Finite-dimensional Problem Research Articles

Nonasymptotic Gaussian Approximation for Inference With Stable Noise

The results of a series of theoretical studies are reported, examining the convergence rate for different approximate representations of $\alpha $ -stable distributions. Although they play a key role in modelling random processes with jumps and discontinuities, the use of $\alpha $ -stable distributions in inference often leads to analytically intractable problems. The LePage series, which is a probabilistic representation employed in this work, is used to transform an intractable, infinite-dimensional inference problem into a finite-dimensional (conditionally Gaussian) parametric problem. A major component of our approach is the approximation of the tail of this series by a Gaussian random variable. Standard statistical techniques, such as Expectation-Maximization (EM), Markov chain Monte Carlo, and Particle Filtering, can then be readily applied. In addition to the asymptotic normality of the tail of this series, we establish explicit, nonasymptotic bounds on the approximation error. Their proofs follow classical Fourier-analytic arguments, using Esseen’s smoothing lemma. Specifically, we consider the distance between the distributions of: $(i)$ the tail of the series and an appropriate Gaussian; $(ii)$ the full series and the truncated series; and $(iii)$ the full series and the truncated series with an added Gaussian term. In all three cases, sharp bounds are established, and the theoretical results are compared with the actual distances (computed numerically) in specific examples of symmetric $\alpha $ -stable distributions. This analysis facilitates the selection of appropriate truncations in practice and offers theoretical guarantees for the accuracy of resulting estimates. One of the main conclusions obtained is that, for the purposes of inference, the use of a truncated series together with an approximately Gaussian error term has superior statistical properties and is likely a preferable choice in practice.

$p$-Safe Analysis of Stochastic Hybrid Processes

We develop a method for determining whether a stochastic system is safe, i.e., whether its trajectories reach unsafe states. Specifically, we define and solve a probabilistic safety problem for Markov processes. Based on the knowledge of the extended generator, we are able to develop an evolution equation, as a system of integral equations, describing the connection between unsafe and initial states. Subsequently, using the moment method, we approximate the infinite-dimensional optimization problem searching for the largest set of safe states by a finite-dimensional polynomial optimization problem. In particular, we address the above safety problem to a special class of stochastic hybrid processes, namely piecewise-deterministic Markov processes. These are characterized by deterministic dynamics and stochastic jumps, where both the time and the destination of the jumps are stochastic. In addition, the jumps can be both spontaneous (in the style of a Poisson process) and forced (governed by guards). In this case, the extended generator of this process and its corresponding martingale problem turn out to be defined on a rather restricted domain. To circumvent this difficulty, we bring the generalized differential formula of this process into the evolution equation and, subsequently, formulate a polynomial optimization.

Dynamic Output Feedback H∞ Control for Linear Parameter-varying Systems with Time-delay

In this paper, we address a synthesis problem of the parameter-dependent output feedback H∞ control for linear parameter-varying systems with time-delay. The scheme adopts the parameter-dependent past state information to construct the dynamic output feedback controller. In this case, on basis of the quadratic Lyapunov functional with parameter-dependence, we analyze the parameter-dependent H∞ stability conditions for the closed-loop time-delayed linear parameter-varying system in terms of linear matrix inequalities. However, this stability condition is of an infinite-dimension. To derive computationally tractable criteria for the dynamic output feedback controller, several slack variables and a convex relaxation technique are employed to have the infinite-dimensional condition of linear matrix inequalities cast into a finite dimensional convex optimization problem. By solving the convex optimization problem, we harvest the dynamic output feedback controller with memory for the time-delayed linear parameter-varying system. Finally, two examples are included to illustrate the effectiveness of the proposed approach.

The Equations Defining Affine Grassmannians in Type A and a Conjecture of Kreiman, Lakshmibai, Magyar, and Weyman

AbstractThe affine Grassmannian of $SL_n$ admits an embedding into the Sato Grassmannian, which further admits a Plücker embedding into the projectivization of Fermion Fock space. Kreiman, Lakshmibai, Magyar, and Weyman describe the linear part of the ideal defining this embedding in terms of certain elements of the dual of Fock space called shuffles, and they conjecture that these elements together with the Plücker relations suffice to cut out the affine Grassmannian. We give a proof of this conjecture in two steps; first, we reinterpret the shuffle equations in terms of Frobenius twists of symmetric functions. Using this, we reduce to a finite-dimensional problem, which we solve. For the 2nd step, we introduce a finite-dimensional analogue of the affine Grassmannian of $SL_n$, which we conjecture to be precisely the reduced subscheme of a finite-dimensional Grassmannian consisting of subspaces invariant under a nilpotent operator.

On the relation between turnpike properties and dissipativity for continuous time linear quadratic optimal control problems

<p style='text-indent:20px;'>The paper is devoted to analyze the connection between turnpike phenomena and strict dissipativity properties for continuous-time finite dimensional linear quadratic optimal control problems. We characterize strict dissipativity properties of the dynamics in terms of the system matrices related to the linear quadratic problem. These characterizations then lead to new necessary conditions for the turnpike properties under consideration, and thus eventually to necessary and sufficient conditions in terms of spectral criteria and matrix inequalities. One of the key novelty of these results is the possibility to encompass the presence of state and input constraints.

Nonlinear optimal control: a numerical scheme based on occupation measures and interval analysis

This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518–538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.

Variational Approach to Data Graduation

The Whittaker-Henderson graduation is a data smoothing method commonly used in actuarial science for mortality table computation. This method is a finite-dimensional minimization problem, which aims to give a smooth approximation of the crude data. Our goal is to treat the Whittaker-Henderson method as a minimization problem in a Sobolev space. We take advantage of some results in the study of Sobolev spaces to analyze the solution to the graduation problem. Solving the arising minimization problem is equivalent to solving a partial differential equation (PDE). To solve the PDE numerically, we use the finite element method (FEM). We apply this variational approach to the data sets found in the literature. In particular, we test our method to check for patterns in (1) mortality rates for Filipino males, (2) global average temperature anomaly, and (3) monthly eBay share price. We also compare the results with that of the Whittaker-Henderson method.

Solvability and Bifurcation of Solutions of Nonlinear Equations with Fredholm Operator

The necessary and sufficient conditions of existence of the nonlinear operator equations’ branches of solutions in the neighbourhood of branching points are derived. The approach is based on the reduction of the nonlinear operator equations to finite-dimensional problems. Methods of nonlinear functional analysis, integral equations, spectral theory based on index of Kronecker-Poincaré, Morse-Conley index, power geometry and other methods are employed. Proposed methodology enables justification of the theorems on existence of bifurcation points and bifurcation sets in the nonstandard models. Formulated theorems are constructive. For a certain smoothness of the nonlinear operator, the asymptotic behaviour of the solutions is analysed in the neighbourhood of the branch points and uniformly converging iterative schemes with a choice of the uniformization parameter enables the comprehensive analysis of the problems details. General theorems and effectiveness of the proposed methods are illustrated on the nonlinear integral equations.

Trajectory Optimization of Autonomous Agents With Spatio-Temporal Constraints

This article addresses the problem of optimally controlling trajectories of autonomous mobile agents (e.g., robots) so as to jointly minimize travel time and energy consumption in the presence of multiple spatio-temporal constraints on these trajectories. In addition to state and input constraints, we impose spatial equality and temporal inequality constraints viewed as interior-point constraints. We address this problem by first identifying the structure of the optimal agent controllable acceleration profile and showing that it is characterized by several parameters subsequently used for trajectory design optimization. Therefore, the infinite dimensional optimal control problem is transformed into a finite dimensional parametric optimization problem. The proposed algorithm is applied to the eco-driving problem of autonomous vehicles approaching multiple signalized intersections. We include simulation results to show quantitatively the advantages of the proposed solution.

Reducing triangular systems of ODEs with rational coefficients, with applications to coupled Regge-Wheeler equations

We concisely summarize a method of finding all rational solutions to an inhomogeneous rational ODE system of arbitrary order (but solvable for its highest order terms) by converting it into a finite dimensional linear algebra problem. This method is then used to solve the problem of conclusively deciding when certain rational ODE systems in upper triangular form can or cannot be reduced to diagonal form by differential operators with rational coefficients. As specific examples, we consider systems of coupled Regge-Wheeler equations, which have naturally appeared in previous work on vector and tensor perturbations on the Schwarzschild black hole spacetime. Our systematic approach reproduces and complements identities that have been previously found by trial and error methods.

The quadratic Wasserstein metric for inverse data matching

This work characterizes, analytically and numerically, two major effects of the quadratic Wasserstein (W2) distance as the measure of data discrepancy in computational solutions of inverse problems. First, we show, in the infinite-dimensional setup, that the W2 distance has a smoothing effect on the inversion process, making it robust against high-frequency noise in the data but leading to a reduced resolution for the reconstructed objects at a given noise level. Second, we demonstrate that, for some finite-dimensional problems, the W2 distance leads to optimization problems that have better convexity than the classical L2 and distances, making it a more preferred distance to use when solving such inverse matching problems.

Modulating traveling fronts for the Swift-Hohenberg equation in the case of an additional conservation law

We consider the one-dimensional Swift-Hohenberg equation coupled to a conservation law. As a parameter increases the system undergoes a Turing bifurcation. We study the dynamics near this bifurcation. First, we show that stationary, periodic solutions bifurcate from a homogeneous ground state. Second, we construct modulating traveling fronts which model an invasion of the unstable ground state by the periodic solutions. This provides a mechanism of pattern formation for the studied system. The existence proof uses center manifold theory for a reduction to a finite-dimensional problem. This is possible despite the presence of infinitely many imaginary eigenvalues for vanishing bifurcation parameter since the eigenvalues leave the imaginary axis with different velocities if the parameter increases. Furthermore, compared to non-conservative systems, we address new difficulties arising from an additional neutral mode at Fourier wave number k=0 by exploiting that the amplitude of the conserved variable is small compared to the other variables.

Designing Heat Concentrators of the Highest Efficiency Using an Optimization Method

Inverse problems for the two-dimensional heat transfer model that arise in the development of technologies of designing thermal concentrators are considered. Using an optimization method, these problems are reduced to finite-dimensional extremum problems, which are solved using particle swarm optimization. It is shown that the concentrators designed with the help of the optimization method are simple for implementation and have the maximum efficiency in the class of devices under consideration.

On convexity of the robust freeway network control problem in the presence of prediction and model uncertainty

In the freeway network control (FNC) problem, the operation of a traffic network is optimized using only flow control. For special cases of the FNC problem, in particular the case when all merging junctions are controlled, there exist tight convex relaxations of the corresponding optimization problem. In practice, many parameters of this optimization problem are not known with certainty, in particular the fundamental diagram and predictions of future traffic demand. This uncertainty poses a challenge for control approaches that pursue a model- and optimization-based strategy. In this work, we propose a robust counterpart to the FNC problem, where we introduce uncertainty sets for both the fundamental diagram and future, external traffic demands and seek to optimize the system operation, minimizing the worst-case cost. For a network with controlled merging junctions, and assuming that certain technical conditions on the uncertainty sets are satisfied, we show that the robust counterpart of the FNC problem can be reduced to a convex, finite-dimensional and deterministic optimization problem, whose numerical solution is tractable.

New numerical process solving nonlinear infinite-dimensional equations

Solving a nonlinear equation in a functional space requires two processes: discretization and linearization. In recent paper Grammant et al. (J Integral Equ Appl 26:413–436, 2014), the authors study the difference between applying them in one and in the other order. Linearizing first the nonlinear problem and discretizing the linear problem will be called option (B). Discretizing first the nonlinear problem and linearizing the discrete nonlinear problem will be called option (C). In this paper, we propose a new numerical process equivalent to the option (B): we linearize first the original nonlinear problem with an alternative linearization scheme than that used in the option (B), then we discretize the resulting iterative linear equations using a projection method to implement the corresponding finite-dimensional problem. The aims of this new process are intended to get weaker the theoretical assumptions and to give a powerful numerical performance. We give sufficient conditions to deal with the convergence results. Finally, as a numerical application, we solve a system of Fredholm equations of the second kind. The accuracy and efficiency of this process are illustrated in some numerical examples.

Exterior powers of the adjoint representation and the Weyl ring of E8

I derive explicitly all polynomial relations in the character ring of E8 of the form χ∧ke8−pk(χ1,…,χ8)=0, where ∧ke8 is an arbitrary exterior power of the adjoint representation and χi is the ith fundamental character. This has simultaneous implications for the theory of relativistic integrable systems, Seiberg–Witten theory, quantum topology, orbifold Gromov–Witten theory, and the arithmetic of elliptic curves. The solution is obtained by reducing the problem to a (large, but finite) dimensional linear problem, which is amenable to an efficient solution via distributed computation.

Monotone Smoothing Splines with Bounds

The problem of monotone smoothing splines with bounds is formulated as a constrained minimization problem of the calculus of variations. Existence and uniqueness of solutions of this problem is proved, as well as the equivalence of it to a finite dimensional but nonlinear optimization problem. A new algorithm for computing the solution which is a spline curve, using a branch and bound technique, is presented. The method is applied to examples in neuroscience and for fitting cumulative distribution functions from data.

Optimal Control of Multiphase Free Boundary Problems for Nonlinear Parabolic Equations

We consider the optimal control of singular nonlinear partial differential equation which is the distributional formulation of the multiphase Stefan type free boundary problem for the general second order parabolic equation. Boundary heat flux is the control parameter, and the optimality criteria consist of the minimization of the $L_2$-norm declination of the trace of the solution to the PDE problem at the final moment from the given measurement. Sequence of finite-dimensional optimal control problems is introduced through finite differences. We establish existence of the optimal control and prove the convergence of the sequence of discrete optimal control problems to the original problem both with respect to functional and control. Proofs rely on establishing a uniform $L_{\infty}$ bound, and $W_2^{1,1}$-energy estimate for the discrete nonlinear PDE problem with discontinuous coefficient.

Critical Cones for Sufficient Second Order Conditions in PDE Constrained Optimization

In this paper, we analyze optimal control problems governed by semilinear parabolic equations. Box constraints for the controls are imposed and the cost functional involves the state and possibly a sparsity-promoting term, but not a Tikhonov regularization term. Unlike finite dimensional optimization or control problems involving Tikhonov regularization, second order sufficient optimality conditions for the control problems we deal with must be imposed in a cone larger than the one used to obtain necessary conditions. Different extensions of this cone have been proposed in the literature for different kinds of minima: strong or weak minimizers for optimal control problems. After a discussion on these extensions, we propose a new extended cone smaller than those considered until now. We prove that a second order condition based on this new cone is sufficient for a strong local minimum.

Verification methods for conic linear programming problems

This paper gives an overview of verification methods for finite dimensional conic linear programming problems. Besides the computation of verified tight enclosures for unique non-degenerate solutions to well-posed conic linear programming instances, we discuss a rigorous treatment of problems with multiple or degenerate solutions. It will be further shown how a priori knowledge about certain boundedness qualifications can be exploited to efficiently compute verified bounds for the optimal objective value. The corresponding approach is applicable even to ill-posed programming problems. Examples from linear and semidefinite programming are used to illustrate the respective approaches and give further explanations. Another topic is the treatment of programming problems whose parameters are subject to uncertainties. Rough but inclusive estimates for the variability of the corresponding programming problems are given, and also a best and worst case analysis is taken into account. At the end of this paper, special consideration is given to typical issues when applying interval arithmetic in the context of conic linear programming. Different ways are shown on how to resolve these issues.