Finite-dimensional Approximation Research Articles

Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386–1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155–177] on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms.

Multiscale Galerkin approximation scheme for a system of quasilinear parabolic equations

We discuss a multiscale Galerkin approximation scheme for a system of coupled quasilinear parabolic equations. These equations arise from the upscaling of a pore scale filtration combustion model under the assumptions of large Damkhöler number and small Péclet number. The upscaled model consists of a heat diffusion equation and a mass diffusion equation in the bulk of a macroscopic domain. The associated diffusion tensors are bivariate functions of temperature and concentration and provide the necessary coupling conditions to elliptic-type cell problems. These cell problems are characterized by a reaction–diffusion phenomenon with nonlinear reactions of Arrhenius type at a gas–solid interface. We discuss the wellposedness of the quasilinear system and establish uniform estimates for the finite dimensional approximations. Based on these estimates, the convergence of the approximating sequence is proved. The results of numerical simulations demonstrate, in suitable temperature regimes, the potential of solutions of the upscaled model to mimic those from porous media combustion. Moreover, distinctions are made between the effects of the microscopic reaction–diffusion processes on the macroscopic system of equations and a purely diffusion system.

Singular perturbation approximation for linear systems with Lévy noise

To solve a stochastic linear evolution equation numerically, finite dimensional approximations are commonly used. For a good approximation, one might end up with a sequence of ordinary stochastic linear equations of high order. To reduce the high dimension for practical computations, we consider the singular perturbation approximation as a model order reduction technique in this paper. This approach is well-known from deterministic control theory and here we generalize it for controlled linear systems with Lévy noise. Additionally, we discuss properties of the reduced order model, provide an error bound, and give some examples to demonstrate the quality of this model order reduction technique.

Spline approximation for systems of linear neutral delay-differential equations

We derive a new finite dimensional semidiscrete approximation scheme for systems of linear neutral delay-differential equations and prove convergence results. Our construction extends to neutral delay equations results which were previously only available for retarded delay equations.

Convex hulls of multidimensional random walks

Let $S_k$ be a random walk in $\mathbb R^d$ such that its distribution of increments does not assign mass to hyperplanes. We study the probability $p_n$ that the convex hull $\operatorname {conv} (S_1, \dots , S_n)$ of the first $n$ steps of the walk does not include the origin. By providing an explicit formula, we show that for planar symmetrically distributed random walks, $p_n$ does not depend on the distribution of increments. This extends the well-known result by Sparre Andersen (1949) that a one-dimensional random walk satisfying the above continuity and symmetry assumptions stays positive with a distribution-free probability. We also find the asymptotics of $p_n$ as $n \to \infty$ for any planar random walk with zero mean square-integrable increments. We further developed our approach from the planar case to study a wide class of geometric characteristics of convex hulls of random walks in any dimension $d \ge 2$. In particular, we give formulas for the expected value of the number of faces, the volume, the surface area, and other intrinsic volumes, including the following multidimensional generalization of the Spitzer–Widom formula (1961) on the perimeter of planar walks: \begin{equation*} \mathbb E V_1 (\operatorname {conv} (0, S_1, \dots , S_n)) = \sum _{k=1}^n \frac {\mathbb E \|S_k\|}{k}, \end{equation*} where $V_1$ denotes the first intrinsic volume, which is proportional to the mean width. These results have applications to geometry and, in particular, imply the formula by Gao and Vitale (2001) for the intrinsic volumes of special path-simplexes, called canonical orthoschemes, which are finite-dimensional approximations of the closed convex hull of a Wiener spiral. Moreover, there is a direct connection between spherical intrinsic volumes of these simplexes and the probabilities $p_n$. We also prove similar results for convex hulls of random walk bridges and, more generally, for partial sums of exchangeable random vectors.

Finite-Dimensional Approximations of the Steklov–Poincare Operator in Periodic Elastic Waveguides

Finite-Dimensional Approximations of the Steklov–Poincare Operator in Periodic Elastic Waveguides

Marginal process framework: A model reduction tool for Markov jump processes.

Markov jump process models have many applications across science. Often these models are defined on a state space of product form and only one of the components of the process is of direct interest. In this paper we extend the marginal process framework, which provides a marginal description of the component of interest, to the case of fully coupled processes. We use entropic matching to obtain a finite-dimensional approximation of the filtering equation, which governs the transition rates of the marginal process. The resulting equations can be seen as a combination of two projection operations applied to the full master equation so that we obtain a principled model reduction framework. We demonstrate the resulting reduced description on the totally asymmetric exclusion process. An important class of Markov jump processes are stochastic reaction networks, which have applications in chemical and biomolecular kinetics, ecological models, and models of social networks. We obtain a particularly simple instantiation of the marginal process framework for mass-action systems by using product Poisson distributions for the approximate solution of the filtering equation. We investigate the resulting approximate marginal process analytically and numerically.

On the path-independence of the Girsanov transformation for stochastic evolution equations with jumps in Hilbert spaces

Based on a recent result in [ 13 ], in this paper, we extend it to stochastic evolution equations with jumps in Hilbert spaces. This is done via Galerkin type finite-dimensional approximations of the infinite-dimensional stochastic evolution equations with jumps in a manner that one could then link the characterisation of the path-independence for finite-dimensional jump type SDEs to that for the infinite-dimensional settings. Our result provides an intrinsic link of infinite-dimensional stochastic evolution equations with jumps to infinite-dimensional (nonlinear) integro-differential equations.

Distributed Multi-Agent Gaussian Regression via Finite-Dimensional Approximations.

We consider the problem of distributedly estimating Gaussian processes in multi-agent frameworks. Each agent collects few measurements and aims to collaboratively reconstruct a common estimate based on all data. Agents are assumed with limited computational and communication capabilities and to gather $M$M noisy measurements in total on input locations independently drawn from a known common probability density. The optimal solution would require agents to exchange all the $M$M input locations and measurements and then invert an $M \times M$M×M matrix, a non-scalable task. Differently, we propose two suboptimal approaches using the first $E$E orthonormal eigenfunctions obtained from the Karhunen-Loève (KL) expansion of the chosen kernel, where typically $E\ll M$E≪M. The benefits are that the computation and communication complexities scale with $E$E and not with $M$M, and computing the required statistics can be performed via standard average consensus algorithms. We obtain probabilistic non-asymptotic bounds that determine a priori the desired level of estimation accuracy, and new distributed strategies relying on Stein's unbiased risk estimate (SURE) paradigms for tuning the regularization parameters and applicable to generic basis functions (thus not necessarily kernel eigenfunctions) and that can again be implemented via average consensus. The proposed estimators and bounds are finally tested on both synthetic and real field data.

Transition Probability of Brownian Motion in the Octant and its Application to Default Modelling

ABSTRACTWe derive a semi-analytical formula for the transition probability of three-dimensional Brownian motion in the positive octant with absorption at the boundaries. Separation of variables in spherical coordinates leads to an eigenvalue problem for the resulting boundary value problem in the two angular components. The main theoretical result is a solution to the original problem expressed as an expansion into special functions and an eigenvalue which has to be chosen to allow a matching of the boundary condition. We discuss and test several computational methods to solve a finite-dimensional approximation to this nonlinear eigenvalue problem. Finally, we apply our results to the computation of default probabilities and credit valuation adjustments in a structural credit model with mutual liabilities.

Finite-Dimensional Approximations of the Steklov–Poincaré Operator for the Helmholtz Equation in Periodic Waveguides

We consider the Dirichlet and Neumann problems for the Laplace operator in periodic waveguides. Integro-differential connections between the solution and its normal derivative, interpreted as a finite-dimensional version of the Steklov–Poincare operator, are imposed on the artificial face of the truncated waveguide. These connections are obtained from the orthogonality and normalization conditions for the Floquet waves which are oscillating incoming/outgoing, as well as exponentially decaying/growing in the periodic waveguide. Under certain conditions, we establish the unique solvability of the problem and obtain error estimates for the solution itself, as well as for scattering coefficients in the solution. We give examples of trapped waves in periodic waveguides.

Weak Universality Theorem on the Approximation of Analytic Functions by Shifts of the Riemann Zeta-Function from a Beatty Sequence

AbstractIn this paper, we prove a discrete analogue of Voronin’s early finite-dimensional approximation result with respect to terms from a given Beatty sequence and make use of Taylor approximation in order to derive a weak universality statement.

Systems of Hydrodynamic Type that Approximate Two-Dimensional Ideal Fluid Equations

Statistical properties of different finite-dimensional approximations of two-dimensional ideal fluid equations are studied. A special class of approximations introduced by A.M. Obukhov (systems of hydrodynamic type) is considered. Vorticity distributions over area and quasi-equilibrium coherent structures are studied. These coherent structures are compared to structures occurring in a viscous fluid with random forcing.

Spectral stability for the wave equation with periodic forcing

We consider the spectral stability problem for Floquet-type systems such as the wave equation vττ=γ2vxx−ψv with periodic forcing ψ. Our approach is based on a comparison with finite-dimensional approximations. Specific results are obtained for a system where the forcing is due to a coupling between the wave equation and a time-period solution of a nonlinear beam equation. We prove (spectral) stability for some period and instability for another. The finite-dimensional approximations are controlled via computer-assisted estimates.

Level set Cox processes

An extension of the popular log-Gaussian Cox process (LGCP) model for spatial point patterns is proposed for data exhibiting fundamentally different behaviors in different subregions of the spatial domain. The aim of the analyst might be either to identify and classify these regions, to perform kriging, or to derive some properties of the parameters driving the random field in one or several of the subregions. The extension is based on replacing the latent Gaussian random field in the LGCP by a latent spatial mixture model specified using a categorically valued random field. This classification is defined through level set operations on a Gaussian random field and allows for standard stationary covariance structures, such as the Matérn family, to be used to model random fields with some degree of general smoothness but also occasional and structured sharp discontinuities.A computationally efficient MCMC method is proposed for Bayesian inference and we show consistency of finite dimensional approximations of the model. Finally, the model is fitted to point pattern data derived from a tropical rainforest on Barro Colorado island, Panama. We show that the proposed model is able to capture behavior for which inference based on the standard LGCP is biased.

Linear predictors for nonlinear dynamical systems: Koopman operator meets model predictive control

This paper presents a class of linear predictors for nonlinear controlled dynamical systems. The basic idea is to lift (or embed) the nonlinear dynamics into a higher dimensional space where its evolution is approximately linear. In an uncontrolled setting, this procedure amounts to numerical approximations of the Koopman operator associated to the nonlinear dynamics. In this work, we extend the Koopman operator to controlled dynamical systems and apply the Extended Dynamic Mode Decomposition (EDMD) to compute a finite-dimensional approximation of the operator in such a way that this approximation has the form of a linear controlled dynamical system. In numerical examples, the linear predictors obtained in this way exhibit a performance superior to existing linear predictors such as those based on local linearization or the so called Carleman linearization. Importantly, the procedure to construct these linear predictors is completely data-driven and extremely simple – it boils down to a nonlinear transformation of the data (the lifting) and a linear least squares problem in the lifted space that can be readily solved for large data sets. These linear predictors can be readily used to design controllers for the nonlinear dynamical system using linear controller design methodologies. We focus in particular on model predictive control (MPC) and show that MPC controllers designed in this way enjoy computational complexity of the underlying optimization problem comparable to that of MPC for a linear dynamical system with the same number of control inputs and the same dimension of the state-space. Importantly, linear inequality constraints on the state and control inputs as well as nonlinear constraints on the state can be imposed in a linear fashion in the proposed MPC scheme. Similarly, cost functions nonlinear in the state variable can be handled in a linear fashion. We treat both the full-state measurement case and the input–output case, as well as systems with disturbances/noise. Numerical examples demonstrate the approach.1

On two-coefficient identification in elliptic variational inequalities

This paper is concerned with elliptic variational inequalities that depend on two parameters. First, we investigate the dependence of the solution of the forward problem on these parameters and prove a Lipschitz estimate. Then, we study the inverse problem of identification of these two parameters and formulate two optimization approaches to this parameter identification problem. We extend the output least-squares approach, provide an existence result and establish a convergence result for finite-dimensional approximation. Further, we investigate the modified output least-squares approach which is based on energy functionals. This latter approach can be related to vector approximation.

Optimization methods for Dirichlet control problems

We discuss several optimization procedures to solve finite element approximations of linear-quadratic Dirichlet optimal control problems governed by an elliptic partial differential equation posed on a 2D or 3D Lipschitz domain. The control is discretized explicitly using continuous piecewise linear approximations. Unconstrained, control-constrained, state-constrained and control-and-state constrained problems are analysed. A preconditioned conjugate method for a reduced problem in the control variable is proposed to solve the unconstrained problem, whereas semismooth Newton methods are discussed for the solution of constrained problems. State constraints are treated via a Moreau–Yosida penalization. Convergence is studied for both the continuous problems and the finite dimensional approximations. In the finite dimensional case, we are able to show convergence of the optimization procedures even in the absence of Tikhonov regularization parameter. Computational aspects are also treated and several numerical examples are included to illustrate the theoretical results.

$$L^{2}$$L2-discretization error bounds for maps into Riemannian manifolds

We study the approximation of functions that map a Euclidean domain \(\Omega \subset {\mathbb {R}}^{d}\) into an n-dimensional Riemannian manifold (M, g) minimizing an elliptic, semilinear energy in a function set \(H\subset W^{1,2}(\Omega ,M)\). The approximation is given by a restriction of the energy minimization problem to a family of conforming finite-dimensional approximations \(S_{h}\subset H\). We provide a set of conditions on \(S_{h}\) such that we can prove a priori \(W^{1,2}\)- and \(L^{2}\)-approximation error estimates comparable to standard Euclidean finite elements. This is done in an intrinsic framework, independently of embeddings of the manifold or the choice of coordinates.

Optimal control problem for a viscoelastic beam and its galerkin approximation

This paper is concerned with the optimal control problem of the vibrations of a viscoelastic beam, which is governed by a nonlinear partial differential equation. We discuss the initial-boundary problem for the cases when the ends of the beam are clamped or hinged. We define the weak solution of this initial-boundary problem. Our control problem is formulated by minimization of a functional where the state of a system is the solution of viscoelastic beam equation. We use the Galerkin method to approximate the solution of our control problem with respect to a spatial variable. Based on the finite dimensional approximation we prove that as the discretization parameters tend to zero then the weak accumulation points of the optimal solutions of the discrete family control problems exist and each of these points is the solution of the original optimal control problem.