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Spam yourself with math PhD qualifying-exam questions.

Stanford Practice, 7 Applied mathematics, gradient descent, convex optimization, subgradient methods

Let $𝐹 (𝑥)$ be a strictly convex function defined for $𝑥 \in ℝ^{𝑛}$ such that for any $𝑀 > 0$ there exists $𝑅 > 0$ so that $𝐹 (𝑥) > 𝑀$ if $| 𝑥 | > 𝑅$ .

(i) Show that $𝐹$ has a unique minimum $𝑥_{0}$ .

(ii) Consider an ordinary differential equation

\frac{d 𝑋}{d 𝑡} = - \nabla 𝐹 (𝑋 (𝑡)), 𝑋 (0) = 𝑥_{1},

with some $𝑥_{1} \in ℝ^{𝑛}$ . Show that $𝐺 (𝑡) = 𝐹 (𝑋 (𝑡))$ is a decreasing function of $𝑡$ .

(iii) Show that $𝑋 (𝑡)$ converges to $𝑥_{0}$ as $𝑡 \to + \infty$ . This is the gradient descent method.

(iv) Consider the function $𝐹 (𝑥) = | 𝑥 |$ which is not strictly convex and is not differentiable at $𝑥 = 0$ . Write down the explicit and implicit first order difference schemes for this ODE with a fixed time step $Δ 𝑡 > 0$ . Does either of them converge to the minimum $𝑥 = 0$ ? How does one cure that issue while keeping the time step fixed?

Stanford Practice, 8 Applied mathematics, Dirichlet energy, first variation, gradient flow, heat equation

Consider the functional

𝐼 (𝑢) = \frac{1}{2} \int_{Ω} {| \nabla 𝑢 (𝑥) |}^{2} d 𝑥,

defined for $𝑢 \in 𝒜︀ = {𝑢 \in 𝐶^{2} (Ω) : 𝑢 = 0 on 𝜕 Ω}$ . Fix one such $𝑢$ , and consider $𝐺 (𝑠; 𝑢, 𝑣) = 𝐼 (𝑢 + 𝑠 𝑣)$ , for various $𝑣 \in 𝒜︀$ .

(i) Find a function $𝑔$ , so that for any $𝑣 \in 𝒜︀$ we have

\frac{d 𝐺}{d 𝑠} (𝑠 = 0; 𝑢, 𝑣) = \int_{Ω} 𝑔 (𝑥) 𝑣 (𝑥) d 𝑥 .

(ii) Relate the heat equation

𝑢_{𝑡} = Δ 𝑢 in Ω, 𝑢 = 0 on 𝜕 Ω, 𝑢 (0, 𝑥) = 𝑓 (𝑥) in Ω

to the gradient descent method for the functional $𝐼 (𝑢)$ starting at $𝑓$ .

Stanford Practice, 9 Applied mathematics, nonlinear Schrödinger equation, energy conservation, mass conservation, blow-up

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 below that the solution remains smooth and decays fast at infinity as long as it exists. Define

𝑀 (𝑡) = \int_{ℝ^{𝑛}} {| 𝑢 (𝑡, 𝑥) |}^{2} d 𝑥,

and

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 .

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

(ii) 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 .

(iii) 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 Spring 2024, 1 Applied mathematics, central limit theorem, convolution, probability density

Let $𝑋_{𝑗}$ , $𝑗 = 1, 2, \dots$ , be real-valued, independent, identically distributed random variables, such that the probability density $𝑝 (𝑥)$ of each $𝑋_{𝑗}$ is a Schwartz class function, and set $𝑍_{𝑛} = (𝑋_{1} + \dots + 𝑋_{𝑛}) / 𝑛$ . Show that the probability distribution function of $𝑍_{𝑛}$ is $𝑝_{𝑍} (𝑥) = 𝑛 [𝑝 ⋆ 𝑝 ⋆ \dots ⋆ 𝑝] (𝑛 𝑥)$ , and there is a constant $𝑠 \in ℝ$ such that for any Schwartz function $𝑔 (𝑥)$ we have $𝔼 (𝑔 (𝑍_{𝑛})) \to 𝑔 (𝑠)$ as $𝑛 \to + \infty$ .

Stanford Spring 2024, 1 Applied mathematics, transport equation, method of characteristics, kinetic equations

Consider the equation $𝜕_{𝑡} 𝑓 + 𝑔 (𝑣) 𝜕_{𝑥} 𝑓 = 0$ with initial condition $𝑓 (0, 𝑥, 𝑣) = 𝑓_{0} (𝑥, 𝑣) \geq 0$ , where $(𝑥, 𝑣) \in ℝ \times ℝ$ .

a. Solve the PDE explicitly for all $𝑡 \geq 0$ .

b. Prove that $𝑓 \geq 0$ for all $𝑡 \geq 0$ .

c. Suppose $𝑔 : ℝ \to ℝ$ satisfies $𝑔^{'} (𝑣) \geq 𝑐 {/ (1 + | 𝑣 |)}^{𝑘}$ . Prove that there exists $𝐶 > 0$ such that

sup_{𝑥} \int_{ℝ} 𝑓 (𝑡, 𝑥, 𝑣) d 𝑣 \leq 𝐶 𝑡^{- 1} \int_{ℝ} sup_{𝑤} [{(1 + | 𝑤 |)}^{𝑘} 𝑓_{0} (𝑦, 𝑤)] d 𝑦, \forall 𝑡 \in (0, \infty) .

Stanford Spring 2024, 2 Applied mathematics, Gagliardo-Nirenberg inequality, Sobolev embedding, scaling arguments

a. Find a unique pair of numbers $𝛼_{𝑛} > 0$ and $𝛽_{𝑛} > 0$ such that an inequality of the form

\int_{ℝ^{𝑛}} {| 𝑢 (𝑥) |}^{2} d 𝑥 \leq 𝐶_{𝑛} {(\int_{ℝ^{𝑛}} {| \nabla 𝑢 (𝑥) |}^{2} d 𝑥)}^{𝛼_{𝑛}} {(\int_{ℝ^{𝑛}} | 𝑢 (𝑥) | d 𝑥)}^{𝛽_{𝑛}}

can possibly hold for all $𝑢 \in 𝒮︀ (ℝ^{𝑛})$ , with a constant $𝐶_{𝑛} > 0$ that depends only on the dimension $𝑛$ .

b. Show that this inequality does hold.

Stanford Spring 2024, 2 Applied mathematics, viscous Burgers' equation, Oleinik inequality, wave equation, energy estimates

a. Let $𝑢 (𝑡, 𝑥)$ be an infinitely differentiable bounded solution to the viscous Burgers' equation

𝜕_{𝑡} 𝑢 + 𝑢 𝜕_{𝑥} 𝑢 = 𝜈 𝜕_{𝑥}^{2} 𝑢,

in dimension one, with some $𝜈 > 0$ . Show that for all $𝑡 > 0$ , $𝑥 \in ℝ$ and $𝑦 > 0$ we have

\frac{𝑢 (𝑡, 𝑥 + 𝑦) - 𝑢 (𝑡, 𝑥)}{𝑦} \leq \frac{1}{𝑡} .

b. Consider the variable coefficient wave equation

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

Suppose that there are constants $𝑐_{1}, 𝑐_{2}, 𝑐_{3} > 0$ such that $𝑐_{1} \leq 𝑐 (𝑡, 𝑥) \leq 𝑐_{2}$ and $| 𝜕_{𝑡} 𝑐 (𝑡, 𝑥) | \leq 𝑐_{3}$ for all $𝑡 \geq 0$ and $𝑥 \in ℝ^{𝑛}$ . Assume also that $𝑢$ is $𝐶^{2}$ and rapidly decaying at infinity. Let

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

Show that there is $𝛼 > 0$ that depends only on $𝑐_{1, 2, 3}$ so that $𝐸 (𝑡) \leq 𝑒^{𝛼 𝑡} 𝐸 (0)$ .

Stanford Spring 2024, 3 Applied mathematics, finite difference method, stability analysis, Fourier multiplier, fourth-order PDE

Consider the equation

𝑢_{𝑡} = - 𝑢_{𝑥 𝑥 𝑥 𝑥}, 𝑥 \in ℝ, 𝑡 > 0

with the initial condition $𝑢 (𝑥, 0) = 𝑣 (𝑥)$ .

a. Write down an explicit-in-time and central-difference-in-space scheme for this equation. Denote the time step by $𝑘$ , the spatial step by $ℎ$ , and the numerical approximation by $𝑈_{𝑗}^{𝑛}$ .

b. Assume that the finite difference method is defined for all $𝑥 \in ℝ$ and consider it as a map $𝐸_{𝑘, ℎ}$ from $𝑈^{𝑛} (𝑥)$ to $𝑈^{𝑛 + 1} (𝑥) = (𝐸_{𝑘, ℎ} 𝑈^{𝑛}) (𝑥)$ . Write down the Fourier multiplier ${\tilde{𝐸}}_{𝑘, ℎ}$ using the convention:

(𝐸_{𝑘, ℎ} 𝑈^{𝑛}) (𝑥) = \sum_{𝑝} 𝑎_{𝑝} 𝑈^{𝑛} (𝑥 - ℎ 𝑝) \Rightarrow {\tilde{𝐸}}_{𝑘, ℎ} (𝜉) = \sum_{𝑝} 𝑎_{𝑝} 𝑒^{- 2 𝜋 𝑖 (𝑝 ℎ) 𝜉} .

c. Find the right condition on $ℎ$ and $𝑘$ so that the method is stable.

Stanford Spring 2024, 3 Applied mathematics, trapezoidal rule, spectral accuracy, Fourier analysis

Consider a $𝐶^{\infty}$ function $𝑓 (𝑥)$ defined on the periodic interval $𝕋 = [0, 1]$ . Let $𝑥_{𝑖} = 𝑖 / 𝑛$ for $𝑖 = 0, 1, \dots, 𝑛 - 1$ and define the quadrature

𝐼_{𝑛} [𝑓] = \frac{1}{𝑛} \sum_{𝑖 = 0}^{𝑛 - 1} 𝑓 (𝑥_{𝑖}) .

Prove that for any integer $𝑘$ there exists constant $𝐶_{𝑘, 𝑓}$ such that for any $𝑛 \geq 1$ we have

| \int_{𝕋} 𝑓 (𝑥) d 𝑥 - 𝐼_{𝑛} [𝑓] | \leq 𝐶_{𝑘, 𝑓} \frac{1}{𝑛^{𝑘}} .

Stanford Spring 2024, 4 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 $𝜂 > 0$ , i.e., $𝑓 (𝑥, 𝑦)$ is uniformly convex in $𝑥$ and concave in $𝑦$ .

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?

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