querent

Spam yourself with math PhD qualifying-exam questions.

Stanford Fall 2024, 2 Applied mathematics, elliptic PDE, well-posedness, maximum principle, eigenvalue problems

(a) Consider Poisson's equation $𝑢_{𝑥 𝑥} + 𝑢_{𝑦 𝑦} = 𝑢$ in a vertical strip $0 < 𝑥 < 𝑎$ , $𝑦 \in ℝ$ , and impose that $𝑢 (𝑥, 𝑦) \to 0$ as $| 𝑦 | \to \infty$ . Two potential problems one may want to solve are Poisson's equation with the boundary conditions

(i) $𝑢 (0, 𝑦) = ℎ (𝑦)$ , $𝑢 (𝑎, 𝑦) = 𝑔 (𝑦)$ , respectively (ii) $𝑢 (0, 𝑦) = ℎ (𝑦)$ , $𝑢_{𝑥} (0, 𝑦) = 𝑘 (𝑦)$ .

You may assume that $𝑔, ℎ, 𝑘$ are Schwartz functions. Which of these boundary value problems are well posed and why?

(b) Consider Poisson's equation $- Δ 𝑢 - 𝛽 𝑢 = 𝑓$ in the ball $𝐵 (0, 𝑅) = {‖ 𝑥 ‖ \leq 𝑅}$ , with the boundary condition $𝑢 = 0$ on the boundary $𝜕 𝐵 (0, 𝑅)$ and some $𝛽 \in ℝ$ fixed. Show that there exists $𝛽_{0} > 0$ so that if $𝛽 < 𝛽_{0}$ and then this problem has at most one solution.

(c) In the setting of part (b) show that there exists $𝛽_{1} > 0$ so that if $𝛽 < 𝛽_{1}$ and $𝑓 (𝑥) \geq 0$ for all $𝑥 \in 𝐵 (0, 𝑅)$ then the unique solution satisfies $𝑢 (𝑥) > 0$ for all $𝑥 \in 𝐵 (0, 𝑅)$ unless $𝑓 \equiv 0$ .

(d) Are the optimal $𝛽_{0}$ and $𝛽_{1}$ from parts (b) and (c) equal to each other?

Stanford Fall 2024, 2 Applied mathematics, conservation laws, finite volume method, Godunov scheme, monotone schemes

Consider the conservation law

𝑢_{𝑡} + {(𝑓 (𝑢))}_{𝑥} = 0, 𝑥 \in ℝ, 𝑡 > 0,

with a smooth convex flux function $𝑓 (𝑢)$ and initial condition $𝑢 (𝑥, 0)$ uniformly bounded. Consider a finite volume method with spatial cells $𝐶_{𝑗} = [𝑥_{𝑗 - 1 / 2}, 𝑥_{𝑗 + 1 / 2}]$ of size $Δ 𝑥$ and time step $Δ 𝑡$ . Let $𝑈_{𝑗}^{𝑛}$ be the cell average.

(a) One can write the finite volume scheme in the form $𝑈_{𝑗}^{𝑛 + 1} = 𝐺 (𝑈_{𝑗 - 1}^{𝑛}, 𝑈_{𝑗}^{𝑛}, 𝑈_{𝑗 + 1}^{𝑛})$ . What is the definition of a monotone scheme? What does monotonicity mean in terms of the numerical flux $\hat{𝑓} (\cdot, \cdot)$ ?

(b) Describe the numerical flux $\hat{𝑓} (\cdot, \cdot)$ of the Godunov scheme.

Stanford Fall 2024, 3 Applied mathematics, wave equation, finite speed of propagation, energy estimates

Assume that $𝑢$ is $𝐶^{2}$ and rapidly decaying at infinity solution to the variable coefficient wave equation

𝑢_{𝑡 𝑡} = \nabla \cdot (𝑐^{2} (𝑡, 𝑥) \nabla 𝑢), 𝑡 > 0, 𝑥 \in ℝ^{𝑛} .

Let the constants $𝑐_{1}, 𝑐_{2}, 𝑐_{3} > 0$ be such that (i) $𝑐_{1} \leq 𝑐 (𝑡, 𝑥) \leq 𝑐_{2}$ , and (ii) $| \nabla 𝑐 (𝑡, 𝑥) | + | 𝑐_{𝑡} (𝑡, 𝑥) | \leq 𝑐_{3}$ for all $𝑡 \geq 0$ and $𝑥 \in ℝ^{𝑛}$ . Suppose that the initial condition satisfies $𝑢 (0, 𝑥) = 𝑢_{𝑡} (0, 𝑥) = 0$ in the ball $𝐵 (0, 𝑅) = {| 𝑥 | \leq 𝑅}$ . Find a constant $𝛾 > 0$ that depends only on $𝑐_{1, 2, 3}$ so that under these assumptions $𝑢 (𝑡, 0) = 0$ for all $𝑡 \leq 𝑅 / 𝛾$ .

Stanford Fall 2024, 3 Applied mathematics, convex-concave optimization, gradient descent ascent, saddle points

Let $𝑓 (𝑥, 𝑦)$ be a function on $ℝ^{2}$ such that $𝑓_{𝑥 𝑥} (𝑥, 𝑦) < - 𝜂$ and $𝑓_{𝑦 𝑦} (𝑥, 𝑦) > 𝜂$ for $𝜂 > 0$ . Consider the max-min problem ${max}_{𝑥} {min}_{𝑦} 𝑓 (𝑥, 𝑦)$ .

(a) Prove that the conditions $𝑓_{𝑥} (𝑥, 𝑦) = 0$ and $𝑓_{𝑦} (𝑥, 𝑦) = 0$ have a unique solution $(𝑥_{*}, 𝑦_{*})$ .

(b) Consider the gradient descent ascent flow

\frac{d 𝑥}{d 𝑡} = + 𝑓_{𝑥} (𝑥, 𝑦), \frac{d 𝑦}{d 𝑡} = - 𝑓_{𝑦} (𝑥, 𝑦) .

Prove that this flow converges to $(𝑥_{*}, 𝑦_{*})$ . What is the convergence rate?

Stanford Fall 2024, 4 Applied mathematics, finite difference method, stability analysis, Fourier multiplier

Consider the equation

𝑢_{𝑡} = - 𝑢_{𝑥 𝑥 𝑥 𝑥}, 𝑥 \in ℝ, 𝑡 > 0 .

(a) Derive a 5-point central difference scheme for $𝑢_{𝑥 𝑥 𝑥 𝑥}$ with the highest order.

(b) Write down a scheme that is implicit in time by using this central difference scheme in $𝑥$ . Please denote the time step by $𝑘$ , the spatial step by $ℎ$ , and the approximation by $𝑈_{𝑗}^{𝑛}$ .

(d) What is the stability condition of this scheme in terms of $𝑘$ and $ℎ$ ?

Stanford Fall 2024, 4 Applied mathematics, least squares estimation, median estimator, optimization

(a) Given the measurements $𝑥_{1}, \dots, 𝑥_{𝑛}$ , consider the $𝐿^{2}$ estimator for the error

𝑒_{2} (𝑥; 𝑥_{1}, \dots, 𝑥_{𝑘}) = \sum_{𝑘 = 1}^{𝑁} {(𝑥 - 𝑥_{𝑘})}^{2} .

Show that the minimizer of $𝑒_{2} (𝑥)$ is

(b) Consider what happens if, instead of the $𝐿^{2}$ error we use the TV error

𝑒_{1} (𝑥; 𝑥_{1}, \dots, 𝑥_{𝑘}) = \sum_{𝑘 = 1}^{𝑁} | 𝑥 - 𝑥_{𝑘} | .

Show that

Stanford Fall 2024, 5 Applied mathematics, Haar wavelets, orthonormal basis, Gaussian processes, L^2 convergence

Set

𝜓 (𝑥) = {\begin{cases} 1 & if 0 \leq 𝑥 < 1 / 2 \\ - 1 & if 1 / 2 \leq 𝑥 < 1 \\ 0 & otherwise \end{cases}

and

𝜓_{𝑗 𝑘} (𝑥) = 2^{𝑗 / 2} 𝜓 (2^{𝑗} 𝑥 - 𝑘), 𝑗, 𝑘 \in ℤ .

Also let $𝑍_{𝑛} (𝜔)$ , $𝑛 \geq 0$ , be a collection of independent Gaussian random variables of mean zero and variance one, that is,

ℙ (𝑍_{𝑛} (𝜔) < 𝑦) = \int_{- \infty}^{𝑦} 𝑒^{- 𝑦^{2}} d 𝑦 / \sqrt{2 𝜋} .

(a) Show that the functions $𝜑_{𝑛} (𝑥) = 𝜓_{𝑗 𝑘} (𝑥)$ for $𝑛 = 2^{𝑗} + 𝑘$ , $0 \leq 𝑘 \leq 2^{𝑗} - 1$ , and $𝜑_{0} (𝑥) = 1$ form an orthonormal basis for $𝐿^{2} [0, 1]$ .

(b) Show that

𝑀 (𝜔) = sup_{𝑛} | 𝑍_{𝑛} (𝜔) | / \sqrt{log 𝑛} < + \infty

almost surely in $𝜔$ .

𝑋 (𝑡; 𝜔) = \sum_{𝑛 = 0}^{\infty} 𝑍_{𝑛} (𝜔) \int_{0}^{𝑡} 𝜑_{𝑛} (𝑠) d 𝑠

converges in $𝐿^{2} (Ω)$ for any fixed $𝑡 \in [0, 1]$ .

Stanford Fall 2024, 5 Applied mathematics, nonlinear Schrödinger equation, energy conservation, blow-up, virial identity

Let $𝑢 (𝑡, 𝑥)$ be a solution to

𝑖 \frac{𝜕 𝑢}{𝜕 𝑡} + \frac{1}{2} Δ 𝑢 = 𝜆 {| 𝑢 |}^{𝑝 - 1} 𝑢, 𝑡 > 0, 𝑥 \in ℝ^{𝑛},

with some $𝑝 > 1$ and a prescribed smooth and compactly supported initial condition $𝑢 (0, 𝑥) = 𝑔 (𝑥)$ . You may assume that the solution remains smooth and decays fast at infinity as long as it exists. Define

𝐸 (𝑡) = \frac{1}{2} \int_{ℝ^{𝑛}} {| \nabla 𝑢 (𝑡, 𝑥) |}^{2} d 𝑥 + \frac{2 𝜆}{𝑝 + 1} \int_{ℝ^{𝑛}} {| 𝑢 (𝑡, 𝑥) |}^{𝑝 + 1} d 𝑥 .

(a) Show that $𝐸 (𝑡) = 𝐸 (0)$ as long as the solution remains smooth and decaying at infinity.

(b) Show that

\frac{d^{2}}{d 𝑡^{2}} \int_{ℝ^{𝑛}} {| 𝑥 |}^{2} {| 𝑢 (𝑡, 𝑥) |}^{2} d 𝑥 = 4 𝐸 (𝑡) + \frac{2 𝜆}{𝑝 + 1} [𝑛 (𝑝 - 1) - 4] \int_{ℝ^{𝑛}} {| 𝑢 (𝑡, 𝑥) |}^{𝑝 + 1} d 𝑥 .

(c) Show that there exists $𝑝_{𝑛}$ so that if $𝜆 = - 1$ and $𝑝 > 𝑝_{𝑛}$ then a solution with $𝐸 (0) < 0$ can not remain smooth and decaying at infinity for all $𝑡 > 0$ .

Stanford Practice, 1 Applied mathematics, Liouville's theorem, divergence-free flows, ODE flow maps

Assume that $div 𝑢 (𝑥) = 0$ for all $𝑥 \in ℝ^{𝑛}$ and consider the ODE

\frac{d 𝑋}{d 𝑡} = 𝑢 (𝑋), 𝑋 (0; 𝑥) = 𝑥 .

Let $𝑆$ be a bounded open set in $ℝ^{𝑛}$ with a smooth boundary and let $𝑆 (𝑡) = {𝑋 (𝑡; 𝑥), 𝑥 \in 𝑆}$ . Show that the $𝑛$ -dimensional volume of $𝑆 (𝑡)$ is equal to that of $𝑆$ .

Stanford Practice, 10 Applied mathematics, least squares estimation, median estimator, optimization

(i) Given the measurements $𝑥_{1}, \dots, 𝑥_{𝑛}$ , consider the $𝐿^{2}$ estimator for the error

𝑒_{2} (𝑥; 𝑥_{1}, \dots, 𝑥_{𝑘}) = \sum_{𝑘 = 1}^{𝑁} {(𝑥 - 𝑥_{𝑘})}^{2} .

Show that the minimizer of $𝑒_{2} (𝑥)$ is

(ii) Consider what happens if, instead of the $𝐿^{2}$ error we use the TV error

𝑒_{1} (𝑥; 𝑥_{1}, \dots, 𝑥_{𝑘}) = \sum_{𝑘 = 1}^{𝑁} | 𝑥 - 𝑥_{𝑘} | .

Show that

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