Ido Bright 

Acting Assistant Professor
Department of Applied Mathematics. 
University of Washington

Interests: Averaging Differential Equations;Singular Perturbations; Occupational Measures; Optimal Control;Infinite Horizon Problems;Shape Optimization. 


ibright (at)


Introduction to Differential Equations and Applications:  AMATH 351


1. Ido Bright, Tight estimates for general averaging applied to almost-periodic differential equations.

Journal of Differential Equations, Volume 246, Issue 7, 1 April 2009, Pages 2922-2937. Read it here 

Estimates are presented for the averaging method of ordinary differential equations. Previous results are improved by relaxing the conditions under which they hold, and by providing tight bounds for the estimations for almost-periodic differential equations and quasi-periodic differential equations. 

2. Zvi Artstein and Ido Bright, Periodic optimization suffices for infinite horizon planar optimal control. 
SIAM Journal on Control and Optimization. Volume 48, Issue 8 (2010), pp. 4963-4986. Read it here 

Under quite general conditions we show that an infinite horizon optimal control problem with a state variable in the plane has a periodic solution. The latter may employ relaxed controls. We examine the possibility to construct then an ordinary control approximation. An application to singularly perturbed control systems is displayed. 

3. Ido Bright, Moving Averages of Ordinary Differential Equations via Convolution. 
Journal of Differential Equations, Volume 250, Issue 3, 1 February 2011, Pages 1267-1284. Read it here 

We introduce an averaging framework, where the solution of a time-varying equation with a small amplitude is approximated by the solution of a slowly varying auxiliary system, generated by convolving the original equation with a kernel function. The effect of the convolution is smoothing of the equation, thus, making it more amenable to numerical computations. We present tight results on the approximation error for general classes of vector fields and kernels. 

4. Ido Bright, A reduction of topological infinite-horizon optimization to periodic optimization in a class of compact 2-manifolds.
Journal of Mathematical Analysis and Applications, Volume 394, Issue 1, 1 October 2012, Pages 84–101. Read it here

We consider a general class of infinite-horizon optimization problems, where the phase space is a two dimensional manifold satisfying the Jordan curve theorem, and the set of feasible curves satisfies general conditions similar to, but weaker than the solution set of a control system. We verify the existence of an optimal periodic solution for this class of problems. Applying this result we obtain two new results: we generalize a previous result for discounted infinite-horizon control problems, and provide a Poincaré–Bendixson type result for continuous generalized semi-flows

5. Zvi Artstein and Ido Bright, On the velocity of planar trajectories.
Nonlinear Differential Equations and Applications NoDEA, April 2013, Volume 20, Issue 2, pp 177-185. Read it here

Consider two trajectories each forming a Jordan curve with a finite length in 
the two dimensional plane. We show that the integral of the velocity of the portion of one 
curve that is included in the inside of the second curve, is bounded by half the length of 
the latter curve. We point out an application of the result to averaging.

 Ido Bright, Guang Lin and Jose Nathan Kutz, Compressive sensing based machine learning strategy for characterizing the flow around a cylinder with limited pressure measurements
Physics of Fluids 25, 127102 (2013). Read it here

Compressive sensing is used to determine the flow characteristics around a cylinder (Reynolds number and pressure/flow field) from a sparse number of pressure measurements on the cylinder. Using a supervised machine learning strategy, library elements encoding the dimensionally reduced dynamics are computed for various Reynolds numbers. Convex L1 optimization is then used with a limited number of pressure measurements on the cylinder to reconstruct, or decode, the full pressure field and the resulting flow field around the cylinder. Aside from the highly turbulent regime (large Reynolds number) where only the Reynolds number can be identified, accurate reconstruction of the pressure field and Reynolds number is achieved. The proposed data-driven strategy thus achieves encoding of the fluid dynamics using the L2 norm, and robust decoding (flow field reconstruction) using the sparsity promoting L1 norm.

Ido Bright and John M Lee, 
A note on flux integrals over smooth regular domains.
Pacific Journal of Mathematics, 272-2 (2014), 305--322. Read it here

We provide new bounds on a flux integral over the portion of the boundary of one regular domain contained inside a second regular domain, based on properties of the second domain rather than the first one. This bound is amenable to numerical computation of a flux through the boundary of a domain, for example, when there is a large variation in the normal vector near a point. We present applications of this result to occupational measures and two-dimensional differential equations, including a new proof that all minimal invariant sets in the plane are trivial.

8. Ido Bright, Planar Infinite-Horizon Optimal Control Problems with Weighted Average Cost and Averaged Constraints, Applied to Cheeger Sets.
Control, Optimisation and Calculus of Variations. 21 (4) 1108-1119 (2015). Read it here


We establish a PoincarÈ-Bendixson type result for a weighted averaged infinite horizon problem in the plane, with and without averaged constraints. For the unconstrained problem, we establish the existence of a periodic optimal solution, and for constrained problem, we establish the existence of an optimal solution that alternates cyclicly between a finite number of periodic curves, depending on the number of constraints. Applications of these results are presented to the shape optimization problems of the Cheeger set and the generalized Cheeger set, and also to a singular limit of the one-dimensional Cahn-Hilliard equation.

9. Ido Bright, Tight Bounds for Averaging Multi-Frequency Differential Inclusions Applied to Control Systems.
Set-Valued and Variational Analysis, 22-4 (2014), 843-857. Read it here


Give We present new tight bounds for averaging differential inclusions, which we apply to multifrequency inclusions consisting of a sum of time-periodic set-valued functions. Specifically we establish an estimates of order  on the approximation error for this averaging problem. These results are then applied to control systems consisting of a sum of time-periodic functions.

Steven L. Brunton, Jonathan H. Tu, Ido Bright, J. Nathan Kutz, 
Compressive sensing and low-rank libraries for classification of bifurcation regimes in nonlinear dynamical systems.
SIAM Journal on Applied Dynamical Systems, 13-4 (2014), 1716–1732.  Read it here

We show that for complex nonlinear systems, model reduction and compressive sensing strategies can be combined to great advantage for classifying, projecting, and reconstructing the relevant low-dimensional dynamics. ℓ2-based dimensionality reduction methods such as the proper orthogonal decomposition are used to construct separate modal libraries and Galerkin models based on data from a number of bifurcation regimes. These libraries are then concatenated into an over-complete library, and ℓ1 sparse representation in this library from a few noisy measurements results in correct identification of the bifurcation regime. This technique provides an objective and general framework for classifying the bifurcation parameters, and therefore, the underlying dynamics and stability. After classifying the bifurcation regime, it is possible to employ a low-dimensional Galerkin model, only on modes relevant to that bifurcation value. These methods are demonstrated on the complex Ginzburg-Landau equation using sparse, noisy measurements. In particular, three noisy measurements are used to accurately classify and reconstruct the dynamics associated with six distinct bifurcation regimes; in contrast, classification based on least-squares fitting (ℓ2) fails consistently.

11. Ido Bright and Monica Torres, The integral of the normal, fluxes over sets of finite perimeter, and applications to averaged shape optimization.
Interfaces and Free Boundaries, To AppearRead it here


Given two intersecting sets of finite perimeter, E_1 and E_2 , with unit normals \nu_1  and \nu_2  respectively, we obtain a bound on the integral of \bnu_{1}  over the reduced boundary of E_{1}  inside E_2. This bound depends only on the perimeter of E_2. For any vector field F:R^{n}\rightarrow R^{n} with the property that F \in L^\infty and div F is a (signed) Radon measure, we obtain bounds on the flux of F over the portion of the reduced boundary of E_1 inside E_2. These results are then applied to averaged shape optimization problems.

12.Ido Bright, On the Value of First Order Singular Optimization Problems.
Proceedings of the IMU-AMS Special Session on Nonlinear Analysis and Optimization. To Appear.


First order singular optimization problems arise in material science in the study of phase transition, and in infinite-horizon optimization. We show that the value of first order problems, which are space-independent, is attained by a sequence of functions converging to a constant function. The proof we present employs occupational measures.

13. Ido Bright, Guang Lin and Jose Nathan Kutz, Classification of Spatio-Temporal Data via Asynchronous Sparse Sampling:  Application to Flow Around a Cylinder


We present a novel method for the classification and reconstruction of time depen- dent, high-dimensional data using sparse measurements, and apply it to the flow around a cylinder. Assuming the data lies near a low dimensional manifold (low-rank dynamics) in space and has pe- riodic time dependency with a sparse number of Fourier modes, we employ compressive sensing for accurately classifying the dynamical regime. We further show that we can reconstruct the full spatio-temporal behavior with these limited measurements, extending previous results of compres- sive sensing that apply for only a single snapshot of data. The method can be used for building improved reduced-order models and designing sampling/measurement strategies that leverage time asynchrony. 

14. Ido Bright, Quinfeng Li and Monica Torres, Occupational Measures and Averaged Shape Optimization


We consider the minimization of averaged shape optimization problems over the class 
of sets of finite perimeter. We use occupational measures, which are probability measures defined in 
terms of the reduced boundary of sets of finite perimeter, that allow to transform the minimization 
in to a linear problem on a set of measures. The averaged nature of the problem allows the optimal 
value to be approximated with sets with unbounded perimeter. In this case, we show that we can 
also approximate the optimal value with convex polytopes with n + 1 faces shrinking to a point. 
We derive conditions under which we show the existence of minimizers and we also analyze the 
apropriate spaces in which to study the problem.


14. Ido Bright, A Short Proof that Minimal Sets of Planar Ordinary Differential Equations are Trivial.


We present a short proof, relaying on the divergence theorem, verifying that minimal sets in the plane are trivial.