, Wave-current interaction on a free surface, STUDIES IN APPLIED MATHEMATICS, Vol: 147, Pages: 1277-1338, ISSN: 0022-2526
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- Citations: 1
Beskos A, Crisan D, Jasra A, et al., 2021, Score-based parameter estimation for a class of continuous-time state space models, SIAM Journal on Scientific Computing, ISSN: 1064-8275
We consider the problem of parameter estimation for a class ofcontinuous-time state space models. In particular, we explore the case of apartially observed diffusion, with data also arriving according to a diffusionprocess. Based upon a standard identity of the score function, we consider twoparticle filter based methodologies to estimate the score function. Bothmethods rely on an online estimation algorithm for the score function of$\mathcal{O}(N^2)$ cost, with $N\in\mathbb{N}$ the number of particles. Thefirst approach employs a simple Euler discretization and standard particlesmoothers and is of cost $\mathcal{O}(N^2 + N\Delta_l^{-1})$ per unit time,where $\Delta_l=2^{-l}$, $l\in\mathbb{N}_0$, is the time-discretization step.The second approach is new and based upon a novel diffusion bridgeconstruction. It yields a new backward type Feynman-Kac formula incontinuous-time for the score function and is presented along with a particlemethod for its approximation. Considering a time-discretization, the cost is$\mathcal{O}(N^2\Delta_l^{-1})$ per unit time. To improve computational costs,we then consider multilevel methodologies for the score function. We illustrateour parameter estimation method via stochastic gradient approaches in severalnumerical examples.
Cass T, Crisan D, Dobson P, et al., 2021, Long-time behaviour of degenerate diffusions: UFG-type SDEs and time-inhomogeneous hypoelliptic processes, Electronic Journal of Probability, Vol: 26, Pages: 1-72, ISSN: 1083-6489
We study the long time behaviour of a large class of diffusion processes on RN, generated by second order differential operators of (possibly) degenerate type. The operators that we consider need not satisfy the Hörmander Condition (HC). Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock. We demonstrate the importance of the class of UFG processes in several respects: i) we show that UFG processes constitute a family of SDEs which exhibit, in general, multiple invariant measures (i.e. they are in general non-ergodic) and for which one is able to describe a systematic procedure to study the basin of attraction of each invariant measure (equilibrium state). ii) We use an explicit change of coordinates to prove that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently “less smooth” than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. iv) We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produce several results on this latter class of processes as well. v) Because processes that satisfy the (uniform) parabolic HC are UFG processes, this paper contains a wealth of results about the long time behaviour of (uniformly) hypoelliptic processes which are non-ergodic.
Street OD, Crisan D, 2021, Semi-martingale driven variational principles, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol: 477, ISSN: 1364-5021
Spearheaded by the recent efforts to derive stochastic geophysical fluid dynamics models, we present a general framework for introducing stochasticity into variational principles through the concept of a semi-martingale driven variational principle and constraining the component variables to be compatible with the driving semi-martingale. Within this framework and the corresponding choice of constraints, the Euler-Poincaré equation can be easily deduced. We show that the deterministic theory is a special case of this class of stochastic variational principles. Moreover, this is a natural framework that enables us to correctly characterize the pressure term in incompressible stochastic fluid models. Other general constraints can also be incorporated as long as they are compatible with the driving semi-martingale.
Crisan D, Dobson P, Ottobre M, 2021, Uniform in time estimates for the weak error of the Euler method for SDEs and a pathwise approach to derivative estimates for diffusion semigroups, Transactions of the American Mathematical Society, Vol: 374, Pages: 3289-3330, ISSN: 0002-9947
We present a criterion for uniform in time convergence of the weak error of the Euler scheme for Stochastic Differential equations (SDEs). The criterion requires (i) exponential decay in time of the space-derivatives of the semigroup associated with the SDE and (ii) bounds on (some) moments of the Euler approximation. We show by means of examples (and counterexamples) how both (i) and (ii) are needed to obtain the desired result. If the weak error converges to zero uniformly in time, then convergence of ergodic averages follows as well. We also show that Lyapunov-type conditions are neither sufficient nor necessary in order for the weak error of the Euler approximation to converge uniformly in time and clarify relations between the validity of Lyapunov conditions, (i) and (ii).Conditions for (ii) to hold are studied in the literature. Here we produce sufficient conditions for (i) to hold. The study of derivative estimates has attracted a lot of attention, however not many results are known in order to guarantee exponentially fast decay of the derivatives. Exponential decay of derivatives typically follows from coercive-type conditions involving the vector fields appearing in the equation and their commutators; here we focus on the case in which such coercive-type conditions are non-uniform in space. To the best of our knowledge, this situation is unexplored in the literature, at least on a systematic level. To obtain results under such space-inhomogeneous conditions we initiate a pathwise approach to the study of derivative estimates for diffusion semigroups and combine this pathwise method with the use of Large Deviation Principles.
Crisan D, Lang O, 2021, Local well-posedness for the great lake equation with transport noise, Revue Roumaine de Mathematiques Pures et Appliquees, Vol: 66, Pages: 131-155, ISSN: 0035-3965
This work is a continuation of the authors' work in [11]. In the equation satisfied by an incompressible uid with stochastic transport is analysed. Here we lift the incompressibility constraint. Instead we assume a weighted incompressibility condition. This condition is inspired by a physical model for a uid in a basin with a free upper surface and a spatially varying bottom topography (see [23]). Moreover, we assume a different form of the vorticity to stream function operator that generalizes the standard Biot-Savart operator which appears in the Euler equation. These two properties are exhibited in the physical model called the great lake equation. For this reason we refer to the model analysed here as the stochastic great lake equation. Just as in [11], the deterministic model is perturbed with transport type noise. The new vorticity to stream function operator generalizes the curl operator and it is shown to have good regularity properties. We also show that the initial smoothness of the solution is preserved. The arguments are based on constructing a family of viscous solutions which is proved to be relatively compact and to converge to a truncated version of the original equation. Finally, we show that the truncation can be removed up to a positive stopping time.
Akyildiz ÖD, Crisan D, Míguez J, 2020, Parallel sequential Monte Carlo for stochastic gradient-free nonconvex optimization, Statistics and Computing, Vol: 30, Pages: 1645-1663, ISSN: 0960-3174
We introduce and analyze a parallel sequential Monte Carlo methodology for the numerical solution of optimization problems that involve the minimization of a cost function that consists of the sum of many individual components. The proposed scheme is a stochastic zeroth-order optimization algorithm which demands only the capability to evaluate small subsets of components of the cost function. It can be depicted as a bank of samplers that generate particle approximations of several sequences of probability measures. These measures are constructed in such a way that they have associated probability density functions whose global maxima coincide with the global minima of the original cost function. The algorithm selects the best performing sampler and uses it to approximate a global minimum of the cost function. We prove analytically that the resulting estimator converges to a global minimum of the cost function almost surely and provide explicit convergence rates in terms of the number of generated Monte Carlo samples and the dimension of the search space. We show, by way of numerical examples, that the algorithm can tackle cost functions with multiple minima or with broad “flat” regions which are hard to minimize using gradient-based techniques.
Cotter C, Crisan D, Holm DD, et al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv
The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.
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