Simultaneous determination of the drift and diffusion coefficients in stochastic differential equations

In this work, we consider a one-dimensional It{\^o} diffusion process X t with possibly nonlinear drift and diffusion coefficients. We show that, when the diffusion coefficient is known, the drift coefficient is uniquely determined by an observation of the expectation of the process during a small time interval, and starting from values X 0 in a given subset of R. With the same type of observation, and given the drift coefficient, we also show that the diffusion coefficient is uniquely determined. When both coefficients are unknown, we show that they are simultaneously uniquely determined by the observation of the expectation and variance of the process, during a small time interval, and starting again from values X 0 in a given subset of R. To derive these results, we apply the Feynman-Kac theorem which leads to a linear parabolic equation with unknown coefficients in front of the first and second order terms. We then solve the corresponding inverse problem with PDE technics which are mainly based on the strong parabolic maximum principle

On the determination of the nonlinearity from localized measurements in a reaction-diffusion equation

This paper is devoted to the analysis of some uniqueness properties of a classical reaction-diffusion equation of Fisher-KPP type, coming from population dynamics in heterogeneous environments. We work in a one-dimensional interval $(a,b)$ and we assume a nonlinear term of the form $u \, (\mu(x)-\gamma u)$ where $\mu$ belongs to a fixed subset of $C^{0}([a,b])$. We prove that the knowledge of $u$ at $t=0$ and of $u$, $u_x$ at a single point $x_0$ and for small times $t\in (0,\varepsilon)$ is sufficient to completely determine the couple $(u(t,x),\mu(x))$ provided $\gamma$ is known. Additionally, if $u_{xx}(t,x_0)$ is also measured for $t\in (0,\varepsilon)$, the triplet $(u(t,x),\mu(x),\gamma)$ is also completely determined. Those analytical results are completed with numerical simulations which show that, in practice, measurements of $u$ and $u_x$ at a single point $x_0$ (and for $t\in (0,\varepsilon)$) are sufficient to obtain a good approximation of the coefficient $\mu(x).$ These numerical simulations also show that the measurement of the derivative $u_x$ is essential in order to accurately determine $\mu(x)$

A "MINLP" Formulation for Optimal Design of a Catalytic Distillation Column Based on a Generic Non Equilibrium Model

This contribution proposes a Mixed Integer Non Linear Programming (MINLP) formulation for optimal design of a catalytic distillation column based on a generic nonequilibrium model (NEQ). The solution strategy for the global optimization combines Simulated Annealing (SA) and Sequential Quadratic Programming (SQP) in order to minimize the objective function. The solution of this MINLP problem yields the optimal values for the temperature, composition and flow rate profiles, tray geometry, column diameter, reflux ratio, reboiler duty, feed tray location, number of trays and catalytic stage location. Hydraulic constraints (entrainment flooding, down-flow flooding, weeping-dumpling) are also considered. For the example, the production of ETBE (Ethyl tert-butyl ether) is presented here

The inverse problem of determining several coefficients in a nonlinear Lotka–Volterra system

International audienceIn this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of a system of two parabolic equations which corresponds to a Lotka-Volterra competition model. Our result gives a sufficient condition for the uniqueness of the determination of four coefficients of the system. This sufficient condition only involves pointwise measurements of the solution (u, v) of the system and of the spatial derivative ∂u/∂x or ∂v/∂x of one component at a single point x 0 , during a time interval (0, ε). Our results are illustrated by numerical computations

An inverse problem involving two coefficients in a nonlinear reaction-diffusion equation

This Note deals with a uniqueness and stability result for a nonlinear reaction-diffusion equation with heterogeneous coefficients, which arises as a model of population dynamics in heterogeneous environments. We obtain a Lipschitz stability inequality which implies that two non-constant coefficients of the equation, which can be respectively interpreted as intrinsic growth rate and intraspecific competition coefficients, are uniquely determined by the knowledge of the solution on the whole domain at two times $t_0$ and $t_1$ and on a subdomain during a time interval which contains $t_0$ and $t_1$. This inequality can be used to reconstruct the coefficients of the equation using only partial measurements of its solution

Uniqueness from pointwise observations in a multi-parameter inverse problem

In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. When the reaction term contains an unknown polynomial part of degree $N,$ with non-constant coefficients $\mu_k(x),$ our result gives a sufficient condition for the uniqueness of the determination of this polynomial part. This sufficient condition only involves pointwise measurements of the solution $u$ of the reaction-diffusion equation and of its spatial derivative $\partial u / \partial x$ at a single point $x_0,$ during a time interval $(0,\epsilon).$ In addition to this uniqueness result, we give several counter-examples to uniqueness, which emphasize the optimality of our assumptions. Finally, in the particular cases N=2 and $N=3,$ we show that such pointwise measurements can allow an efficient numerical determination of the unknown polynomial reaction term

Coefficient determination via asymptotic spreading speeds

International audienceIn this paper, we analyze the inverse problem of determining the reaction term f (x, u) in reaction-diffusion equations of the form ∂tu − D∂xxu = f (x, u), where f is assumed to be periodic with respect to x ∈ R. Starting from a family of exponentially decaying initial conditions u0,λ, we show that the solutions uλ of this equation propagate with constant asymptotic spreading speeds wλ. Our main result shows that the linearization of f around the steady state 0, ∂u f (x, 0), is uniquely determined (up to a symmetry) among a subset of piecewise linear functions, by the observation of the asymptotic spreading speeds wλ

Fos but not Cart (cocaine and amphetamine regulated transcript) is overexpressed by several drugs of abuse: a comparative study using real-time quantitative polymerase chain reaction in rat brain.

International audienceIt has been reported that cocaine and amphetamine-regulated transcript (Cart) peptides can increase locomotor activity and produce a conditioned place preference. To establish whether or not Cart can be consider as a valuable marker of addiction we performed a comparative study of the expression of Cart and Fos genes by several drugs of abuse. This was achieved using real-time quantitative polymerase chain reaction in four rat brain structures: prefrontal cortex, caudate putamen, nucleus accumbens and hippocampus. As expected, a significant induction of the immediate early gene Fos was observed after acute administration of morphine, cocaine, 3, 4-methylenedioxymethamphetamine and Delta(9)-Tetrahydrocannabinol. On the contrary none of these drugs was able to produce a significant change in Cart mRNA levels demonstrating that the expression of this gene is not modulated by drugs of abuse in these brain structures

Coronary artery bypass surgery in high-risk patients

BACKGROUND: In high-risk coronary artery bypass patients; off-pump versus on-pump surgical strategies still remain a matter of debate, regarding which method results in a lower incidence of perioperative mortality and morbidity. We describe our experience in the treatment of high-risk coronary artery patients and compare patients assigned to on-pump and off-pump surgery. METHODS: From March 2002 to July 2004, 86 patients with EuroSCOREs > 5 underwent myocardial revascularization with or without cardiopulmonary bypass. Patients were assigned to off-pump surgery (40) or on-pump surgery (46) based on coronary anatomy coupled with the likelihood of achieving complete revascularization. RESULTS: Those patients undergoing off-pump surgery had significantly poorer left ventricular function than those undergoing on-pump surgery (28.6 ± 5.8% vs. 40.5 ± 7.4%, respectively, p < 0.05) and also had higher Euroscore values (7.26 ± 1.4 vs. 12.1 ± 1.8, respectively, p < 0.05). Differences between the two groups were nonsignificant with regard to number of grafts per patient, mean duration of surgery, anesthesia and operating room time, length of stay intensive care unit (ICU) and rate of postoperative atrial fibrillation CONCLUSION: Utilization of off-pump coronary artery bypass graft (CABG) does not confer significant clinical advantages in all high-risk patients. This review suggest that off-pump coronary revascularization may represent an alternative approach for treatment of patients with Euroscore ≥ 10 and left ventricular function ≤ 30%