Finite-dimensional Approximation Research Articles

Solution to the inverse Wulff problem by means of the enhanced semidefinite relaxation method

Abstract We propose a novel method of resolving the optimal anisotropy function. The idea is to construct the optimal anisotropy function as a solution to the inverse Wulff problem, i.e. as a minimizer for the anisoperimetric ratio for a given Jordan curve in the plane. It leads to a nonconvex quadratic optimization problem with linear matrix inequalities. In order to solve it we propose the so-called enhanced semidefinite relaxation method which is based on a solution to a convex semidefinite problem obtained by a semidefinite relaxation of the original problem augmented by quadratic-linear constraints. We show that the sequence of finite-dimensional approximations of the optimal anisoperimetric ratio converges to the optimal anisoperimetric ratio which is a solution to the inverse Wulff problem. Several computational examples, including those corresponding to boundaries of real snowflakes, and discussion on the rate of convergence of numerical method are also presented in this paper.

Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations

Based on a recent result on linking stochastic differential equations on ℝd to (finite-dimensional) Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensional stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.

Blow-up in systems with nonlinear viscosity

Sufficient conditions for the blow-up of solutions of the hydrodynamic systems proposed by Ladyzhenskaya in 1966 with nonlinear viscosity and exterior sources are obtained. Questions relating to local solvability and uniqueness are answered using the finite-dimensional Galerkin approximation method The energy method, which was first applied to hydrodynamic systems by Korpusov and Sveshnikov, is used to obtain estimates of the blow-up time and blow-up rate. The determining role of nonlinear exterior sources, not viscous or hydrodynamic nonlinearity, on the occurrence of the blow-up effect is shown.

Active Noise Control in a Duct With Flow

Active noise control in a one dimensional acoustic duct, in which fluid medium inside the duct has a mean flow velocity, is studied. The acoustic duct model with general boundary conditions is solved in Laplace domain and infinite dimensional system transfer functions are obtained. For controller designs, appropriate microphone, and noise canceling source locations are determined. Low order finite dimensional transfer function approximations of actual system transfer functions are obtained. It is found that, in a selected frequency range, approximations represent actual system in a satisfactory way. By using approximated system transfer functions, finite dimensional, low order, optimal H2 and H∞ controllers are synthesized via linear matrix inequalities method. Closed loop frequency response and time domain simulations show that the controllers successfully suppress unwanted sound, which propagates along the duct.

Blind deconvolution for distributed parameter systems with unbounded input and output and determining blood alcohol concentration from transdermal biosensor data

We develop a blind deconvolution scheme for input–output systems described by distributed parameter systems with boundary input and output. An abstract functional analytic theory based on results for the linear quadratic control of infinite dimensional systems with unbounded input and output operators is presented. The blind deconvolution problem is then reformulated as a series of constrained linear and nonlinear optimization problems involving infinite dimensional dynamical systems. A finite dimensional approximation and convergence theory is developed. The theory is applied to the problem of estimating blood or breath alcohol concentration, (respectively, BAC or BrAC) from biosensor-measured transdermal alcohol concentration (TAC) in the field. A distributed parameter model with boundary input and output is proposed for the transdermal transport of ethanol from the blood through the skin to the sensor. The problem of estimating BAC or BrAC from the TAC data is formulated as a blind deconvolution problem. A scheme to identify distinct drinking episodes in TAC data based on a Hodrick Prescott filter is discussed. Numerical results involving actual patient data are presented.

An Analysis of Infinite Dimensional Bayesian Inverse Shape Acoustic Scattering and Its Numerical Approximation

We present and analyze an infinite dimensional Bayesian inference formulation, and its numerical approximation, for the inverse problem of inferring the shape of an obstacle from scattered acoustic waves. Given a Gaussian prior measure on the shape space, whose covariance operator is the inverse of the Laplacian, the Bayesian solution of the inverse problem is the posterior measure given by the Radon--Nikodym derivative with respect to the Gaussian prior measure. The well-posedness of the Bayesian formulation in infinite dimensions is proved, including the justification of the Radon--Nikodym derivative and the continuous dependence of the posterior measure on the observation data via the Hellinger distance. The proof is made possible by a suitable shape parametrization in a Banach space setting and the regularity of the forward solution with respect to the smoothness of the shape. This also facilitates proving the Lipschitz continuity of the observation operator with respect to the scatterer shape via the shape derivative of the forward solution. Next, a finite dimensional approximation to the Bayesian posterior is proposed and the corresponding approximation error is quantified. The approximation strategy involves a Nyström scheme for approximating a boundary integral formulation of the forward Helmholtz problem and a Karhunen--Loève truncation for approximating the prior measure. Weak convergence of the resulting finite dimensional approximation, as well as convergence in the Hellinger distance, are investigated. In particular, we determine the convergence rate as a function of the number of Nyström quadrature points and the number of truncated terms in the Karhunen--Loève series. Finally, we estimate the error between the exact posterior moments, e.g., posterior mean and variance, and their finite dimensional approximate counterparts in terms of the errors due to forward equation approximation and prior approximation. The main result of this work is that the convergence rate for approximating the Bayesian inverse problem is spectral, and this directly inherits the spectral convergence rates of the approximations of both the prior and the forward problems.

Asymptotic Behavior of the Coupled Nonlinear Schrödinger Lattice System

This paper studies asymptotic behavior of solutions for the coupled nonlinear Schrödinger lattice system. We obtain the existence and stability of compact attractor by means of tail estimates method and finite-dimensional approximations.

Myopic models of population dynamics on infinite networks

Reaction-diffusion equations are treated on infinite networks using semigroup methods.To blend high fidelity local analysis with coarse remote modeling,initial data and solutions come from a uniformly closed algebra generated by functionswhich are flat at infinity. The algebra is associated with a compactification of the networkwhich facilitates the description of spatial asymptotics. Diffusive effects disappear at infinity,greatly simplifying the remote dynamics. Accelerated diffusion models with conventionaleigenfunction expansions are constructed to provide opportunities for finite dimensional approximation.

Sampling and Reconstruction of Solutions to the Helmholtz Equation

We consider the problem of reconstructing general solutions to the Helmholtz equation $\Delta u+\lambda^2 u=0$, for some fixed $\lambda>0$, on some domain $\Omega\subset \R^2$ from the data of these functions at scattered points $x_1,\dots,x_n\subset \Omega$. This problem typically arises when sampling acoustic fields with $n$ microphones for the purpose of reconstructing this field over a region of interest $\Omega$ that is contained in a larger domain $D$ (i.e., a room).% in which the acoustic field is defined. In many applied settings, the boundary conditions satisfied by the acoustic field on $\partial D$ are unknown as well as the exact shape of $D$. Our reconstruction method is based on the approximation of a general solution $u$ by linear combinations of Fourier-Bessel functions or plane waves $e_{\bk}(x):=e^{i \bk \cdot x}$ with $|\bk|=\lambda$. We study two different ways of discretizing the infinite dimensional space $V_\lambda$ of solutions to the Helmholtz equation, leading to two different types of finite dimensional approximation subspaces, and we analyze the convergence of the least squares estimates to $u$ in these subspaces based on the samples $(u(x_i))_{i=1,\dots,n}$. Our analysis describes the amount of regularization that is needed to guarantee the convergence of the least squares estimate towards $u$, in terms of a condition that depends on the dimension of the approximation subspace and the sample size $n$. This condition also involves the distribution of the samples and reveals the advantage of using non-uniform distributions that have more points near or on the boundary of $\Omega$. Numerical illustrations show that our approach compares favorably with reconstruction methods using other basis functions, and other types of regularization.

IDA-PBC control for the coupled plasma poloidal magnetic flux and heat radial diffusion equations in tokamaks

The IDA-PBC control of plasma dynamics in a tokamak is investigated. It is based on a model made of the two coupled PDEs of resistive diffusion for the magnetic poloidal flux and of radial thermal diffusion. The used Thermal-Magneto-Hydro-Dynamics (TMHD) couplings are the Lorentz forces (with non-uniform resistivity) and the bootstrap current. The control model is obtained with the coupling of the two finite dimensional approximations obtained from the two diffusion models, using two geometric reduction schemes. A feedforward control is used to ensure the compatibility with the actuator physical ability. Then, an IDA-PBC (Interconnection and Damping Assignment - Passivity Based Control) controller is proposed for the coupled model to improve the system stabilization and convergence speed. The obtained numerical results are validated against the simulation data obtained from the RAPTOR (RApid Plasma Transport simulatOR) code for the TCV (Tokamak of Configuration Variable at CRPP, EPFL, Lausanne, Switzerland) tokamak real-time control system.

Оценка точности по состоянию конечномерных аппроксимаций задач оптимизации для полулинейных эллиптических уравнений с разрывными коэффициентами и решениями

Оценка точности по состоянию конечномерных аппроксимаций задач оптимизации для полулинейных эллиптических уравнений с разрывными коэффициентами и решениями

Large deviations for nonlocal stochastic neural fields.

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a Q-Wiener process, and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers’ law for neural fields poses substantial difficulties, but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multiscale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Mathematics Subject Classification (2000): 60F10, 60H15, 65M60, 92C20.

Handling Convexity-Like Constraints in Variational Problems

We provide a general framework to construct finite-dimensional approximations of the space of convex functions, which also applies to the space of $c$-convex functions and to the space of support functions of convex bodies. We give precise estimates of the distance between the approximation space and the admissible set. This framework applies to the approximation of convex functions by piecewise-linear functions on a mesh of the domain and by other finite-dimensional spaces such as tensor-product splines. We show how these discretizations are well suited for the numerical solution of problems of calculus of variations under convexity constraints. Our implementation relies on proximal algorithms and can be easily parallelized, thus making it applicable to large-scale problems in dimension two and three. We illustrate the versatility and the efficiency of our approach on the numerical solution of three problems in calculus of variation: three-dimensional denoising, the principal agent problem, and optimizat...

Smooth finite-dimensional approximations of distributed optimization problems via control discretization

Approximating finite-dimensional mathematical programming problems are studied that arise from piecewise constant discretization of controls in the optimization of distributed systems of a fairly broad class. The smoothness of the approximating problems is established. Gradient formulas are derived that make use of the analytical solution of the original control system and its adjoint, thus providing an opportunity for algorithmic separation of numerical optimization and the task of solving a controlled initial-boundary value problem. The approximating problems are proved to converge to the original optimization problem with respect to the functional as the discretization is refined. The application of the approach to optimization problems is illustrated by solving the semilinear wave equation controlled by applying an integral criterion. The results of numerical experiments are analyzed.

On the attainable region for process networks

In this work the attainable region (AR) concept for process networks with outlet flow rate specifications is introduced for the first time. For process unit models to which the infinite dimensional State‐space conceptual framework is applicable, it is shown that identification of AR boundary membership is equivalent to feasibility assessment of an infinite linear program (ILP). A number of important AR properties are then theoretically established, including AR convexity, and representation of the AR in a concentration state space of reduced dimension. Finite dimensional approximations of the aforementioned ILP are then employed in creating increasingly accurate approximations of the AR. A case study for the vapor‐liquid equilibrium‐based separation of a ternary azeotropic mixture is used to illustrate the proposed method. The quantified two‐ and four‐Dimensional ARs indicate that acetone mole fractions above 0.79 (acetone/methanol azeotrope) are attainable for the considered outlet flow rate ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 193–212, 2014

Structure-preserving approximation of distributed-parameter second-order systems using Krylov subspaces

In this article, the approximation of linear second-order distributed-parameter systems (DPS) is considered using a Galerkin approach. The resulting finite-dimensional approximation model also has a second-order structure and preserves the stability as well as the passivity. Furthermore, by extending the Krylov subspace methods for finite-dimensional systems of second order to DPS, the basis vectors of the Galerkin projection are determined such that the transfer behaviour of the DPS can be approximated by using moment matching. The structure-preserving approximation of an Euler–Bernoulli beam with Kelvin–Voigt damping demonstrates the results of the article.

Variational discretization for rotating stratified fluids

In this paper we develop and test a structure-preserving discretization scheme for rotating and/or stratified fluid dynamics. The numerical scheme is based on a finite dimensional approximation of the group of volume preserving diffeomorphisms recently proposed in [25, 9] and is derived via a discrete version of the Euler-Poincare variational formulation of rotating stratified fluids. The resulting variational integrator allows for a discrete version of Kelvin circulation theorem, is applicable to irregular meshes and, being symplectic, exhibits excellent long term energy behavior. We then report a series of preliminary tests for rotating stratified flows in configurations that are symmetric with respect to translation along one of the spatial directions. In the benchmark processes of hydrostatic and/or geostrophic adjustments, these tests show that the slow and fast component of the flow are correctly reproduced. The harder test of inertial instability is in full agreement with the common knowledge of the process of development and saturation of this instability, while preserving energy nearly perfectly and respecting conservation laws.

Complexity analysis of accelerated MCMC methods for Bayesian inversion

The Bayesian approach to inverse problems, in which the posterior probability distribution on an unknown field is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computationally feasible for partial differential equation (PDE) inverse problems. Balancing the sources of error arising from finite-dimensional approximation of the unknown field, the PDE forward solution map and the sampling of the probability space under the posterior distribution are essential for the design of efficient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown diffusion coefficient. We provide complexity analyses of several Markov chain Monte Carlo (MCMC) methods for the efficient numerical evaluation of expectations under the Bayesian posterior distribution, given data δ. Particular attention is given to bounds on the overall work required to achieve a prescribed error level ε. Specifically, we first bound the computational complexity of ‘plain’ MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDE. Our (new) work versus accuracy bounds show that the complexity of this approach can be quite prohibitive. Two strategies for reducing the computational complexity are then proposed and analyzed: first, a sparse, parametric and deterministic generalized polynomial chaos (gpc) ‘surrogate’ representation of the forward response map of the PDE over the entire parameter space, and, second, a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the forward PDE. For both of these strategies, we derive asymptotic bounds on work versus accuracy, and hence asymptotic bounds on the computational complexity of the algorithms. In particular, we provide sufficient conditions on the regularity of the unknown coefficients of the PDE and on the approximation methods used, in order for the accelerations of MCMC resulting from these strategies to lead to complexity reductions over ‘plain’ MCMC algorithms for the Bayesian inversion of PDEs.

Numerical analysis for the approximation of optimal control problems with pointwise observations

In this paper, we study the numerical methods for optimal control problems governed by elliptic PDEs with pointwise observations of the state. The first order optimality conditions as well as regularities of the solutions are derived. The optimal control and adjoint state have low regularities due to the pointwise observations. For the finite dimensional approximation, we use the standard conforming piecewise linear finite elements to approximate the state and adjoint state variables, whereas variational discretization is applied to the discretization of the control. A priori and a posteriori error estimates for the optimal control, the state and adjoint state are obtained. Numerical experiments are also provided to confirm our theoretical results. Copyright © 2013 John Wiley & Sons, Ltd.

An approximation to Wiener measure and quantization of the Hamiltonian on manifolds with non-positive sectional curvature

This paper gives a rigorous interpretation of a Feynman path integral on a Riemannian manifold M with non-positive sectional curvature. An L2 Riemannian metric GP is given on the space of piecewise geodesic paths HP(M) adapted to the partition P of [0,1], whence a finite-dimensional approximation of Wiener measure is developed. It is proved that, as mesh(P)→0, the approximate Wiener measure converges in an L1 sense to the measure exp{−2+3203∫01Scal(σ(s))ds}dν(σ) on the Wiener space W(M) with Wiener measure ν. This gives a possible prescription for the path integral representation of the quantized Hamiltonian, as well as yielding such a result for the natural geometric approximation schemes originating in Andersson and Driver (1999) [3] and followed by Lim (2007) [34].