arXiv subjects

Jeffrey Humpherys

Publications and source records attributed to Jeffrey Humpherys.

At least 19 recordsLinked to original sources

arXiv2026

Riccati Stability Without Auxiliary Matrices

In the 2004 collection \emph{Unsolved Problems in Mathematical Systems and Control Theory}, Erik Verriest posed the problem of characterizing Riccati stability ``without invoking additional matrices.'' We give such a characterization through a scalar invariant of the resolvent family. The invariant is defined by covariance balances using at most $n^2+1$ frequency--direction pairs. A finite-dimensional separation argument shows that this covariance radius equals the optimal common ellipsoidal norm of the resolvent family. Combined with the strict bounded real lemma, this yields a necessary and sufficient condition for Riccati stability involving only the Hurwitz property of the first matrix and a single scalar inequality. The invariant reduces to the ordinary spectral radius for one matrix, lies between the pointwise spectral-radius and unscaled small-gain levels of the resolvent family, and coincides with the operator-space spectral radius of Shalit and Shamovich for the natural resolvent function space.

math.OC↗

arXiv2026

Proof of a Conjecture of De Cock and De Moor

De Cock and De Moor proposed a conjecture connecting two seemingly different viewpoints in stochastic subspace identification, one based on Lyapunov equations and the other on principal angles and canonical correlations. The conjecture was recorded as Problem 9.1 of \emph{Unsolved Problems in Mathematical Systems and Control Theory}. We give a direct finite-dimensional proof under the natural nonresonance condition, without requiring stability or diagonalizability. The key mechanism is the rank-one perturbation, which exposes a hidden Cauchy-matrix structure and reduces the problem to rational interpolation. A density and continuity argument then removes the generic spectral assumptions. The result strengthens the original statement. The eigenvalues agree with algebraic multiplicity, a nonsingularity assumption of the original formulation becomes automatic, and on a dense open set of parameters the two matrices are similar rather than merely cospectral. While this manuscript was being prepared, Gillberg and Löfberg independently posted a proof based on a Lyapunov-kernel identity and the classical $AB$--$BA$ principle. The proof given here was developed independently and follows a different route.

math.OC↗

arXiv2018

Tandem Blocks in Deep Convolutional Neural Networks

Due to the success of residual networks (resnets) and related architectures, shortcut connections have quickly become standard tools for building convolutional neural networks. The explanations in the literature for the apparent effectiveness of shortcuts are varied and often contradictory. We hypothesize that shortcuts work primarily because they act as linear counterparts to nonlinear layers. We test this hypothesis by using several variations on the standard residual block, with different types of linear connections, to build small image classification networks. Our experiments show that other kinds of linear connections can be even more effective than the identity shortcuts. Our results also suggest that the best type of linear connection for a given application may depend on both network width and depth.

stat.ML↗

arXiv2017

Euler vs. Lagrange: The role of coordinates in practical Evans-function computations

The Evans function has become a standard tool in the mathematical study of nonlinear wave stability. In particular, computation of its zero set gives a convenient numerical method for determining the point spectrum of the associated linear operator (and thus the spectral stability of the wave in question). We report on an unexpected complication that frustrates this computation for viscous shock profiles in gas dynamics. Although this phenomenon---related to the choice of Eulerian or Lagrangian coordinate system used to describe the gas---is present already in the one-dimensional setting, its implications are especially important in the multidimensional case where no computationally viable Lagrangian description of the gas is readily available. We introduce new "pseudo-Lagrangian" coordinates that allow us to overcome this difficulty, and we illustrate the utility of these coordinates in the setting of isentropic gas dynamics in two space dimensions.

math.AP↗

arXiv2017

Forward Thinking: Building and Training Neural Networks One Layer at a Time

We present a general framework for training deep neural networks without backpropagation. This substantially decreases training time and also allows for construction of deep networks with many sorts of learners, including networks whose layers are defined by functions that are not easily differentiated, like decision trees. The main idea is that layers can be trained one at a time, and once they are trained, the input data are mapped forward through the layer to create a new learning problem. The process is repeated, transforming the data through multiple layers, one at a time, rendering a new data set, which is expected to be better behaved, and on which a final output layer can achieve good performance. We call this forward thinking and demonstrate a proof of concept by achieving state-of-the-art accuracy on the MNIST dataset for convolutional neural networks. We also provide a general mathematical formulation of forward thinking that allows for other types of deep learning problems to be considered.

stat.ML↗

arXiv2017

Forward Thinking: Building Deep Random Forests

The success of deep neural networks has inspired many to wonder whether other learners could benefit from deep, layered architectures. We present a general framework called forward thinking for deep learning that generalizes the architectural flexibility and sophistication of deep neural networks while also allowing for (i) different types of learning functions in the network, other than neurons, and (ii) the ability to adaptively deepen the network as needed to improve results. This is done by training one layer at a time, and once a layer is trained, the input data are mapped forward through the layer to create a new learning problem. The process is then repeated, transforming the data through multiple layers, one at a time, rendering a new dataset, which is expected to be better behaved, and on which a final output layer can achieve good performance. In the case where the neurons of deep neural nets are replaced with decision trees, we call the result a Forward Thinking Deep Random Forest (FTDRF). We demonstrate a proof of concept by applying FTDRF on the MNIST dataset. We also provide a general mathematical formulation that allows for other types of deep learning problems to be considered.

stat.ML↗

arXiv2017

Balanced flux formulations for multidimensional Evans function computations for viscous shocks

The Evans function is a powerful tool for the stability analysis of viscous shock profiles; zeros of this function carry stability information. In the one-dimensional case, it is typical to compute the Evans function using Goodman's integrated coordinates [G1]; this device facilitates the search for zeros of the Evans function by winding number arguments. Although integrated coordinates are not available in the multidimensional case, we show here that there is a choice of coordinates which gives similar advantages.

math.AP↗

arXiv2016

Multidimensional stability of large-amplitude Navier-Stokes shocks

Extending results of Humpherys-Lyng-Zumbrun in the one-dimensional case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the multidimensional stability of planar Navier--Stokes shocks across the full range of shock amplitudes, including the infinite-amplitude limit, for monatomic or diatomic ideal gas equations of state and viscosity and heat conduction coefficients $μ$, $μ+η$, and $ν=κ/c_v$ in the physical ratios predicted by statistical mechanics, with Mach number $M>1.035$. Our results indicate unconditional stability within the parameter range considered; this agrees with the results of Erpenbeck and Majda for the corresponding inviscid case of Euler shocks. Notably, this study includes the first successful numerical computation of an Evans function associated with the multidimensional stability of a viscous shock wave. The methods introduced may be used in principle to decide stability for any $γ$-law gas, $γ>1$, and arbitrary $μ>|η|\ge 0$, $ν>0$ or, indeed, for shocks of much more general models and equations, including in particular viscoelasticity, combustion, and magnetohydrodynamics (MHD).

math.AP↗

arXiv2013

Viscous hyperstabilization of detonation waves in one space dimension

It has long been a standard practice to neglect diffusive effects in stability analyses of detonation waves. Here, with the principal aim of quantifying the impact of these oft-neglected effects on the stability characteristics of such waves, we use numerical Evans-function techniques to study the (spectral) stability of viscous strong detonation waves---particular traveling-wave solutions of the Navier-Stokes equations modeling a mixture of reacting gases. Remarkably, our results show a surprising synergy between the high-activation-energy limit typically studied in stability analyses of detonation waves and the presence of small but nonzero diffusive effects. While our calculations do show a modest delay in the onset of instability in agreement with recently reported calculations by direct numerical simulation of the physical equations, our approach, based on the Evans function, also provides additional spectral information. In particular, for each of the families of detonation waves in our computational domain, we find a completely unexpected kind of hysteresis in the limit of increasing activation energy; that is, our calculations suggest that, whenever diffusive effects are present, there is a return to stability as unstable eigenvalues return to the stable complex half plane.

math.AP↗

arXiv2013

Stability of viscous weak detonation waves for Majda's model

Continuing the program initiated by Humpherys, Lyng, & Zumbrun [17] for strong detonation waves, we use a combination of analytical and numerical Evans-function techniques to analyze the spectral stability of weak detonation waves in a simplified model for gas-dynamical combustion. Combining these new spectral stability results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [22], we conclude that these waves are nonlinearly stable. The principal novelty of this analysis is the treatment of weak detonation waves. In contrast to the case of strong detonation waves, weak detonation waves are undercompressive and the stability of these waves is delicate and has not been treated by standard weighted-norm techniques. The present analysis thus provides a case study illustrating the flexibility and power of the Evans-function- based approach to stability. As in the case of strong detonations, we find that all tested waves are spectrally stable, hence nonlinearly stable.

math.AP↗

arXiv2013

Stability of viscous detonations for Majda's model

Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of Majda's scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an efficient energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [J. Differential Equations 233 (2007), no. 2, 654-698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of the Majda model without a smallness assumption. Notably, our results indicate that, for the simplified Majda model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to "galloping" or "pulsating" solutions as is observed in the full reactive Navier-Stokes equations. This answers in the negative a question posed by Majda as to whether the scalar detonation model captures this aspect of detonation behavior.

math.AP↗

arXiv2010

Efficient numerical stability analysis of detonation waves in ZND

As described in the classic works of Lee--Stewart and Short--Stewart, the numerical evaluation of linear stability of planar detonation waves is a computationally intensive problem of considerable interest in applications. Reexamining this problem from a modern numerical Evans function point of view, we derive a new algorithm for their stability analysis, related to a much older method of Erpenbeck, that, while equally simple and easy to implement as the standard method introduced by Lee--Stewart, appears to be potentially faster and more stable.

math.NA↗

arXiv2009

Computing expected transition events in reducible Markov chains

We present a closed-form, computable expression for the expected number of times any transition event occurs during the transient phase of a reducible Markov chain. Examples of events include time to absorption, number of visits to a state, traversals of a particular transition, loops from a state to itself, and arrivals to a state from a particular subset of states. We give an analogous expression for time-average events, which describe the steady-state behavior of reducible chains as well as the long-term behavior of irreducible chains.

math.PR↗

arXiv2009

One-dimensional stability of parallel shock layers in isentropic magnetohydrodynamic

Extending investigations of Barker, Humpherys, Lafitte, Rudd, and Zumbrun for compressible gas dynamics and Freistühler and Trakhinin for compressible magnetohydrodynamics, we study by a combination of asymptotic ODE estimates and numerical Evans function computations the one-dimensional stability of parallel isentropic magnetohydrodynamic shock layers over the full range of physical parameters (shock amplitude, strength of imposed magnetic field, viscosity, magnetic permeability, and electrical resistivity) for a $γ$-law gas with $γ\in [1,3]$. Other $γ$-values may be treated similarly, but were not checked numerically. Depending on magnetic field strength, these shocks may be of fast Lax, intermediate (overcompressive), or slow Lax type; however, the shock layer is independent of magnetic field, consisting of a purely gas-dynamical profile. In each case, our results indicate stability. Interesting features of the analysis are the need to renormalize the Evans function in order to pass continuously across parameter values where the shock changes type or toward the large-amplitude limit at frequency $λ=0$ and the systematic use of winding number computations on Riemann surfaces.

math.AP↗

arXiv2008

On the shock wave spectrum for isentropic gas dynamics with capillarity

We consider the stability problem for shock layers in Slemrod's model of an isentropic gas with capillarity. We show that these traveling waves are monotone in the weak capillarity case, and become highly oscillatory as the capillarity strength increases. Using a spectral energy estimate we prove that small-amplitude monotone shocks are spectrally stable. We also show, through the use of a novel spectral energy estimate, that monotone shocks have no unstable real spectrum regardless of amplitude; this implies that any instabilities of these monotone traveling waves, if they exist, must occur through a Hopf-like bifurcation, where one or more conjugate pairs of eigenvalues cross the imaginary axis. We then conduct a systematic numerical Evans function study, which shows that monotone and mildly oscillatory profiles in an adiabatic gas are spectrally stable for moderate values of shock and capillarity strengths. In particular, we show that the transition from monotone to non-monotone profiles does not appear to trigger any instabilities.

math.AP↗

arXiv2007

Stability of strong ideal-gas shock layers

Extending recent results in the isentropic case, we use a combination of asymptotic ODE estimates and numerical Evans-function computations to examine the spectral stability of shock-wave solutions of the compressible Navier--Stokes equations with ideal gas equation of state. Our main results are that, in appropriately rescaled coordinates, the Evans function associated with the linearized operator about the wave (i) converges in the large-amplitude limit to the Evans function for a limiting shock profile of the same equations, for which internal energy vanishes at one endstate; and (ii) has no unstable (positive real part) zeros outside a uniform ball $|λ|\le Λ$. Thus, the rescaled eigenvalue ODE for the set of all shock waves, augmented with the (nonphysical) limiting case, form a compact family of boundary-value problems that can be conveniently investigated numerically. An extensive numerical Evans-function study yields one-dimensional spectral stability, independent of amplitude, for gas constant $γ$ in $[1.2, 3]$ and ratio $ν/μ$ of heat conduction to viscosity coefficient within $[0.2,5]$ ($γ\approx 1.4$, $ν/μ\approx 1.47$ for air). Other values may be treated similarly but were not considered. The method of analysis extends also to the multi-dimensional case, a direction that we shall pursue in a future work.

math-ph↗

arXiv2007

Spectral stability of noncharacteristic isentropic Navier-Stokes boundary layers

Building on work of Barker, Humpherys, Lafitte, Rudd, and Zumbrun in the shock wave case, we study stability of compressive, or "shock-like", boundary layers of the isentropic compressible Navier-Stokes equations with gamma-law pressure by a combination of asymptotic ODE estimates and numerical Evans function computations. Our results indicate stability for gamma in the interval [1, 3] for all compressive boundary-layers, independent of amplitude, save for inflow layers in the characteristic limit (not treated). Expansive inflow boundary-layers have been shown to be stable for all amplitudes by Matsumura and Nishihara using energy estimates. Besides the parameter of amplitude appearing in the shock case, the boundary-layer case features an additional parameter measuring displacement of the background profile, which greatly complicates the resulting case structure. Moreover, inflow boundary layers turn out to have quite delicate stability in both large-displacement and large-amplitude limits, necessitating the additional use of a mod-two stability index studied earlier by Serre and Zumbrun in order to decide stability.

math.AP↗

arXiv2007

Stability of viscous shocks in isentropic gas dynamics

In this paper, we examine the stability problem for viscous shock solutions of the isentropic compressible Navier--Stokes equations, or $p$-system with real viscosity. We first revisit the work of Matsumura and Nishihara, extending the known parameter regime for which small-amplitude viscous shocks are provably spectrally stable by an optimized version of their original argument. Next, using a novel spectral energy estimate, we show that there are no purely real unstable eigenvalues for any shock strength. By related estimates, we show that unstable eigenvalues are confined to a bounded region independent of shock strength. Then through an extensive numerical Evans function study, we show that there is no unstable spectrum in the entire right-half plane, thus demonstrating numerically that large-amplitude shocks are spectrally stable up to Mach number $M\approx 3000$ for $1 \le γ\leq 3$. This strongly suggests that shocks are stable independent of amplitude and the adiabatic constant $γ$. We complete our study by showing that finite-difference simulations of perturbed large-amplitude shocks converge to a translate of the original shock wave, as expected.

math.AP↗